-1ASSIGNMENT #101
THINGS FOR YOU TO READ AND USE
A Forensics Problem that may be Solvable Using Things you have Learned
in this Course
Problem VII of assignment #100 asked you to consider whether it would be possible to
determine the number of hours since the death of a murder victim if two temperature
readings of the victim's body are taken by investigators after the body is found.
Suppose it were assumed that at the time of death the victim's body temperature was the
“normal” 98.6˚ Fahrenheit. Suppose also that the time the body is found, the air
temperature was stored in memory cell V. Finally, suppose that it is assumed that this
ambient temperature has changed very little since the time of death. If these assumptions
are made we might be able to use the same type of model used previously for data points
related to a heated object cooling in an environment with a stable ambient temperature.
This means we would work with the assumption that an image formula for the model
function, f, would have the form f (x) = a ⋅ b x + c .
In the statement of the problem two temperatures are given. By making a good choice of
times to which these temperatures are paired, it would be possible to find numerical
values for the constants “a,” “b,” and “c.”
As you do problem V of this assignment, you will be asked to determine what coefficients
you can from the assumptions. You will then be challenged to consider whether or not
there is enough information to determine values for any remaining coefficients and the
base. This is not an easy problem, but it is hoped it will capture your interest and
encourage you to speculate and explore.
A Reminder About Tangent Line Slopes as Estimates of Change
The name of this text is The Mathematical Analysis of Change. Tangent line slopes are
used to predict change. Here is yet another presentation of that fact.
Tangent lines are themselves the graphs of linear functions. Suppose T (x) = f ′(c)⋅ x + b
is an image formula for function T whose graph is the line tangent to the graph of another
function, f, at the point (c, f (c)) . If I represents a positive real number, the difference
between the second coordinate represented by T(c+I) and the second coordinate
represented by T(c) has the following equivalent representations:
T(c + I) − T(c) = [ f ′(c) ⋅ (c + I) + b ]− [ f ′(c) ⋅ c + b]
= [ f ′(c) ⋅ c + f ′(c) ⋅ I + b ]− [ f ′(c) ⋅ c + b] = f ′(c) ⋅ I
MAC Assignment #101
CHMS
-2This result can be expressed in the following ways:
#1: When the first coordinate of a point on the graph of T increases by I, the
second coordinate changes by f ′(c)⋅ I .
#2: When moving to the right on the graph of T, changes in the second
coordinates are directly proportional to changes in the first coordinate and the
constant of variation is the slope of the graph of T.
Thus, while moving to the right along a tangent line, changes in the second coordinates
can be computed as the product of the slope of that tangent line and the corresponding
changes in the first coordinates.
Tangent lines to the graphs of the types of model functions we have been using in this
course tend to follow those graphs rather closely when there is a “slight” movement to the
right away from the point of tangency. For this reason, computed changes of second
coordinates of points on tangent lines approximate changes of second coordinates of
points on the graphs of the functions to which those lines are tangent. This fact justifies
an expectation that scatter plots made by using Euler's Method can show points near the
graph of the solution of the differential equation used to compute the tangent line slopes.
The key is to use a horizontal increment which is relatively small so that the estimated
changes in the second coordinates can be expected to be quite accurate.
__________
I.
A. Give three partial sums which approximate the “infinite” decimal 0.9
B. Give the least upper bound of the set of partial sums which approximate 0.9 .
II. This problem is about the function L described by the differential equation
L ′(t) = 0.004 ⋅ L(t) ⋅ (80 − L(t)). Begin by using PROGRAM EULER to generate a set of
30 points which should be close to the graph of function L. Use (0,5) as the initial point
and 1 as the increment size.
A. Give a differential equation for a vertically scaled exponential function, P,
whose graph you expect would rather closely approximate the “smiling” portion
of the graph of L.
B. Give a command for generating list #3 so that the list #1, list #3 points are the
images of the list #1, list #2 points under transformation T with image formula
 1
T(t,y) =  t,  .
 y
C.
MAC Assignment #101
CHMS
-31. Give an equation which describes the horizontal asymptote implied by
the differential equation for L.
2. Give an equation which describes a horizontal asymptote for a function
which would model the list #1, list #3 data points.
3. Give a command that can be used to change the list #3 points so that the
list #1, list #3 points can be modeled by a function which has the t − axis
as a horizontal asymptote.
D. Actually use the commands you gave as your answers for “B” and “C,3” above
to generate a new list #3. Then, after examining a scatter plot of the current list
#1, list #3 data points, use an appropriate regression command to help create an
image formula for a function f which will model these data points. Give that
image formula on your homework paper with the coefficients accurate to four
decimal places. Also transfer to the Y2 position in the [Y=] list an expression
from which the graph of f can be generated.
E. Give a command which will restore the list #3 entries to what they were before
you did “D” right above. Modify your Y2 entry to take this change into account.
Check your work by using the command to restore list #3 and then viewing a
graph generated from the modified Y2 entry together with a list #1, list #3 scatter
plot.
F. Give an entry for Y3 in terms of Y2 from which a graph can be generated which
visually does an excellent job of fitting points of a list #1, list #2 scatter plot. For
a hint, see the image formula for transformation T in part “B” above.
III. Copy this equation and supply an expression in the form of the sum of two addends
for the denominator of the fraction to the right of the equal sign:
1
1 =
a ⋅b x +
C
IV. Suppose h(x) =
C
3
:
f (x)
A. Use the Chain Rule to give an expression for h ′(x) in the form of the product
of three factors, one of which is a power of f (x) .
B. Give an equivalent to your answer for “A” right above which does not contain
a negative exponent.
V. Here is a partial restatement of the forensics problem from assignment #100. Suppose
police investigators find the body of a murder victim and measure its temperature at 94.3˚
MAC Assignment #101
CHMS
-4Fahrenheit. The body was found in a room where the air temperature was 68˚. The
problem is to determine the time of death in relation to the time the body was found.
A. What assumption could be made so that it would be possible to determine the
x
value of “c” in an image formula of the form T(x) = a ⋅ b + c for a model
function, T, for the (the time in hours since the investigators made the first
temperature measurement, the temperature of the body after than many hours)
data points.
B. Based on your answer for “A,” give a value for the coefficient “c” for the
x
image formula T(x) = a ⋅ b + c
C. You have more than one choice of a useful model for answering the question
about the time of death. Give an ordered pair on the graph of the model you
would choose so that you could determine the numerical value of the vertical scale
factor for your model.
D. Using the values for the coefficients “a” and “c” from work above, is there
enough information given above to determine the value of the coefficient “b” in
x
the image formula T(x) = a ⋅ b + c ?
VI. If T ′(c) = 0.6 , give the tangent line slope estimate of the change in the second
coordinates of points on the graph of T when the first coordinate makes the described
change:
A. The first coordinate changes from “c” to “c+1.”
B. The first coordinate changes from “c” to “c+0.3.”
C. The first coordinate changes from “c” to “c+0.04.”
D. The first coordinate changes from “c” to “c+I.” where “I” represents a “typical”
positive real number.
x
VII. Suppose the model function T with image formula T(x) = 30 ⋅(0.3) + 70 is used to
model the change in temperature of an object when it is brought into a room with a
constant ambient temperature. The data points are described as (the time in minutes since
the object was brought into the room, the Fahrenheit temperature of the object at that
time).
A. What is the initial temperature of the object implied by the image formula for
T?
B. What is the ambient air temperature of the room implied by the image formula
for T?
MAC Assignment #101
CHMS
-5C. Enter an expression for Y1 from which a graph of T could be generated. Then
compute, accurate to three decimal places, the change in temperature of the object
that took place during the fourth minute of its exposure to the air in the room.
D.
1. Give an expression for T ′(x) on your homework paper. Also make an
entry for Y2 from which a graph of T ′ can be generated.
2. If Y2(a) is the tangent line slope estimate of your answer for “C” right
above, what is the value of “a”?
3. Using your value for “a” right above, give the tangent line slope
estimate of your answer for “C” accurate to three decimal places.
E. Give, accurate to three decimal places, the tangent line slope estimate of the
change of temperature of the object during the first 6 seconds of its fourth minute
of exposure to the air.
VIII.
A. In each case, give two of the many possible non-equivalent expressions for
h(x) .
x
1. When h ′(x) = 3.
2. When h ′(x) = 3⋅ x + e .
B. True or False: The same value for h(3) − h(2) can be computed from either of
your answers for “A,1” right above.
IX. Shown below is the graph of a model velocity function, v:
(4, 3)
(0, 3)
velocity in
feet per
minute
time in minutes
A. Give the missing numbers for the right column of this table:
MAC Assignment #101
CHMS
-6-
elapsed
time in
minutes
0
1
2
3.5
4
cumulative change of
position after the
passage of that much
time
B. Give the area of the region bounded above by the graph of v, on the left by the
line described by t = 0, on the right by the line described by t = 4 and below by
the segment of the horizontal axis from (0,0) to (4,0).
X.
(4, 5)
a portion of
the graph of f
(1,1)
Here are two proposed tables for some points on the graph of function f from the
illustration at the bottom of the previous page:
table #1
pre-image image
1
1.0
2
1.6
3
3.1
4
5.0
table #2
pre-image image
1.0
1
2.4
2
3.7
3
5.0
4
A. Give the number of the correct statement:
#1: Table #1 is consistent with the graph while Table #2 is not.
#2: Table #1 is not consistent with the graph while Table #2 is.
#3: Both tables are consistent with the graph.
#4: Neither table is consistent the graph.
MAC Assignment #101
CHMS
-7B. Copy and complete this table of images for f ′ , the derivative of the function f
in the illustration. The entries you make in the right column of the table below
should be consistent with the graph of f as well as your answer for “A.”
pre-image image in f ′
1
2
3
4
MAC Assignment #101
CHMS
-8-
ASSIGNMENT #102
THINGS FOR YOU TO READ AND USE
Coordinating Tangent Line Slopes with Changes in Second Coordinates
In doing problem X of assignment #101 you found that this graph and this table were
consistent:
(4, 5)
pre-image
1
2
3
4
a portion of
the graph of f
(1,1)
image
1.0
1.6
3.1
5.0
The graph is concave up and increasing. Consistent with this, the table of values shows
that as the first coordinates increase by one the corresponding changes in the second
coordinates increase in magnitude.
You have recently been reminded of the relationship between changes in first coordinates,
changes in second coordinates, and slopes when moving on a tangent line. Here is an
example of how that relationship applies in this case:
C
(3, 3.1)
B
(2, 1.6)
(3, ?)
A
a portion of the line tangent
to the graph of f at (2, 1.6)
The coordinates of points A and C are based on the table values. As one moves on the
tangent line in the illustration from point A to point B, the change in the first coordinates
is 1, so the change in the second coordinates is equal to
MAC Assignment #102
CHMS
-9-
1.(the slope of the tangent line) = 1. f ′(2) .
In this case, this change must be less than the change in the second coordinates in moving
from point A to point C. This means that 1⋅ f ′(2) = f ′(2) must be less than (3.1-1.6) =
1.5. In this and subsequent assignments you will be asked to provide tables of values for
derivatives of certain functions. You will need to coordinate the values you provide with
information you will be given about changes in the second coordinates of points on
graphs of those functions.
__________
I. Here again is a statement of the forensics problem first posed in Assignment #100.
Suppose police investigators find the body of a murder victim and measure its
temperature at 94.3˚ Fahrenheit. The body was found in a room where the air
temperature was 68˚. The problem is to determine the time of death in relation to the
time the body was found. We will use a function T with an image formula of the form
x
T(x) = a ⋅ b + c as a model for the data points described as (time in hours since
investigators first measured the temperature, the temperature of the body after that much
time).
A. Assume that you decide that the point (0,94.3) will be on the graph of T.
Assume also that the air temperature has been constantly 68˚ Fahrenheit since the
time of death. Based on this assumptions, give exact values for the constants “a”
and “c” in the image formula for T.
B. Suppose the investigators took a second measure of the body temperature onehalf hour after the first measure and found that it had dropped from 94.3˚ to 93.7˚.
1. Based on T(x) = a ⋅ b + c , your answer for “A” and this new fact, create
an equation that will have the value of “b” as a solution.
x
2. By some convenient method, determine the value of “b” accurate to four
decimal places.
D. Create a new equation which will have a solution that will help answer the
question, “How much time elapsed between the time of the murder and the time
the police investigators first measured the temperature of the body?” Assume that
the temperature of the victim at the time of death was the normal 98.6˚ Fahrenheit.
II. This problem is about the function L described by the differential equation
L ′(t) = 0.003 ⋅ L(t) ⋅ (120 − L(t)). Begin by using PROGRAM EULER to generate a set of
31 points (how many steps does this require?) which should be close to the graph of
function L. Use (0,6) as the initial point and 1 as the increment size.
MAC Assignment #102
CHMS
-10A. Give a differential equation for a vertically scaled exponential growth function
whose graph you would expect to approximate the “smiling” portion of the graph
of L.
B. Give an equation describing the horizontal asymptote implied by the
differential equation for L.
C. Give an equation describing the horizontal asymptote for a model for the data
generated by PROGRAM EULER after it has been mapped with transformation T
 1
with image formula T(t,y) =  t,  .
 y
D. Give a command or give commands that can be used to generate list #3 so that
the list #1, list #3 data points can be modeled very closely with a vertically scaled
exponential function, f. As you do this, be sure to take your answer for “C” right
above into account. On your homework paper, give an image formula for
function f with the coefficients rounded accurate to four decimal places. Also
transfer an expression to the Y2 equation from which a graph of f can be
generated.
E. Copy this connection diagram and fill in the two missing image equations:
y = f (t)
1 

T1 (t, y) = t, y +

120 
 1
T2 (t,y) =  t,
 y
F. Based on your work in “E” right above, give an entry for Y3 from which a
graph can be generated which will do an excellent job of covering and connecting
the points on a list #1, list #2 scatter plot. Check this work by viewing a graph
generated from your entry together with the appropriate scatter plot.
III. Suppose h(x) =
20
:
x
20⋅ a ⋅ b + 1
A. Give an equivalent expression for h(x) in which there is a negative integer
exponent and no fraction.
B. Use the Chain Rule to give an expression for h ′(x) in the form of the product
x
of seven factors, one of which is an integer power of 20 ⋅ a ⋅ b + 1 .
C. Give an equivalent to your answer for “B” right above which does not contain a
negative exponent.
MAC Assignment #102
CHMS
-11IV. Suppose that the model function T with image formula T(x)= − 41⋅ (0.75)x + 74 is
used to model the change of temperature of an object after it is exposed to the air in a
room. The time is measured in minutes.
A. What is the initial temperature of the object implied by the image formula for
T?
B. What is the ambient air temperature of the room implied by the image formula
for T?
C.
1. On your homework paper, sketch that portion of the graph of the model
function T which has first coordinates in the interval [0,30]. You should
be able to do this without the help of your calculator. Label the
coordinates of the intersection of this graph with the vertical coordinate
axis.
2. Based on your sketch, would you expect tangent line slope estimates of
changes in temperature to be too large or too small?
D. Compute, accurate to three decimal places, the tangent line slope estimate of
the change of temperature of the object during the fifth minute that it was exposed
to the air.
E. According to the model function T, what would be the change in the
temperature of the object during the fifth minute it was exposed to the air? Give
your answer accurate to three decimal places.
F. Give, accurate to three decimal places, the tangent line slope estimate of the
change in temperature of the object during the first 6 seconds of the fifth minute it
was exposed to the air.
G. According to the model function T, what would be the change in temperature
of the object during the first 6 seconds of the fifth minute the object was exposed
to the air? Give your answer accurate to three decimal places.
V. Recall that model velocity functions are the derivatives of associated model position
functions. Suppose that function v with image formula v(t) = ln(0.5)⋅(− 6)⋅ (0.5)t is a
model velocity function. Suppose also that the unit designation for function v images is
“feet per second.”
A. Give, accurate to the nearest thousandth, the tangent line slope estimate of the
change of position during the time interval [3,3.5]. The endpoints of the interval
[3,3.5] are measures of time in seconds.
MAC Assignment #102
CHMS
-12B. Give, accurate to the nearest thousandth, the tangent line slope estimate of the
change of position during the time interval [3.5,4]. Again, the endpoints of the
interval are measures of time in seconds.
VI.
(4, 5)
a portion of
the graph of f
(1,1)
Here are two proposed tables for some points on the graph of function f from the
illustration at the bottom of the previous page:
table #1
pre-image image
1
1.0
2
1.6
3
3.1
4
5.0
table #2
pre-image image
1.0
1
2.8
2
4.2
3
5.0
4
A. Give the number of the correct statement:
#1: Table #1 is consistent with the graph while Table #2 is not.
#2: Table #1 is not consistent with the graph while Table #2 is.
#3: Both tables are consistent with the graphs.
#4: Neither table is consistent the graph.
B. Copy and complete this table of images for f ′ , the derivative the function f
whose graph is in the illustration. If you found that either or both of the tables
from “A” right above are consistent with the graph, then your entries for the
images for f ′ should also be consistent with that table or with those tables. There
are many ways to do this consistent with the graph shown. You just need to find
one of them.
MAC Assignment #102
CHMS
-13-
pre-image image in f ′
1
2
3
4
VII. For this problem consider P(t) = 5000 ⋅1.03 to be a model for the population of a
suburb of a growing community. The tines are measured in years.
t
A.
1. Create an equation which will have as a solution the time it will take for
the population to double from 5000 to 10,000.
2. Give an exact representation of the time it will take for the population to
double from 5000 to 10,000.
B. Give an expression for P ′(t) .
C. Give an appropriate unit designation for the values of P ′(t) .
D. Copy and complete the following table giving the entries accurate to the
nearest ten thousandth:
rate of change
time (in years) population of the population
0
ln(2)
ln(1.03)
2⋅
ln(2)
ln(1.03)
E. Give the number of any of the following statement which are true:
#1: The numbers in the population column vary directly as the numbers in
the time column.
#2: The numbers in the rate of change column vary directly as the numbers
in the time column.
#3: The numbers in the rate of change column vary directly as the numbers
in the population column.
MAC Assignment #102
CHMS
-14-
MAC Assignment #102
CHMS
-15-
ASSIGNMENT #103
THINGS FOR YOU TO READ AND USE
A Conjecture about the Form of Image Formulas for Logistic Functions
In problem III of assignment #102 you were given the differential equation
L ′(t) = 0.003⋅ L(t)⋅ (120 − L(t)) for a logistic function, L, and you were asked to use
PROGRAM EULER to generate some data points. While the PROGRAM EULER data
points will not be exactly on the graph of the logistic function, we can expect them to be
close enough to that graph to give an accurate impression of its basic shape.
The scatter plot below was made after the PROGRAM EULER data points were mapped
1
with the transformation T1 with image formula T1 (x, y) = (x, ) :
y
This pattern of data points in the scatter plot resembles one we would expect for (time,
temperature) data points for an object cooling to an ambient room temperature.
The ExpReg command in the calculator produces the numbers for an image formula for
an exponential function which is vertically scaled, but not translated. An exponential
function which is only vertically scaled will have a graph which is asymptotic to the
x − axis . Therefore the points in the scatter plot were translated down a little to make a
vertically scaled exponential model more appropriate.
We determined the vertical translation number by noting first that the differential
equation, L ′(t) = 0.003⋅ L(t)⋅ (120 − L(t)), implies that the PROGRAM EULER data
points should become approximately asymptotic to the horizontal line described by
1
y = 120 . Since the transformation T1 with image formula T1 (x, y) = (x, ) was used to
y
map the PROGRAM EULER points onto those in the scatter plot, it seems reasonable to
1
suggest that y =
would be a good approximation of an equation for a horizontal
120
asymptote for the points on the scatter plot.
MAC Assignment #103
CHMS
-161
), the data points
120
were modeled rather successfully with a vertically scaled exponential function, f, with an
x
image formula of the form f (x) = a ⋅ b . To get the form of an equation for the original
PROGRAM EULER data points we reversed the sequence of transformations we had
used to make the use of ExpReg reasonable. Here is a connection diagram incorporating
this sequence of transformations:
After using the translation T2 with image formula T2 (x, y) = (x,y −
an equation which
x
y = a⋅b
describes a vertically
scaled, vertically
translated exponential
1
x
function
y = a⋅b +
120
an equation which
describes the reciprocal of
1
a vertically scaled, vertically
y=
translated exponential
a ⋅ bx + 1
function
120
T3 ( x, y ) = ( x, y +
T4 ( x , y) = ( x ,
1
)
120
1
)
y
The fit of the graph of the function at the end of the connection diagram to the
PROGRAM EULER data points was visually quite impressive. This suggests the
following conjecture about the form for an image formula for a logistic function.
If the graph of a logistic model function, L, is to have a horizontal asymptote
described by y = C, it seems reasonable to conjecture that function L
1
has an image formula of the form L(x) =
1
a ⋅b x +
C
__________
I. Describe the assumptions you would need to make and the information you would need
if you were to be able to use things you have learned in this course to determine the time
of death of a murder victim.
II. Suppose police investigators found the body of a murder victim in a room with an
ambient temperature of 72˚ Fahrenheit. Then measured the temperature of the body to be
92.6˚. Twenty minutes later they measured the temperature again and found it to be
91.8˚.
A. Assuming that the (time in hours since the first measure was taken by police
investigators, temperature of the body) data points is modeled with a vertically
scaled, vertically translated exponential function, T, give an image formula for T.
Give exact values for the vertical scale factor and vertical translation number.
Give the base accurate to four decimal places.
MAC Assignment #103
CHMS
-17-
B. Create an equation whose solution will help the investigators estimate the
elapsed time between the death of the victim and the taking of the first
temperature measurement. Assume that at the time of death the victim had a
normal body temperature of 98.6˚.
C. Based on the model from part “A” and the assumptions stated in previous
portions of this problem, how many hours before the investigators made their first
temperature measurement was the victim murdered? Give your answer accurate
to the nearest tenth.
III. Suppose h(x) =
C
:
x
C⋅ a ⋅ b + 1
A. Give an equivalent expression for h(x) in which there is a negative integer
exponent and no fraction.
B. Use the Chain Rule to give an expression for h ′(x) in the form of the product
x
of seven factors, one of which is a power of C ⋅ a ⋅ b + 1 .
C. Give an equivalent to your answer for “B” right above which does not contain a
negative exponent.
IV.
(1, 5)
a portion of
the graph of f
(4,1)
Give the right columns of these tables in a way which is consistent with the graph in the
illustration above. Also, take care that the entries in the table for f ′ are consistent with
changes in the second coordinates in your table for f.
pre-image image in f
1
2
3
4
MAC Assignment #103
CHMS
pre-image image in f ′
1
2
3
4
-18-
V. Suppose that (c,T(c)) is a “typical” point on the graph of a model function T for data
about the changing temperature of an object. Suppose also that T ′(c) represents the slope
of the line tangent to the graph of T at the “typical” point (c,T(c)) . Give the tangent line
slope estimate of the change of temperature over the time interval [c, c+I]. Here “I”
represents the length of a time interval.
VI. Suppose that the model function T with image formula T(x)= − 30 ⋅ (0.88)x + 70 is
used to model the change of temperature in degrees Fahrenheit of an object after it is
exposed to the air in a room. The time is measured in minutes.
A. What is the initial temperature of the object implied by the image formula for
T?
B. What is the ambient air temperature of the room implied by the image formula
for T?
C.
1. On your homework paper, sketch that portion of the graph of T which
has first coordinates in the interval [0,30]. You should be able to do this
without the help of your calculator. Label the coordinates of the
intersection of this graph with the vertical coordinate axis.
2. Based on your sketch, would you expect tangent line slope estimates of
changes in temperature to be too large or too small?
D.
1. Give an expression for T ′(x).
2. Give an appropriate unit designation for the values of the rate of change
function, T ′ .
3. Compute, accurate to three decimal places, the tangent line slope
estimate of the change of temperature of the object during the seventh
minute that it was exposed to the air.
4. Give an equivalent to your answer for “D,1” right above which contains
T(x).
E.
1. Copy and complete the following table. Give the entries in the rate of
change column accurate to the nearest hundred-thousandth.
MAC Assignment #103
CHMS
-19-
difference between the
object temperature and
the ambient temperature
rate of change of
the temperature
-30
-15
-7.5
2. True or False: The rates of change of temperature vary directly as the
differences between the object temperature and the ambient temperature.
VII. Suppose for some model position function s, s ′(t) = t + 1 is an image formula for its
derivative, the associated model velocity function. The values of t are measures of time
in seconds and the values of s are position numbers in meters.
A. Give the tangent line slope estimate of the change of position during the time
interval [0,0.5]. Keep in mind that the first coordinates on the graph of s are
measures of time in seconds. Here the change in the first coordinates is 0.5
second, not 1 second.
B. Give the tangent line slope estimate of the change of position during the time
interval [0.5,1.0].
C. Create from your answers for “A” and “B” an estimate of the change of
position during the time interval [0,1].
D. Give two non-equivalent expressions for s(t) based on the fact that
s ′(t) = t + 1.
E. Use one of your expressions from “D” right above to compute the change of
position during the time interval [0,1].
MAC Assignment #103
CHMS
-20-
ASSIGNMENT #104:
I.
A. If the derivative of function h has image formula h ′(x) = 2 ⋅ x and the graph of
h passes through (0,1), give an expression for h(x) .
B. If the derivative of function k has image formula k ′(x) = 2 ⋅ x and the graph of
k passes through (1,4), give an expression for k(x).
II. Suppose for some model position function s, s ′(t) = 2 ⋅t is an image formula for its
derivative, the associated model velocity function. The value of t are measures of time in
minutes. The values of s are position numbers in yards.
A. Give the tangent line slope estimate of the change of position during the time
interval [1,1.2].
B. Give the tangent line slope estimate of the change of position during the time
interval [1.2,1.4].
C. Create from your answers for “A” and “B” an estimate of the change of
position during the time interval [1,1.4].
D.
1. Give an expression for s(t) which meets the condition that s(0) = 5.
2. Use your expression from “D,1” right above to compute s(1.4) - s(1.0).
III. Suppose during the early stages in the growth of the population of a bacteria colony
t
the population is modeled by the function P with image formula P(t) = 1200 ⋅(1.14) .
Consider the pre-images for function P to be measures of time in hours.
A. Make a sketch showing the basic shape of the graph of P. On the sketch, label
the coordinates of the intersection of the graph with the vertical coordinate axis.
B. Based on the shape of the graph of P, would you expect tangent line slope
estimates of changes in population to be too big or too small?
C.
1. Give an expression for P ′(t) .
2. Give an appropriate unit designation for the values of P ′ .
MAC Assignment #104
CHMS
-213. Give an equivalent to your answer for “C,1” above which contains P(t) .
D.
1. Copy and complete the table below. Give the rates of change accurate
to the nearest ten-thousandth.
the population of
the bacteria colony
rate of change of
the bacteria population
1200
1800
2400
3600
2. True or False: the rates of change of the population vary directly as the
size of the population.
IV.
A. Give the number of the statement below which accurately states the
relationship known as Newton's Law of Cooling:
#1: The rates of temperature change vary directly as the time the cooling
process has been taking place.
#2: The rates of temperature change vary directly as the temperature of the
cooling object.
#3: The rates of temperature change vary directly as the differences
between the object temperature and the ambient temperature.
V.
(1, 5)
a portion of
the graph of f
(4, 1)
Give numbers for the right columns of these tables in a way which is consistent with the
graph in the illustration above. Also, the values of f ′ in the second table should be
consistent with the changes in the values of f given in the first table.
MAC Assignment #104
CHMS
-22-
pre-image image in f ′
1
2
3
4
pre-image image in f
1
2
3
4
VI.
B
C
A
O
the unit circle
A. If angle COA in the illustration above has a measure of 27˚, give an exact
representation of the coordinates of point C.
B. Suppose that segment OB has length 100 (the size of the unit circle has been
exaggerated). Give an exact representation of the coordinates of point B.
C. Suppose an object is launched into the air from the ground with a speed of 100
feet per second. Suppose also that the object is launched at an angle of 27˚ to the
ground. After one second, what would be the exact distance along the ground
from the launch point to a point directly below the flying object?
VII. In each case, give an expression for h ′( x ):
A. When h( x ) = cos(2 ⋅ x ) + 7.
2⋅x
B. When h(x) = 5
.
5
C. When h(x) = (2⋅ x) (notice the placement of the parentheses).
D. When h( x ) = ln(2 ⋅ x ) .
VIII.
A. Give the coordinates of the point of intersection between the base “e”
exponential function, exp, and the y − axis.
B. Give the coordinates of the point of intersection between the natural logarithm
function, ln, and the x − axis . Hint, how are the graphs of exp and ln related?
C. Give a numerical expression for ln(1).
MAC Assignment #104
CHMS
-23-
ASSIGNMENT #105
THINGS FOR YOU TO READ AND USE
A Reminder about Newton’s Law of Cooling
Newton's Law of Cooling can be expressed in a sentence as:
The rates at which an object cools or warms toward an assumed constant ambient
temperature vary directly as the differences between the changing temperatures of the
object and the ambient temperature.
Remember that when two columns of numbers vary directly as one another, when a
number in one of the columns changes by a certain factor, the corresponding number in
the other column changes by the same factor. Here is a simple illustration of that type or
relationship between changes:
column #1 column #2
-3.6
9.0
-3.0
7.5
-2.4
6.0
-1.2
3.0
In moving from row #1 of column #1 to row #4 of column #1, the entry changes by a
1
1
factor of . Likewise, the entry in row #4 of column #2 is
of the entry of row #1 of
3
3
that column. The table would be consistent with Newton's Law of Cooling if the numbers
in column #1 were differences between the temperature of an object and a constant
ambient temperature while the numbers in column #2 were rates of change of the
temperature.
__________
I.
A
(0,130)
B
(?, 100)
(0,70)
Shown above is a portion of the graph of a temperature function, T, for an object which it
is assumed is cooling in a way consistent with Newton's Law of Cooling. The line
through (0,70) is an horizontal asymptote for the graph of the temperature function.
MAC Assignment #105
CHMS
-24-
A. Suppose the line tangent to the graph of T at A = (0,130) has a slope of -2.4
degrees per second. What is the slope of the line tangent to the graph of T at point
B?
B. What would be the slope of the line tangent to the graph of T at the point with
second coordinate 80?
II Suppose the change in temperature of an object is modeled by the function T with
x
image formula T(x) = 60 ⋅(0.9) + 70 . The first coordinates of points on the graph of T
are measures of time in minutes and the second coordinates are measures of temperature
in degrees Fahrenheit.
A. Sketch a graph of the model function T. Label the coordinates of the
intersection of this graph with the vertical coordinate axis. Also, if there is an
asymptote for the model function, include it in your sketch and describe it with an
equation written by your sketch of it.
B. Based upon your sketch of the graph of T, would you expect tangent line slope
estimates of changes of in temperature to be too large or too small? In answering
this question, be careful about the signs of the tangent line slopes.
C.
1. Give an expression for T ′( x ).
2. Give an appropriate unit designation for the values of T ′ .
3. Give an equivalent to your answer for “C,1” right above which contains
“ T(x) .”
4. Based on your answer for “C,3” right above and consistent with
Newton's Law of Cooling, copy and complete this table. Give exact
values for the entries in the rate of change column.
difference between the
object temperature and
the ambient temperature
60
30
15
5
III.
MAC Assignment #105
CHMS
rate of change of
the temperature
-25-
(7, 9)
(4, 5)
(1, 1)
a portion of
the graph of f
A. Consistent with the graph of function f in the illustration, sketch that portion of
the graph of f ′ which has first coordinates in the interval [0.5, 7.5].
B. Consistent with the graph you sketched for “A” right above, sketch that portion
of the graph of f ′′ which has first coordinates in the interval [0.5, 7.5]. Keep in
mind that f ′′ is the derivative of f ′ .
C. Consistent with the illustration above give the right column of each table.
Assume that the point with first coordinate 4 is “special.”
pre-image image in f
pre-image image in f ′
1
2
3
4
5
6
7
1
2
3
4
5
6
7
IV. Here is a slightly modified version of an illustration from assignment #104:
B
C
O
the unit circle
A
Again assume that angle COA has a measure of 27˚. The illustration is to be related to
launching an object from the ground at an angle of 27˚ to the horizon, with an initial
speed of 100 feet per second.
A.
MAC Assignment #105
CHMS
-261. The arrow from O to A represents the “horizontal component” of the
initial velocity. Give the exact length of that arrow.
2. The arrow from A to B represents the “vertical component” of the initial
velocity. Give the exact length of that arrow.
B. Set your calculator to Degree and Par(ametric) MODES and then set X1T
=100*cos(27)*T and Y1T=100*sin(27)*T. Now push [2nd] and [WINDOW] and
set TblStart = 0 and ∆ Tbl = 0.5. Then push [2nd] and [GRAPH]. Fill in the
numbers for the right column of the table at the top of the next page with the
entries accurate to the nearest hundredth.
time in seconds
since the object
was launched
distance from the launch point
to a point directly under
the launched object
0
0.5
1.0
C. Push [WINDOW] and set Tmin = 0, Tmax = 3, and Tstep = 0.2. Also set the
viewing window to show points whose first and second coordinates lie in the
intervals [0,265] and [0,136], respectively. Then push [GRAPH]. Do you think
the figure generated by your calculator is an accurate simulation of the flight of an
object launched at an angle of 27˚ to the horizon with an initial velocity of 100
feet per second?
V.
−
A. Suppose function v has a derivative with image formula v ′(t ) = a(t ) = 32 and
v (0) = 6. Give an expression for v (t ).
B. Suppose the model position function s is associated with the model velocity
function, v for which you gave an expression as your answer for “A.” If s (0) = 0 ,
give an expression for s (t ) .
VI. Here is an illustration of a Ferris wheel with a radius of 40 feet. The center of the
wheel is 45 feet above the ground and the wheel is rotating counterclockwise at the
constant rate of two revolutions per minute. Think of point A as moving with the Ferris
wheel while point R is a stationary reference point.
MAC Assignment #105
CHMS
-27-
A
R
A. Without using your calculator, give the coordinates of at least five data points
described as (the number of seconds since point A moved past reference point R,
the distance from point A straight down to the ground). The first coordinate of
each of your data points should be in the interval [0,30].
B. Sketch what you think would be the shape the graph of a good model function
for all the data points described as (the number of seconds since point A moved
past reference point R, the distance from point A straight down to the ground).
Do this for the time interval [0,60] where the numbers in the interval are measures
of time in seconds.
VII. Shown at the top of the next page is a portion of the graph of a sine wave function.
Before proceeding with this problem, set your calculator to Radian MODE.
(7.5, 40)
(30, 0)
A. For the sine wave function a portion of whose graph is shown above:
1. Give the fundamental period.
2. Give the amplitude.
B. Begin a connection diagram with your choice of either “ y = cos( x )” or
“ y = sin(x ) ” and end it with an equation for the graph in the illustration. This is
an important review exercise, so check your work by making an entry in the [Y=]
list based on your answer and see if an appropriate graph can be generated from it.
MAC Assignment #105
CHMS
-28You should also check that this entry is consistent with the fact that the point (7.5,
40) is on the graph.
VIII. In each case, give an expression for h ′( x ):
 2 ⋅ π 
A. When h( x ) = 60 ⋅ cos
⋅ x + 5.
 40

MAC Assignment #105
CHMS
4
B. When h(x) = ln(x ) .
-29-
ASSIGNMENT #106
I. For this entire problem you will be working with the family of model position functions
all of which are associated with the model velocity function, v. The unit designations for
the points on the graphs of the associated position functions are seconds for the first
coordinates and feet for the second coordinates. The model velocity function v has
v (t ) = 2 ⋅ t + 1 as an image formula.
A.
1. Suppose model position function s is one of many associated with v.
Give the slope of the line tangent to the graph of function s at the point (1,
s(1)). Give the unit designation of your answer.
2. Give the tangent line slope estimate of the change of position over the
time interval [1,1.3].
3. Give the slope of the line tangent to the graph of function s at the point
(1.3, s(1.3)). Give the unit designation.
4. Give the tangent line slope estimate of the change of position over the
time interval [1.3,1.6].
5. Compute from your answers for “2” and “4” above an estimate of the
change of position over the time interval [1,1.6].
B.
1. Based on the image formula given for function v in the introduction to
this problem, copy and complete s1 (t ) = so that it becomes an image
formula for the model position function associated with v for which
s1 (0) = 7 .
2. Use your answer for “1” right above to compute the change of position
over the time interval [1,1.6].
3. Again, based on the image formula given for function v in the
introduction to this problem, copy and complete s 2 ( t ) = so that it becomes
an image formula for the model position function associated with v for
which s 2 (0) = ¯10 .
4. Use your answer for “3” right above to compute the change of position
over the time interval [1,1.6].
II. Assume that an object is released to fall toward the ground in the Earth's gravitational
field and that the acceleration induced by that field is constantly ¯ 32 feet per second per
second.
MAC Assignment #106
CHMS
-30-
A. Give an image formula for the model acceleration function, a. Begin this
image formula with “ a( t ) = .”
B. Give an image formula for the model velocity function, v, associated with the
model acceleration function, a, from “A” right above. Assume an initial velocity
of 100 ⋅ sin(27Þ) feet per second for the object whose motion is being modeled.
C. Give an image formula for the model position function, s, associated with the
model velocity function from “B” right above. Assume that an initial position of
0 feet for the object whose motion is being modeled.
III. Here again is an illustration from assignment #104:
B
C
O
A
the unit circle
Again assume that angle COA has a measure of 27˚. The illustration is to be related to
launching an object from the ground at an angle of 27˚ to the horizon, with an initial
speed of 100 feet per second. The aim of this problem is to generate a realistic parametric
simulation of the actual flight of the launched object. The T values with which the
calculator works will be considered as measures of time in seconds since the object was
launched from point O.
A. Give an entry for X1T from which can be generated the horizontal distances
from point O to a point right under the flying object at the various times used as T
values.
B. Give an entry for Y1T from which can be generated the vertical distances from
the flying object to points on the ground right under it at the various times used as
T values. This entry should take into consideration that the object is moving in
the Earth's gravitational field immediately as it is launched. See Problem II of this
assignment to get an idea of an expression that can be somehow included in the
Y1T expression to take this “falling” into account.
Test your entries by viewing a simulation of the flight of the object using your calculator.
Be sure it is set to degree MODE. If you succeed, you should see a simulated flight with
a duration of about 3 seconds. The flight path should have a familiar shape. Set Tmin =
0, Tmax = 3.0, and TStep = 0.1.
IV.
MAC Assignment #106
CHMS
-31(1, 9)
a portion of
the graph of f
(4, 5)
(7, 1)
A. Consistent with the graph of function f in the illustration, sketch that portion of
the graph of f ′ which has first coordinates in the interval [0.5, 7.5].
B. Consistent with the graph you sketched for “A” right above, sketch that portion
of the graph of f ′′ which has first coordinates in the interval [0.5, 7.5]. Keep in
mind that f ′′ is the derivative of f ′ .
C. Consistent with the illustration given and with the sketches you made, give the
right column of each table. Assume that the point with first coordinate 4 is
“special.”
pre-image image in f ′
1
2
3
4
5
6
7
pre-image image in f
1
2
3
4
5
6
7
V.
f ′(x) > 0
1
7
Recall that a “zero” of a function is a pre-image which has an image of 0 in that function.
In the above illustration the vertical segments over “1” and “7” indicate that they are
zeros of f ′ .
A. Copy the illustration onto your homework paper and between the vertical
segments add either “ f ↑ ” or “ f ↓ ” to indicate what is implied by f ′(x) > 0
about the behavior of the graph of function f.
B. There are many different possible functions named “f” consistent with the
information conveyed by the diagram you have just made on your homework
paper. Sketch a portion of the graph of one such function. The points on your
sketch should have first coordinates in the interval [0,8]. All the images indicated
by your sketch should be positive.
MAC Assignment #106
CHMS
-32-
C. Sketch a portion of a graph of another such function. Again, the points on your
sketch should have first coordinates in the interval [0,8]. This time some of the
images indicated by your sketch should be negative while others are positive.
VI. Here again is an illustration of a Ferris wheel with a radius of 40 feet. The center of
the wheel is 45 feet above the ground and the wheel is rotating counterclockwise at the
constant rate of two revolutions per minute. Remember to think of point A as moving
with the Ferris wheel while point R is a stationary reference point.
A
R
A. Sketch a model for data points described as (the number of seconds since point
A moved past reference point R, the distance from point A straight down to the
ground). Do this for data points with first coordinates in the interval [0,60]. Also
label on your graph the coordinates of one inflection point, one relative maximum
point, and one relative minimum point.
B. Create a connection diagram which begins with an equation describing a basic
sine wave (either “ y = cos(x)” or “ y = sin(x) ”) and ends with an equation which
describes the curve you sketched as your answer for “A.” Check your answer
with a graph generated by your calculator. Also check that the second coordinates
for the points you labeled are consistent with your equation.
C. If the model function whose graph is described by the equation at the end your
connection diagram from “B” right above, is given the name “g,” give an
expression for g ′(x).
D.
1. Create an equation whose solutions will be those times when point A is
exactly 65 feet above the ground.
2. Use the intersect command to help give any times in the interval [0,30]
when point A is exactly 65 feet above the ground. Give these times
accurate to the nearest hundredth.
E. Give the exact coordinates of at least five of the data points described as (the
number of seconds since point A moved past reference point R, the horizontal
MAC Assignment #106
CHMS
-33displacement of point A relative to an imagined vertical line through the center of
the Ferris wheel). Use positive numbers for displacements to the right of the
vertical line and negative numbers for displacements to the left of the vertical
reference line. The first coordinates of all these points should be in the interval
[0,30].
F. Sketch what you think would be the shape the graph of a good model function
for all the data points described as (the number of seconds since point A moved
past reference point R, the horizontal displacement of point A relative to an
imagined vertical line through the center of the Ferris wheel). Do this for the time
interval [0,60] where the numbers in the interval are measures of time in seconds.
MAC Assignment #106
CHMS
-34-
ASSIGNMENT #107
THINGS FOR YOU TO READ AND USE
Modeling the Motion of an Object Moving in the Earth’s Gravitational Field
When you study physics you will learn a theory which states that bodies falling freely
close to the surface of the Earth and in the Earth's gravitational field are under the
influence of forces which vary directly as their masses. Furthermore, according to the
theory, the constant of variation, that is, the slope of the line of (mass, force) data points,
is the same regardless of the mass. In the theory this constant is referred to as the
acceleration of the falling body due to gravity. For freely falling objects near the surface
of the Earth, this acceleration is measured to have an absolute value of about 32 feet per
second per second.
When free fall motion is analyzed mathematically, it is necessary to be clear how the
position of an object is measured. One choice is to measure the position of the falling
object as the distance from it to the surface of the earth. When this is done, during the
time the object is moving toward the Earth, the position function is decreasing, so the
velocity function has all negative images. As the object falls toward the Earth we would
expect the absolute value of the velocities to increase. This means that the negative
velocities decrease away from zero. Thus the velocity function is also decreasing. This
means that its derivative, the acceleration function, should have negative images.
Since the acceleration of a falling object moving toward the Earth has an absolute value
of about 32 feet per second per second, when the position of a falling object is measured
as its distance from the surface of the earth, a(t)= − 32 is often used as an image formula
for the model acceleration function. From this starting point, a “climb of a ladder” to an
associated model velocity function and then to an associated model position function may
be possible. Here is an example:
The first step up the ladder: Since a(t) = v ′(t)= − 32, the model velocity function,
v, associated with the model acceleration function, a, must be in the family of
functions which have image formulas of the form v(t)= − 32 ⋅ t + C . Now, suppose
that it is known by some means that v(0) = 4.5. The only value for C consistent
with this fact is 4.5, so v(t)= − 32 ⋅ t + 4.5 .
The second step up the ladder: Since v(t)= − 32 ⋅ t + 4.5 , the position functions, s,
associated with v, must be a member of the family of functions with image
formulas of the form s(t)= − 16 ⋅t 2 + 4.5⋅ t + C . If now we know the image of
some particular time in function s, the value for C can be determined. For
example, if s(0) = 100, the image formula for s must be
s(t)= − 16 ⋅t 2 + 4.5⋅ t + 100 .
__________
MAC Assignment #107
CHMS
-35I. All the parts of this problem are about the function T with image formula
T(x)= − 25 ⋅(0.95) x + 70 . Think of this function as a model for (time measured in
minutes, temperature in degrees Fahrenheit) for an object placed into a room where the
ambient temperature differs from the initial object temperature. Enter an expression for
Y2 from which a graph of T can be generated.
A.
1. What is the ambient temperature implied by the image formula for T?
2. Does the image formula imply that the object temperature will drop
toward the ambient temperature or rise toward the ambient temperature?
B.
1. Give an expression for T ′(x) which also contains T(x).
2. Suppose a table is made in which the numbers in column #1 are
differences between the temperature of the object and the ambient
temperature and the numbers in column #2 are the corresponding rates of
change of the temperature.
a. If the number in row #1 of column #1 is -30, what would be the
number in row #1 of column #2?
b. If a scatter plot were made of the column #1, column #2
numbers, what would be the slope of the line which would pass
through all the points of the scatter plot?
II. Use the following assumptions to “climb the ladder” (see the Things for You to Read
and Use section for the meaning of this phrase) from an image formula for a model
acceleration function to an image formula for a model position function:
Assumption #1: The position of the moving object is the distance of that object
from the surface of the Earth and the acceleration due to gravity has an absolute
value of 32 feet per second per second.
Assumption #2: The object has an initial velocity represented by 40 ⋅ sin(55Þ)
Assumption #3: The object has an initial position of 150 feet.
A. Give an image formula for the model acceleration function, a.
B. Give an image formula for the associated model velocity function, v.
C. Give an image formula for the associated model position function, s.
MAC Assignment #107
CHMS
-36-
D. Create an equation which has as a solution the time it will take in seconds for
the object whose motion is being modeled to reach the ground.
E. Use the intersection command to help determine, accurate the nearest
hundredth of a second, the time it will take for the object to reach the surface of
the Earth.
III.
P = origin of a coordinate
system for a frame of reference
face of
a 150 foot
cliff
arrow representing
an initial velocity making
an angle of 55Þ with
the horizon. The initial speed
is 40 feet per second.
The object of this problem is to create a parametric simulation of the flight of an object
launched with an initial speed of 40 feet per second from the edge of a 150 foot vertical
cliff. The launch angle is 55˚ relative to the horizon. Point P in the illustration is given
coordinates (0,0) as the origin of a coordinate system which creates a frame of reference
for simulating the motion. Set your calculator to Degree MODE.
A. Give an entry for X1T from which can be generated the horizontal distances
from point P to a point right under the flying object at the various times used as T
values.
B. Give an entry for Y1T from which can be generated the vertical distances from
the flying object to points on the ground below the cliff at the various times used
as T values. This entry should take into consideration both the initial vertical
component of the motion and the fact that the object will under go a constant
vertical acceleration of ¯ 32 feet per second per second due to its presence in the
Earth's gravitational field. You might find it useful to refer to problem II right
above.
Test your entries by actually generating a parametric simulation of the flight of the object.
The y − axis can be used as the face of the cliff and the x − axis as the ground under the
cliff. If you are successful, the simulated flight should last a little over 4.2 seconds and
MAC Assignment #107
CHMS
-37the trajectory of the object should have a familiar shape. Viewing window--[0,110],
[0,225].
IV. Here once again is the illustration of a Ferris wheel with a radius of 40 feet. The
center of the wheel is 45 feet above the ground and the wheel is rotating counterclockwise
at the rate of two revolutions per minute. Set your calculator to Radian MODE.
A
R
A. Sketch a model for data points described as (the number of seconds since point
A moved past reference point R, the horizontal displacement of point A relative to
an imagined vertical line through the center of the Ferris wheel). Do this for data
points with first coordinates in the interval [0,60]. Also label on your graph the
coordinates of one inflection point, one relative maximum point, and one relative
minimum point.
B. Create a connection diagram which begins with an equation describing a basic
sine wave (either “ y = cos( x )” or “ y = sin(x ) ”) and ends with an equation which
describes the curve you sketched as your answer for “A.” Check your answer
with a graph generated by your calculator. Also check that the second coordinates
for the points you labeled are consistent with your equation.
C. If the model function whose graph is described by the equation at the end of
your connection diagram from “B” right above is given the name “f,” give an
expression for f ′( x ).
D. Suppose function g is a model for the data points described as (the number of
seconds since point A moved past reference point R, the distance from point A
straight down to the ground):
1. Give an image formula for function g.
2. Use the intersect command to help determine the first time in the
interval [0,30] when point A will be exactly 70 feet above the ground.
Store the most accurate approximation possible of this number in a
memory cell identified as “M” in your calculator. Also give it on your
homework paper accurate to the nearest hundredth.
MAC Assignment #107
CHMS
-383. Make an entry in the [Y=] list based on g ′( x ) and then give, accurate to
the nearest thousandth, the rate at which the distance of point A from the
ground is changing when point A is exactly 70 feet above the ground for
the first time. Also store this number in your calculator in a memory cell
identified as “V.”
V.
f ′( x ) > 0
f ′( x ) < 0
f ′( x ) > 0
7
1
Recall that a “zero” of a function is a pre-image which has an image of 0 in that function.
In the above illustration the vertical segments over “1” and “7” indicate that they are
zeros of f ′ .
A. Copy the illustration onto your homework paper and above each inequality
about the signs of images in f ′ add either “ f ↑ ” or “ f ↓ ” to indicate what is
implied about the behavior of the graph of function f.
B. There are many different possible functions named “f” consistent with the
information conveyed by the diagram you have just made on your homework
paper. Sketch a portion graph of one such function. The points on your sketch
should have first coordinates in the interval [0,8]. Some of the images indicated
by your sketch should be negative while others are positive.
C. On the same coordinate system you used for your sketch for “B” add a sketch
of the graph of a portion of a second possible function f. Again, the points on
your sketch should have first coordinates in the interval [0,8]. Assume that this
second graph has no more inflection points than does the first.
VI.
(g, h)
(a,b)
(c, d)
(e, f)
(j, k)
Consider the curve above to be the graph of a function f whose mathematical domain is
the interval [a, j].
A. Consistent with the given curve, sketch a graph of f ′ . Label the coordinates
of any and all x − intercepts of the graph of f ′ .
B. Consistent with the given curve, sketch a graph of f ′′ . Label the coordinates
of any and all x − intercepts of the graph of f ′′ .
VII.
MAC Assignment #107
CHMS
-396
A. If f (x) = x and g( x ) = x , show how to use the Product Rule to get an
expression for h ′( x ) when h( x ) = f ( x ) ⋅ g( x ) .
P
B. Give an equivalent to your answer for “A” right above in the form a ⋅ x where
“a” and “P” are real numbers.
1
x3
5
C. If f ( x) =
and g(x) = x , show how to use the Chain Rule to get an
expression for h ′( x ) in the form of the product of four factors when
h( x ) = g( f ( x )) .
P
D. Give an equivalent to your answer for “C” right above in the form a ⋅ x where
“a” and “P” are real numbers.
MAC Assignment #107
CHMS
-40-
ASSIGNMENT #108
THINGS FOR YOU TO READ AND USE
Introducing Number Line Diagrams for the Derivative and Second
Derivative of a Family of Functions
a portion of the graph of f ,′ the
derivative of several different
possible functions, f
(7, 0)
(4.8, 0)
(1, 0)
Shown about is a portion of the graph of the derivative of several possible functions. The
graph implies certain features that would be shared by all the members of the family of
functions which have f ′ as their derivative. One way to become clear about these
implications is to create number line diagrams.
Here first is a number line diagram for f ′ . The notations about positive and negative
images in f ′ were made by observing whether the graph of f ′ was above or below the
horizontal coordinate axis before, between, and after the zeros of that function. In the
diagram “f ”is used as the name of a typical member of the family of function which have
f ′ as their derivative.
DONE SECOND: indications of the implied direction of the graph
of any function f whose derivative is f ′
f↑
f↑
f ′(x) > 0
f ′(x) > 0
1
f↓
7
f ′(x) < 0
readings of the sign of the second
DONE FIRST : coordinates on the graph of f ′
The number line diagram for f ′ leaves unanswered some questions about details of the
graph of f. For example, over the interval when f is increasing, is it increasing in a
MAC Assignment #108
CHMS
-41“frown,”, or is it increasing in a “smile,” or are there inflection points separating frowning
portions from smiling portions? To resolve such questions, we can proceed to create a
number line diagram for f ′′ , the second derivative of function f.
DONE LATER:
readings of the
DONE FIRST : direction of
the graph of f ′
indications of the implied “mood” of the graph
of any function f whose derivative is f ′
f′↓
f′↓
f′↑
f ′′( x ) < 0
f ′′( x ) > 0
1
4.8
f ′′( x ) < 0
ALSO DONE LATER: indications of the sign of second coordinates
on the graph of the second derivative of any function f whose derivative is f ′
Please note that a “smiling” mood is referred to in most calculus texts as “concave up.”
A “frowning” mood is referred to as “concave down.”
There are many functions, f, whose graphs are consistent with all the information in the
number line diagrams created for f ′ and f ′′ . The curve below shows the features they
must all share when the first coordinates span the interval [1,7]. In the illustration, “f” is
used as a “generic” name for the members of a very large family of anti-derivatives for
function f ′ .
(7, f (7))
increasing and concave down
to a relative maximum point
increasing and concave
up to an inflection point
decreasing and
concave down
(4.8, f (4.8))
(1, f (1))
increasing and concave down
to a point through which the
tangent line is horizontal
__________
I. All the parts of this problem are about the function L with image formula
x
 0.08 
L(x) =¯5000 ⋅ 1+
+ 10000 . Think of this function as a model for (the number of

12 
monthly payments made, the unpaid balance of a loan after that many monthly payments).
Enter an expression for Y2 from which a graph of L can be generated.
MAC Assignment #108
CHMS
-42-
A. Give an expression for L ′( x ) . Also enter an expression for Y1 from which the
graph of L ′ can be generated.
B.
1. Give, accurate to four decimal places, the slope of the line tangent to the
graph of L at the point (14, L(14)).
2. Give, accurate to four decimal places, the tangent line slope estimate of
the change in the unpaid balance over the time interval [14,15].
3. Give, accurate to four decimal places, the slope of the line tangent to the
graph of T at the point (15, T(15)).
4. Give, accurate to four decimal places, the tangent line slope estimate of
the change in the unpaid balance over the time interval [15,16].
5. Use your answers for “2” and “4” right above to give an estimate,
accurate to four decimal places, of the change in the unpaid balance over
the time interval [14,16].
II. Given a( t )= − 32 as an image formula for a model acceleration function and the
conditions that v (0) = 90 ⋅ sin(65Þ) and s (0) = 300 for the associated model velocity and
position functions, v and s, create an expression for s (t ) .
III.
arrow representing
an initial velocity making
an angle of 65Þ with
the horizon. The initial speed
is 90 feet per second.
origin for a
frame of reference
face of
a 300 foot
cliff
The object of this problem is to create a parametric simulation of the flight of an object
launched with an initial speed of 90 feet per second from the edge of a 300 foot vertical
cliff. The launch angle is 65˚ relative to the horizon. Set your calculator to degree
MODE.
MAC Assignment #108
CHMS
-43A. Give an entry for X1T from which can be generated the horizontal distances
from point O to a point right under the flying object at the various times used as T
values.
B. Give an entry for Y1T from which can be generated the vertical distances from
the flying object to points on the ground directly below the object at the various
times used as T values. This entry should take into consideration both the initial
vertical component of the motion and the fact that the object will under go a
constant vertical acceleration of ¯ 32 feet per second per second due to its presence
in the Earth's gravitational field.
Test your entries by actually generating a parametric simulation of the flight of the object.
The y − axis can be used as the face of the cliff and the x − axis as the ground under the
cliff. If you are successful, the simulated flight should last a little over 7.5 seconds and
the trajectory of the object should have a familiar shape.
C. After you are successful at getting a simulation of the flight of the object on the
screen, push [TRACE] and move the cursor to a point of your choice on the curve.
Then record the “T” value on the screen on your homework paper. Then push
[2nd] and [TRACE] and select 4: dx/dt . Then push [ENTER] one more time and
record on your homework paper the value shown for dx/dt .
D. Repeat the process described in “C” above four times, and on your homework
paper fill in four rows of a copy of this table:
the value of T for the point the value the calculator gives for
dx/dt accurate to the nearest hundredth
to which you traced
E. What do you think the number or numbers recorded in the right column of your
table from “D” right above mean?
IV. Here once again is the illustration of a Ferris wheel with a radius of 40 feet. The
center of the wheel is 45 feet above the ground and the wheel is rotating counterclockwise
at the rate of two revolutions per minute. Before answering these questions, set your
calculator to Radian MODE.
MAC Assignment #108
CHMS
-44-
A
R
A. Suppose function f is a model for the data points described as (the number of
seconds since point A moved past reference point R, the horizontal displacement
of point A relative to an imagined vertical line through the center of the Ferris
wheel).
1. Give an image formula for function f.
2. Give an expression for an entry in the [Y=] list from which a graph of
the derivative of f could be generated. Also make this entry in the [Y=]
list.
3. You should still have stored in a memory cell identified as “M” a very
good approximation of the number of seconds it would take point A to
first reach a position 70 feet above the ground after passing reference point
R. Give, accurate to the nearest thousandth, the rate at which point A is
moving horizontally relative to the imagined vertical reference line
through the center of the Ferris wheel at the time stored in memory cell M.
Store this rate in a memory cell identified as “H.”
B. Suppose an object is released to the side of the seat attached to the Ferris wheel
at point A when point A is 70 feet above the ground. Assume the object is held
out away from the Ferris wheel so the object can move without making contact
with it. After some careful thought, sketch the path you think the object would
follow from the time it was released until it struck the ground.
3⋅ x
V. Given h ′( x ) = 5 ⋅ x + 3 ⋅ e :
A. Give one possible expression for h( x ) .
B. Give an expression for h( x ) which is not equivalent to the expression you gave
as your answer for “A” right above.
C. Give an image formula for a transformation, T, which will map your answer for
“A” onto your answer for “B.”
VI.
MAC Assignment #108
CHMS
-45-
(3, 0)
(9, 0)
a portion of the
graph of f ′
(6.6, f ′(6.6))
A. Make a number line diagram for f ′ .
B. Consistent with the graph in the illustration, create a number line diagram for
f ′′ which has first coordinates in the interval [0,10].
C. Consistent with your number line diagrams for graphs of f ′ and f ′′ , sketch a
curve showing the features the diagrams indicate must be shared by all antiderivatives of f ′ . Use “f” as a “generic” name for these anti-derivatives and on
your curve sketch use notation of the form (a, f (a)) to indicate the locations of all
relative minimum and relative maximum points and any and all inflection points.
Numerical values should be used for “a.”
D. What type of transformation do you think would map any member of the
family of anti-derivatives for f ′ onto any other member of that same family?
MAC Assignment #108
CHMS
-46-
ASSIGNMENT #109
THINGS FOR YOU TO READ AND USE
The Meaning of the dx/dt Number or Numbers that can be Displayed by
your Calculator
In problem IV of assignment #108 you were asked to create a parametric simulation of
the flight of an object launched out from the edge of a cliff at an angle of 65˚ to the
horizon. Since the initial speed was given as 90 feet per second, the horizontal
component of the initial velocity was represented by 90 ⋅ cos(65Þ). Here is an illustration
which highlights the effects of this horizontal component of the launch velocity:
simulated flight
path
d1
d2
d3
point at the base of the
cliff used as the origin to
establish a frame of reference
With the frame of reference established as in the illustration the positive numbers, d1, d2,
and d3 measure the distances from a point right under the flying object to a point at the
base of the cliff. These distances can be paired with the measures of time since the object
was launched to form data points like these: (t1, d1), (t2, d2), and (t3, d3). If function f is
a model function whose graph passes through all such data points, then
f (t) = 90 ⋅ cos(65Þ)⋅t is an image formula for f. Note the close relationship between this
image formula and the equation X1T = 90 ∗ cos(65Þ)∗ T which is the first coordinate
entry made in parametric mode in preparation for viewing a simulation of the flight of the
object.
MAC Assignment #109
CHMS
-47The function f which models the horizontal displacements of the moving object relative to
the base of the cliff is linear. Its graph is a line through the origin with a slope
represented by 90 ⋅ cos(65Þ), the horizontal component of the initial velocity.
f ′(t) = 90⋅ cos(65Þ) is an image formula for the derivative of f. 90 ⋅ cos(65Þ)≈ 38.04 and
this number was given over and over again when the command “dx/dt” was used at
several points along the path of the simulated motion. This reveals the following about
notation used in describing parametric motion:
If function f models the horizontal displacement relative
to a frame of reference of an object whose motion is being modeled parametrically,
dx/dt is an alternative to f ′(t) . Both notations are widely used.
The “d” in the “dx/dt” notation suggests the word “derivative.” It also suggests that the
average rate of change in the horizontal displacement from a point such as (t1, d1) to the
d 2 − d1 d 2 − d1
point such as (t2, d2) is computed as
.
is a ratio of a difference of
t 2 − t1
t 2 − t1
displacements and a difference of times.
__________
I. All the parts of this problem are about the function T with image formula
T(x) = 32 ⋅(0.95) x + 72 . Think of this function as a model for (time measured in
minutes, temperature in degrees Fahrenheit) for an object placed into a room where the
ambient temperature differs from the initial object temperature. Enter an expression for
Y2 from which a graph of T can be generated.
A.
1. What is the ambient temperature implied by the image formula for T?
2. Does the image formula imply that the object temperature will drop
toward the ambient temperature or rise toward the ambient temperature?
B. If a scatter plot were made of data points described as (the difference between
the object temperature and the ambient temperature at a given time, the rate of
change of the temperature at that time), what would be the slope of a line passing
through all the points of the scatter plot?
II.
MAC Assignment #109
CHMS
-48-
arrow representing
an initial velocity making
an angle of 40Þ with
the horizon. The initial speed
is 75 feet per second.
face of
a 250 foot
cliff
P
The object of this problem is create a parametric simulation of the flight of an object
launched with an initial speed of 75 feet per second from the edge of a 250 foot vertical
cliff. The launch angle is 40˚ relative to the horizon. Think of point P as the origin of a
coordinate system which establishes a frame of reference.
A. Give an entry for X1T from which can be generated the horizontal distances
from point P to a point right under the flying object at the various times used as T
values.
B. Give an entry for Y1T from which can be generated the vertical distances from
the flying object to points on the ground below the cliff at the various times used
as T values. Again, this entry should take into consideration both the initial
vertical component of the motion and the fact that the object will under go a
constant vertical acceleration of ¯ 32 feet per second per second due to its presence
in the Earth's gravitational field.
C.
1. Create an image formula for a function f by equating f(t) to your
expression for X1T.
2. Give an expression for f ′(t ) .
D.
1. Create an image formula for a function g by equating g(t) to your
expression for Y1T.
2. Give an expression for g ′(t ).
E. Set your calculator to Function MODE before proceeding with this problem.
1. Use your calculator's intersection command to help give the smallest
positive solution of the equation g(t) =0, where the g(t) expression is from
“D,1.” Give this solution accurate to three decimal places.
MAC Assignment #109
CHMS
-492. Give the exact solution of the equation g ′(t ) = 0 where the g ′(t )
expression is from “D, 2.”
3. Accurate to three decimal places, how many seconds will pass from the
time the object is launched until it strikes the ground?
4. Accurate to three decimal places, how many seconds will pass from the
time the object is launched until it reached its greatest distance above a
point on the ground right below it?
Test your entries by actually generating a parametric simulation of the flight of the object.
The y − axis can be used as the face of the cliff and the x − axis as the ground under the
cliff. If you are successful, the simulated flight should last a little over 5.7 seconds and
the trajectory of the object should have a familiar shape.
F. After you are successful at getting a simulation of the flight of the object on the
screen, push [TRACE] and move the cursor to a point of your choice on the curve.
Then record the “T” value on the screen on your homework paper. Then push
[2nd] and [TRACE] and select 3: dy/dt. Then push [ENTER] one more time and
record on your homework paper the value shown for dy/dt accurate to three
decimal places.
G. Repeat the process described in “C” above and on your homework paper, fill in
four rows of a copy of this table:
the value of
T for the point
to which you
traced
the value the
calculator gives
for dy/dt accurate
to the nearest
hundredth
H. What do you think the number or numbers recorded in the right column of your
table from “D” right above mean?
III. Here again is the illustration of a Ferris wheel with a radius of 40 feet. The center of
the wheel is 45 feet above the ground and the wheel is rotating counterclockwise at the
rate of two revolutions per minute.
MAC Assignment #109
CHMS
-50-
A
R
Suppose an object is released to the side of the seat attached to the Ferris wheel at point A
when point A is 70 feet above the ground. Assume the object is held out away from the
Ferris wheel so the object can move without making contact with it. The object of this
problem is to move toward generating a parametric simulation of the motion of the
object.
A. As a first step, a frame of reference must be chosen. For the vertical axis a line
parallel to the vertical line through the center of the Ferris wheel will be chosen.
The horizontal axis will be a line at the level of the ground lying in a plane
parallel to the plane containing one rim of the wheel.
Recall that
2 ⋅π
f (t ) = 40 ⋅ sin(
⋅t ) is an image formula for the data points described as (the
30
time in seconds since point A passed reference point R, the displacement of point
A relative to a vertical line passing through the center of the wheel. Recall also
 2 ⋅ π 
that g( t )= − 40 ⋅ cos
⋅ t + 45 is a model for the data points described as (the
 30 
time in seconds since point A passed reference point R, the displacement of point
A relative the point on the ground right below it at that time). Also remember that
 2 ⋅ π 
the smallest positive solution of the equation g( t )= − 40 ⋅ cos
⋅ t + 45 = 70 is
 30 
stored in your calculator's memory as M.
Set your calculator to radian MODE. Now use the above information to help
calculate, accurate to three decimal places, the coordinates of the released object
at the time it is released. Store the most accurate values possible for these
coordinates in your calculator's memory as “F” and “S” respectively.
B. Calculate and store in your calculator’s memory as H the rate of change of the
horizontal displacement at time M. Also, record this rate of change on your
homework paper accurate to the nearest hundredth.
C. Calculate and store in your calculator’s memory as V the rate of change of the
vertical displacement at time M. Again, record this rate of change on your
homework paper accurate to the nearest hundredth.
MAC Assignment #109
CHMS
-51D. “Climb a little ladder” to get an expression for s1 (t ) from s1′ (t ) = v1 (t ) = H and
s1 (0) = F . Consider “H” and “F” to be real number constants.
E. “Climb a two step ladder” to get an expression for s 2 ( t ) from
s ′2′( t ) = a2 ( t )= − 32 , v2 (0) = V , and s 2 (0) = S . Consider “V” and “S” to be real
number constants.
IV. Shown below is the graph of the derivative of each member of a large family of antiderivatives.
(3, f ′(3))
(0, 0)
(4, 0)
(2, 0)
(1, f ′(1))
graph off ′, the derivative
of each member of a family of
possible anti-derivatives
A. Make a number line diagram for f ′ which overlaps slightly the interval [0,4].
B. Consistent with your answer for “A” and consistent with the sketch, make a
number line diagram for f ′′ .
C. Sketch a curve which shows all the features which your diagrams indicate
should be shared by the graph of each member of the family of anti-derivatives of
f ′ . Use “f” as a generic name for members of this family, and label “key” points
on the curve you sketch with notation of the form (a, f (a)) . Use numerical values
for “a.”
V. Suppose h(x) =
3
x4
is considered to have the form h(x) = g( f (x)) with f (x) =
1
x4 :
A. Give an expression for g(x) .
B. Give an expression for h ′(x) in the form of the product of four factors, two of
which are powers of x.
P
C. Give an equivalent to your answer for “B” in the form a ⋅ x .
VI. If h(x) =
3
x4
x
 3
⋅   , give an expression for h ′(x) in the form of the sum of two
 4
products.
MAC Assignment #109
CHMS
-52VII. There are many, many possible non-equivalent expressions for h(x) consistent with
h ′(x) = 3⋅ x + 1. Give any two of them.
MAC Assignment #109
CHMS
-53-
ASSIGNMENT #110
THINGS FOR YOU TO READ AND USE
The Meaning of the dy/dt Number that can be Displayed by your Calculator
v1
v2
simulated flight
path
v3
v4
point at the base of the
cliff used as the origin to
establish a frame of reference
Shown on the vertical axis in the above illustration are four points whose second
coordinates give the vertical displacement of an object which has been projected out from
the edge of a cliff. These displacements give a measure of the distance from the object to
a point on the ground right beneath it.
In problem II of assignment #109 the height of the cliff was 250 feet, the initial velocity
of the launched object was 75 feet per second, and the launch angle was 40˚ relative to
the horizon. We can “climb the ladder” from g ′′(t ) = ¯32 to an expression for g( t ) which
meets the conditions g′(0) = 75 ⋅sin(40Þ) and g(0) = 250 in these two steps:
Step 1: Set g ′(t ) = ¯32 ⋅ t + C and then choose the value 75⋅ sin(40Þ) for “C” to
meet the condition g′(0) = 75 ⋅sin(40Þ).
2
Step 2: Set g( t ) = ¯16 ⋅ t + 75⋅sin(40Þ)⋅ t + C and then choose the value 250 for
this new “C” to meet the condition g(0) = 250 .
After taking these steps we have an image formula for function g which is a model for the
data points which include the points (t1,v1), (t2, v2), (t3, v3), and (t4, v4). The numbers
MAC Assignment #110
CHMS
-54identified as dy/dt which you took from your calculator as you did parts “E” and “F” of
problem II of assignment #109 are all images in the function g ′ with image formula
g′(t)= − 32⋅ t + 75⋅ sin(40Þ). Your work on problem II was intended to introduce you to
the following use of notation in calculus.
If function g models the vertical displacement relative
to a frame of reference of an object whose motion is being modeled parametrically,
dy/dt is an alternative to g ′(t ). Both notations are widely used.
__________
I. Here once again is the illustration of a Ferris wheel with a radius of 40 feet. The center
of the wheel is 45 feet above the ground and the wheel is rotating counterclockwise at the
rate of two revolutions per minute.
A
R
Suppose an object is released to the side of the seat attached to the Ferris wheel at point A
when point A is 70 feet above the ground. Assume the object is held out away from the
Ferris wheel so the object can move without making contact with it. The object of this
problem is to generate a parametric simulation of the motion of the object. You should
have stored in your calculator the following:
In memory cell M: the number of seconds that it takes point A to first reach a
points 70 feet above the ground after it passes point R.
In memory cells F and S: very good estimates of the coordinates of point A at time
M relative to the coordinate system frame of reference described in problem III of
assignment #109.
In memory cells H and V: very good estimates of the horizontal and vertical
components of the initial velocity of the object at time M. These components are
the rates of change of the horizontal and vertical displacements, respectively.
MAC Assignment #110
CHMS
-55Now create expressions for X1T and Y1T from which a simulation of the motion of the
object after its release can be generated. You might find it useful to look to parts “B” and
“C” of problem III of assignment #109 to get hints about how these expressions can be
created using the names of memory cells in which key numbers are stored. You should
try the simulation. It will take a little thought to determine a good window setting for
viewing it.
II. Use a( t )= − 32 as an image formula for a model acceleration function and
v (0) = 80 ⋅ sin(70Þ) and s (0) = 200 as initial conditions for the associated model velocity
and position functions to “climb the ladder” to an expression for s (t ) .
III. Suppose an object is projected upward and outward from the edge of a cliff with an
initial speed of 80 feet per second in a direction which makes an angle of 70˚ with an
imaginary extension beyond the edge of the cliff of the horizontal ground back from the
edge.
70Þ
arrow representing
an initial velocity
in a direction making
an angle of 70Þ with
the horizon. The initial
speed is 80 feet per second.
face of
a 200 foot
cliff
P
A. Suppose point P at the foot of the cliff in the illustration is the origin for a
frame of reference for the study of the flight of the object. Assume that time is
measured in seconds. Give entries for X1T and Y1T which can be used to
generate a parametric simulation of the flight of the object. Check that your
calculator is in degree MODE. Then test your answer by having your calculator
generate a simulation of the flight of the object. You may need to experiment
with the window settings to get a view of the full flight.
B. Create an equation which will have as one of its solutions the time it will take
in seconds for the object to strike the ground at the end of its flight.
C. Place your calculator in Function MODE and use the intersection command to
calculate the relevant solution of the equation you created as your answer for “B”
right above. Give that solution accurate to the nearest hundredth of a second.
Also store the most accurate value possible in your calculators memory as “T.”
MAC Assignment #110
CHMS
-56D. Use your stored answer from “C” to help calculate the distance of the point
where the object will strike the ground from a point at the base of the cliff directly
under the launch point.
E. Suppose the graph of function g passes through the points described as (the
time in seconds since the launch, the distance from the object straight down to the
ground at that time). Give an expression for g( t ).
F. For function g from “E” right above, give an expression for g ′(t ). Recall that
the numbers that can be computed from this expression are also represented by the
dy
traditional notation,
.
dt
G. Create from your answer for “F” an equation which will have as one of its
solutions the amount of time it will take for the launched object to reach its
maximum height over the ground at the base of the cliff. Now use your calculator
to help get an estimate of this time accurate to the nearest hundredth of a second.
IV. Shown below is the graph of the derivative of each member of a large family of antiderivatives.
(0, f ′(0))
graph of f ′, the derivative
(6, f ′(6))
of each member of a family of
possible anti-derivatives
(1, 0)
(3.5, 0)
(5, 0)
A. Make a number line diagram for f ′ which overlaps slightly the interval [0, 6].
B. Consistent with your answer for “A” and consistent with the sketch, make a
number line diagram for f ′′ .
C. Sketch a curve which shows all the features which your diagrams indicate
should be shared by the graph of each member of the family of anti-derivatives of
f ′ . Use “f” as a generic name for members of this family, and label “key” points
on the curve you sketch with notation of the form (a, f (a)) . Use numerical values
for “a.”
3.52+ 0.01 − 3.52 −0.01
V. Here is a symmetric difference quotient:
.
2 ⋅ 0.01
A. Suppose that this symmetric difference quotient is seen to have the form
f (c + h) − f (c − h)
:
2⋅h
MAC Assignment #110
CHMS
-57-
1. Give an expression for f ( x) . Also enter as Y1 an expression from
which a graph of function f could be generated.
2. Give a value for “c.”
3. Make a sketch of a portion of the graph of f so that you can show two
points, the line through which has as its slope the value represented by the
symmetric difference quotient. Label the coordinates of these two points.
B. Rewrite the original difference quotient so that the numerator is the difference
of two products of two factors each.
C. Use a distributive form to help give an equivalent to your answer for “B” right
above in the form of the product of two factors, one of which is a power of 3.5
while the other is in the form of a fraction.
D. Recall that you have entered an expression for Y1 from which a graph of
function f could be generated.
1. Now give a nDeriv command for computing the value of the factor in
the form of a fraction from your answer for “C.”
2. The exact value computed from a use of the nDeriv command you just
gave as your answer for “1” would be equal to the slope of the line passing
through two points on the graph of f different from the two points whose
coordinates you labeled as you did part “A, 3” of this problem. Make a
sketch of a portion of the graph of f on which you label the coordinates of
these two new points.
3. The nDeriv command you gave as your answer for “D,1” right above
estimates the slope of a line tangent to the graph of f at some point to
which the two points you just labeled are horizontally symmetric.
a. Give the exact coordinates of that point.
b. Give two other nDeriv commands which in succession will give
better estimates of the slope of this tangent line.
c. Give an expression representing the exact value of the slope of
this tangent line.
E. Consistent with your answer for “A, 3,” give an expression for f ′( x ).
−
VI. Suppose h(x) = x
MAC Assignment #110
4
3
is considered to have the form h( x ) = g( f ( x )) with f ( x) =
CHMS
1
x3 :
-58-
A. Give an expression for g( x ) .
B. Give an expression for h ′( x ) in the form of the product of four factors, two of
which are powers of x.
P
C. Give an equivalent to your answer for “B” in the form a ⋅ x .
−
VII. If h(x) = x
4
3
x
 4
⋅   , give an expression for h ′(x) in the form of the sum of two
 3
products.
VIII. There are many, many possible non-equivalent expressions for h(x) consistent with
h ′(x) = 5⋅ x 2 + cos(x) . Give any two of them.
MAC Assignment #110
CHMS
-59-
ASSIGNMENT #111
THINGS FOR YOU TO READ AND USE
A Reminder about Symmetric Difference Quotients
B = (c + h, f (c + h))
A = (c − h, f (c − h))
T = (c, f (c))
line tangent
to the graph of
f at (c, f (c)) . The
slope of this
tangent line is f ′(c)
The line passing through points A and B in the illustration above looks to be
approximately parallel to the line tangent to the graph function f at the point (c, f (c)) .
f (c + h) − f (c − h) f (c + h) − f (c − h)
Therefore,
=
, the slope of the line through points
(c + h) − (c − h)
2⋅h
A and B, approximates f ′(c) , the slope of the tangent line.
Notice that the change in the first coordinate in moving from point A to point T is h. h is
also the change in the first coordinate in moving from point T to point B. This means that
point T is positioned so that it is horizontally symmetric to points A and B. This suggests
f (c + h) − f (c − h)
the reason why
is referred to as a symmetric difference quotient.
2⋅h
If function f is an exponential function, the symmetric difference quotient has these
equivalents:
.
The rightmost expression in the string above reveals that the slope of the line tangent to
c
the graph of the exponential function at (c, f (c)) = (c, b ) is approximated by the product
c
of the second coordinate of the point of tangency, b , and a new symmetric difference
b 0+ h − b 0− h f (0 + h ) − f (0 − h )
quotient. The new symmetric difference quotient is
=
.
2⋅h
2 ⋅h
This new difference quotient therefore estimates the slope of the line tangent to the graph
of f at (0, f (0)) = (0,1). You should recall the fascinating fact that the slope of the tangent
line through (0,1) is exactly ln(b) for every possible base, b, for an exponential function.
__________
MAC Assignment #111
CHMS
-60-
I. Shown below is the graph of the derivative of each member of a large family of antiderivatives.
7.1
1
10
A. Make a number line diagram for f ′ which overlaps slightly the interval [1,10].
B. Consistent with your answer for “A” and consistent with the sketch, make a
number line diagram for f ′′ .
C. Sketch a curve which shows all the features which your diagrams indicate
should be shared by the graph of each member of the family of anti-derivatives of
f ′ . Use “f” as a generic name for members of this family, and label “key” points
on the curve you sketch with notation of the form (a, f (a)) . Use numerical values
for “a.”
0.81.7 + 0.03 − 0.81.7 − 0.03
II. Here is a symmetric difference quotient:
.
2 ⋅ 0.03
A. Suppose that this symmetric difference quotient is seen to have the form
f (c + h) − f (c − h)
:
2⋅h
1. Give an expression for f ( x) . Also enter as Y1 an expression from
which a graph of function f could be generated.
2. Give a value for “c.”
3. Make a sketch of a portion of the graph of f so that you can show two
points on the line that has as its slope the value represented by the
symmetric difference quotient. Label the coordinates of these two points.
B. Rewrite the original difference quotient so that the numerator is the difference
of two products of two factors each.
C. Use a distributive form to help give an equivalent to your answer for “B” right
above in the form of the product of two factors, one of which is a power of 0.8
while the other is in the form of a fraction.
MAC Assignment #111
CHMS
-61-
D. Recall that you have entered an expression for Y1 from which a graph of
function f could be generated.
1. Now give a nDeriv command for computing the value of the factor in
the form of a fraction from your answer for “C.”
2. The exact value computed from a use of the nDeriv command you just
gave as your answer for “1” would be equal to the slope of the line passing
through two points on the graph of f different from the two points whose
coordinates you labeled as you did part “A,3” of this problem. Make a
sketch of a portion of the graph of f on which you label the coordinates of
these two new points.
3. The nDeriv command you gave as your answer for “D,1” right above
estimates the slope of a line tangent to the graph of f at some point to
which the two points you just labeled are horizontally symmetric.
a. Give the exact coordinates of that point.
b. Give two other nDeriv commands which in succession will give
better estimates of the slope of this tangent line.
c. Give an expression representing the exact value of the slope of
this tangent line.
E. Consistent with your answer for “A, 3,” give an expression for f ′( x ).
III. There are many different vertically scaled, vertically translated exponential functions
whose graphs would have the features shown in the illustration below. Assume that the
horizontal line through (25,71) is a horizontal asymptote for the graphs of these functions.
(25, 71)
(0, 50)
A. Give image formulas for functions f and g if these are two of the many possible
vertically scaled, vertically translated exponential functions whose graphs would
have the features shown.
MAC Assignment #111
CHMS
-62-
B. What additional information would be sufficient for you to choose only one
vertically scaled, vertically translated exponential function whose graph has the
features shown?
IV. Suppose an object is projected upward and outward from the edge of a cliff with an
initial speed of 90 feet per second in a direction which makes an angle of 75˚ with an
imaginary extension beyond the edge of the cliff of the horizontal ground back from the
edge.
75Þ
arrow representing
the initial velocity of
an object launched at
an angle of 75Þ with
the horizon. The initial
speed of the object is
90 feet per second.
face of
a 280 foot
cliff
A. Suppose vertical position of the object is considered to be its distance
measured in feet to the ground below the cliff. Assume that time is measured in
seconds. Give entries for X1T and Y1T which can be used to generate a
parametric simulation of the flight of the object.
B. Create an equation which will have as one of its solutions the time which will
elapse from when the object is launched until it strikes the ground. Use the most
efficient means you can to determine this time accurate to the nearest hundredth of
a second.
C. Use your answer for “B” to help determine, accurate to the nearest tenth, the
distance of the strike point of the object at the end of its flight from a point at the
foot of the cliff directly below the launch point.
D. Create an equation which will have as one of its solutions the time it will take
after the launch for the object to attain its greatest distance above the ground.
Give that time accurate to the nearest hundredth of a second. Also store it in your
calculator's memory as “T.”
E. Use the value stored as “T” to determine, accurate to the nearest foot, the
greatest distance of the object from the ground directly below it attained during
the flight.
MAC Assignment #111
CHMS
-63Get you calculator to simulate the flight of the object. Be careful about the MODE
settings. Also, use some of the answers above to help determine your WINDOW settings.
If you are successful, you should be able to see the entire flight on the first viewing.
2
= (sin( x )) 5
V. Suppose h( x )
is considered to have the form h( x ) = g( f ( x )) where
function g is a power function.
A. Give expressions for f ( x) and g( x ) .
B. Give an expression for h ′( x ).
VI. In each case below there are many, many possible non-equivalent expressions for
h( x ) consistent with the given expression for h ′( x ). Give any two of them.
−
2
A. When h ′( x ) = ⋅ x
5
3
5 .
−
2
B. When h ′( x ) = ⋅ (sin( x ))
3
1
3 ⋅ cos( x ).
sin( x )
VII. Suppose h( x ) = e
is considered to have the form h( x ) = g( f ( x )) where function
g is an exponential function.
A. Give expressions for f ( x) and g( x ) .
B. Give an expression for h ′( x ).
VIII. In each case below there are many, many possible non-equivalent expressions for
h( x ) consistent with the given expression for h ′( x ). Give any two of them.
x
A. When h ′(x) = ln(3) ⋅3 .
sin(x )
B. When h ′(x) = ln(3) ⋅3
⋅ cos(x) .
IX. Recall that the base e exponential function is often referred to as “exp.” Suppose
h( x ) = exp(ln(x )).
A. Give an equivalent expression for h( x ) which contains only one letter.
B. Use your answer for “A” to give an expression for h ′( x ) which contains at
most one letter.
C. Use the Chain Rule to give an expression for h ′( x ) in the form of a product of
two factors. This expression should contain only one “ ′ ” mark.
MAC Assignment #111
CHMS
-64-
D. Create a new equation by using the two expressions for h ′(x) from your
answers for “B” and “C” right above.
E. Use your answer to “D” to help get an expression for l n′( x ).
MAC Assignment #111
CHMS
-65-
ASSIGNMENT #112
THINGS FOR YOU TO READ AND USE
An Example to Remind You about Computing Estimates of Change from
Tangent Line Slopes
(b, f (b))
a portion of the line tangent
to the graph of f at the point
(a, f (a)). The slope of this line
is represented by f ′(a).
(a + I, f (a + I))
(a, f (a))
a portion of the graph
of function f
The illustration should help remind you that as one moves from the point (a, f (a)) to the
point (a + I, f (a + I)) , the change in the second coordinate can be estimated using the
slope of the tangent line to the graph of f at the point (a, f (a)) . That estimate of change
(in this case an overestimate) is equal to the product f ′(a) ⋅I.
___________
I. Suppose a clock is started as a car leaves a stop sign at an intersection. The position of
the car is chosen to be its distance from a cross walk line in the intersection from which
the car departed. Over the course of the next two minutes, the driver increases the speed
of the car to 35 miles per hour, holds that speed for a while and then comes to a stop at
the next intersection where there is a traffic light showing red. Assume that the motion of
the car is rectilinear.
A. Over the time interval [0, 2] sketch what could be the graph of a good model
function, v, made up of the (time in minutes since the car left the first intersection,
the velocity of the car) pairs. As you do this, think about whether it makes sense
to avoid having any abrupt changes of direction anywhere along the graph.
B. Consistent with the graph you made as your answer for “A” right above, create
a number line diagram for the model velocity function, v.
C. Consistent with work you have already done in this problem, sketch a graph of
the model acceleration function, a, associated with the model velocity function, v.
D. Now, consistent with the work you have already done, sketch a graph of the
model position function associated with the model velocity and acceleration
functions.
E. Suppose the model position function associated with the model velocity and
acceleration functions from parts “A” and “C” is given the name “s.” Give the
number of any of the statements below that are true.
MAC Assignment #112
CHMS
-66-
#1: According to the models, the distance traveled by the car over the time
interval [0, 2] is equal to s (2) − s (0) .
#2: According to the models, the distance traveled by the car over the time
interval [0, 2] is equal to s ′(2) − s ′(0) .
II. Here is a table of pre-images and images for some function, f. Assume that the graph
of f has no inflection points on the interval [0.5, 4.5].
pre-images
for function f
1
2
3
4
images for
function f
2
3.2
4.7
6.8
A. Consistent with the table, sketch what could be that portion of the graph of f
made up of points with first coordinates in the interval [0.5, 4.5].
B. Here is a proposed table of pre-images and images for function f ′ , the
derivative of function f. If any of the images in the table are inconsistent with
either the table for f or your sketch for “A,” give that image or those images and
give a reason why you consider it or them inconsistent.
pre-images
images for
for function f ′ function f ′
1
2
3
4
1.3
1.4
2.4
2.1
III. Suppose a family is deciding whether or not to obtain a home mortgage for $90,000.
They have decided that they want to pay off the mortgage in fifteen years with equal
monthly payments. After checking with various lenders they find that the best annual
interest rate they can get is 5.3%
A. Create a difference equation with an initial condition which can be used to
compute the month by month unpaid balance of the mortgage. Use the letter “P”
to represent the size of the monthly payments.
B. Suppose function f is a model for data points described as (the number of
months since the loan was made, the unpaid balance of the loan):
MAC Assignment #112
CHMS
-671. Give the form of an expression for f ( x) . Where possible, give exact
numerical values. Otherwise use letters in this expression.
2. Give the exact coordinates of two points you know must be on the graph
of the model function, f.
IV. Suppose h( x ) = ln(sin( x )) and the domain of h is the open interval (0,π). If h is seen
as having the form h( x ) = g( f ( x )) where function f is a function whose graph can be
mapped onto itself with many different horizontal translations:
A. Give expressions for f ( x) and g( x ) .
B. Give an expression for h ′( x ).
V. In each case, give an expression for h ′( x ).
A. When h( x ) = ln(3 ⋅ x ).
B. When h( x ) = x ⋅ ln(4 ⋅ x) .
VI. In each case below, there are many, many possible non-equivalent expressions for
h( x ) consistent with the given expression for h ′( x ). Give any two of them.
A. When h ′( x ) =
1
⋅2 .
2⋅x
B. When h ′( x ) = 5 ⋅
1
⋅ cos( x ). In this case, the domain of h is (2.π, 3.π).
sin( x )
VII.
(24, 107)
Distance in feet from
point A to the point on the
ground right beneath it
(0,7)
time in seconds since point A
passed reference point R
A. The curve in the illustration above is a portion of the graph of a sine wave
function. (0,7) is a relative minimum point and (24,107) is a relative maximum
MAC Assignment #112
CHMS
-68point. Give an image formula for a two-dimensional scale change which would
map the graph described by y = cos( x ) onto a sine wave which has its inflection
points on the x-axis, a relative minimum point on the y − axis , the same amplitude
and fundamental period as the sine wave in the illustration.
B. Create a connection diagram which begins with y = cos( x ) and ends with an
equation for the graph of the sine wave function a portion of whose graph is
shown in the illustration above.
A
R
C. Suppose the sine wave function whose graph is partially shown in the
illustration for parts “A” and “B” is related to the motion of the Ferris wheel in the
illustration right above.
1. What is the implied time for the Ferris wheel to make one complete
revolution?
2. What is the implied distance from reference point R to a point on the
ground directly beneath it?
3. What is the implied radius of the Ferris wheel?
D. Working from the partial graph of the sine wave function shown in the
illustration for parts “A” and “B,” sketch a graph showing one cycle of the
derivative of that function.
E. Suppose the sine wave function whose graph is partially shown in the
illustration for parts “A” and “B” is named “g.” Create an image formula for the
function whose images will give the rates of change of the distances from point A
to the point on the ground directly beneath it as the Ferris wheel turns.
F. There is a smallest positive time when the rate of change of the distance from
point A to a point on the ground directly beneath it is greatest.
1. Give that time.
2. Give the exact rate of change of that distance at that time.
MAC Assignment #112
CHMS
-69-
ASSIGNMENT #113
I. As in a problem in the last assignment, suppose a clock is started as a car leaves a stop
sign at an intersection. This time the position of the car is chosen to be its distance from
a cross walk line in the next intersection. Over the course of the next two minutes, the
driver increases the speed of the car to 35 miles per hour, holds that speed for a while and
then comes to a stop at the next intersection where there is a traffic light showing red.
A. Over the time interval [0,2] sketch what could be the graph of a good model
function for the (time since the car left the first intersection, the velocity of the
car) data points. As you do this, think about whether it makes any sense to avoid
having any abrupt changes of direction anywhere along the graph.
B. Consistent with the graph you made as your answer for “A” right above, create
a number line diagram for the model velocity function, v.
C. Consistent with work you have already done in this problem, sketch a graph of
the model acceleration function, a, associated with the model velocity function, v.
D. Now, consistent with the work you have already done, sketch a graph of the
model position function associated with the model velocity and acceleration
functions.
E. Suppose the model position function associated with the model velocity and
acceleration functions from parts “A” and “C” is given the name “s.” Give the
number of any of the statements below that are true.
#1: According to the models, the distance traveled by the car over the time
interval [0, 2] is equal to s (2) − s (0) .
#2: According to the models, the distance traveled by the car over the time
interval [0, 2] is equal to s ′(2) − s ′(0) .
II. Here are tables of pre-images and images for some function, f, and its derivative, f ′ .
Assume that the graph of f has no inflection points on the interval [0.5, 4.5].
pre-images
for function f
images for
function f
1
2
3
4
3.0
2.2
0.8
¯ 0.9
pre-images
images for
for function f ′ function f ′
1
2
3
4
¯ 0.7
¯ 1.2
¯ 1.5
¯ 2.1
If possible, sketch what could be that portion of the graph of f made up of points with first
coordinates in the interval [0.5, 4.5] which is consistent with both tables.
MAC Assignment #113
CHMS
-70-
III. There are many different vertically scaled, vertically translated exponential functions
whose graphs would have the features shown in the illustration below. Assume that the
horizontal line through (25,66) is a horizontal asymptote for the graphs of these functions.
(24, 66)
(0, 40)
A. Give image formulas for functions f and g if these are two of the many possible
vertically scaled, vertically translated exponential functions whose graphs would
have the features shown.
B. Suppose it is also known that the graph of the function in the illustration passes
through the point (17, 58):
1. Create an equation which would have as a solution the base for a model
function having the features shown in the illustration and passing through
this additional point.
2. Use the intersect command to get a value for the base of the exponential
model. You call this value to the home screen by pushing [X,T,θ ] and
then pushing [ENTER]. Do this and store the result as “B.” Also give this
value accurate to three decimal places on your homework paper.
3. Now make use of the natural log function to write a replacement for
your “B,1” equation which does not contain an exponent.
4. “Solve” your equation from “3” right above for the natural logarithm
image of the base. Give this value on your homework paper accurate to
four decimal places.
5. Use your answer for “4” right above to get a value for the base of the
exponential model. Then recall your stored value from “2” and compare
the results. If they do not match accurate to at least five decimal places,
rework what is necessary to overcome the discrepancy. If you get a very
close match, report this fact on your homework paper.
MAC Assignment #113
CHMS
-71-
IV. Suppose a family will initiate a 15 year, $90,000 home mortgage at an annual interest
rate of 7.15%. The loan will be paid off in equal monthly payments.
A. Create a difference equation with an initial condition which can be used to
compute the month by month unpaid balance of the mortgage. Use the letter “P”
to represent temporarily the size of the monthly payments.
B. Suppose function f is a vertically scaled, vertically translated exponential
function model for data points described as (the number of months since the loan
was made, the unpaid balance of the loan):
1. Create a system whose solution will give the vertical scale factor and
translation number for an expression for f ( x) . Store these values in your
calculator's memory. Also, give them on your homework paper accurate
to the nearest hundredth.
2. Make an entry for Y1 from which the graph of f can be generated. Use
memory cell names in this expression. Then use this expression to help
give the unpaid balance of the mortgage right after the first payment has
been made. Store this value as “F” in your calculator's memory. Also,
give this value on your homework paper accurate to the nearest cent.
C. Use your answer for part “A” of this problem to create an equation which can
be solved for a numerical value for “P,” the size of the monthly payment. Store
this value in your calculator's memory as P. Also give this value of P on your
homework paper accurate to the nearest cent.
D. Put your calculator into sequence MODE and make an entry for u(n) based on
your answer for “A.” Use the letter “P” in this entry so your calculator will make
use of your computed value for the size of the monthly payment. Then push
WINDOW and make the appropriate settings for nMin and u(nMin) for
computing unpaid balances for this mortgage. At the home screen key in u(180)
and push [ENTER]. Give the computed value on your homework paper. You
should be aware that the result provides a check of your work in this problem.
E. Create an equation whose solution will help answer the following question:
“How many monthly payments will have been made before the unpaid loan
balance first drops below $45,000?” You may use the names of memory cells in
this equation.
F. Use the intersection command to determine the number of monthly payments
that will be made before the unpaid balance first drops below $45,000.
Remember to return your calculator to Func MODE before using the program.
MAC Assignment #113
CHMS
-72-
 −π π 
V. Suppose h( x ) = ln(cos( x )) and the domain of h is  ,  . If h is considered to have
 2 2
the form h( x ) = g( f ( x )) where function f is a function whose graph can be mapped onto
itself with many different horizontal translations:
A. Give expressions for f ( x) and g( x ) .
B. Give an expression for h ′( x ).
VI. In each case, give an expression for h ′( x ).
A. When h( x ) = x ⋅ ln(2) .
x
B. When h( x ) = ln(2 ).
x
2
C. When h(x) = 2 ⋅ln( x ) .
VII. In each case below there are many, many possible non-equivalent expressions for
h( x ) consistent with the given expression for h ′( x ). Give any two of them.
A. When h ′( x ) = 8.3 ⋅
1
⋅ 3.
3⋅ x
B. When h ′( x ) = ln(4) ⋅
1 x
⋅4 .
4x
VIII.
A
R
Suppose that the Ferris wheel in the illustration above has a radius of 50 feet and it
completes one full rotation every 48 seconds. Assume that reference point R is on a
vertical reference line through the center of the wheel and that point R is seven feet above
the ground.
A. Suppose that function f is a model for data points described as (the time since
point A moved pass fixed reference point R, the horizontal displacement of point
A relative to the vertical reference line through the center of the wheel). Consider
displacements to the right of the vertical reference line as positive and
displacements to the left of the reference line as negative. Sketch one cycle of the
graph of function f. On your sketch, label the coordinates of one relative
minimum point, one relative maximum point, and one inflection point.
MAC Assignment #113
CHMS
-73B. Create a connection diagram which begins with either y = cos( x ) or y = sin(x )
and ends with an equation describing the graph of function f from “A” right
above. Check your answer by making computations based on the points whose
coordinates you labeled on your sketch for “A.”
C.
1. Sketch a graph of one cycle of the derivative of function f from “A.”
2. Based on your sketch, give the rate of change of the horizontal
displacement of point A relative to the vertical reference line exactly 12
seconds after point A moved past point R.
D. In the last assignment you worked with a model function for the data points
described as (the time since point A moved pass reference point R, the vertical
distance from point A to a point on the ground directly beneath it). If g is the
 2 ⋅π 
name of that model function, it has image formula g( x ) = − 50 ⋅ cos
⋅ x  + 57 .
 48

1. Create an equation which will have as a solution the smallest positive
time when point A will be exactly 80 feet above a point on the ground
directly beneath it.
2. Use the intersect command to determine, as closely as the calculator
can, the smallest positive time when the vertical displacement of point A
relative to the ground is 80 feet. Recall that this value can be accessed by
entering “X” at the home screen and pushing [ENTER]. After recalling
this value, store it in your calculator's memory in a memory cell named
“Q.”
IX. “Climb the ladder” from the image formula a( t )= − 32 for a model acceleration
function to an image formula for the associated position function which meets the
2⋅π
 2 ⋅π 
conditions s ′(0) =
⋅ 50 ⋅ sin
⋅ R and s (0) = 80. Assume that “R” represents a
 48

48
real number constant.
MAC Assignment #113
CHMS
-74-
ASSIGNMENT #114
I. Shown below is the graph of the derivative of each member of a large family of antiderivatives.
(3.5, f ′(3.5))
(2, 0)
(8.5, f ′(8.5))
(10, 0)
(6, 0)
A. Make a number line diagram for f ′ .
B. Consistent with your answer for “A” and consistent with the sketch, make a
number line diagram for f ′′ .
C. Sketch a curve which shows all the features which your diagrams indicate
should be shared by the graph of each member of the family of anti-derivatives of
f ′ . Use “f” as a generic name for members of this family, and label “key” points
on the curve you sketch with notation of the form (a, f (a)) . Use numerical values
for “a.”
II.
(1, 2)
a portion of the
graph of function f
(4,¯ 1)
(7,¯ 4)
A. Consistent with the graph of function f in the illustration, sketch that portion of
the graph of f ′ which has first coordinates in the interval [0.5, 7.5].
MAC Assignment #114
CHMS
-75B. Consistent with the graph you sketched for “A” right above, sketch that portion
of the graph of f ′′ which has first coordinates in the interval [0.5, 7.5]. Keep in
mind that f ′′ is the derivative of f ′ .
C. Consistent with the illustration given and with the sketches you made, give the
right column of each table. Assume that the point with first coordinate 4 is
“special.”
pre-images
for function f
images for
function f
1
2
3
4
5
6
7
pre-images
for function f ′
images for
function f ′
1
2
3
4
5
6
7
III. Suppose a family borrows $8500 to help pay for a car. The annual interest rate is
7.65% and the loan is to be paid off in equal monthly payments over a four year period.
A. Create a difference equation with an initial condition which can be used to
compute the month by month unpaid balance of the car loan.
B. Create an image formula for a model function, f, for the (number of payments
made, the unpaid balance of the loan after that many payments) data points. In
this image formula, give the exact base and give any coefficients accurate to the
nearest hundredth. Also make use of your calculator's memory so that further
computations give results as accurate as possible.
C. Use your results for “A” and “B” to compute the size of the monthly payment
accurate to the nearest cent. Also store the most accurate possible result in your
calculator's memory using a memory cell named “P.”
D. Compute the change in the unpaid balance from the time the 36th payment is
made until right after the 37th payment is made. Give this result on your
homework paper accurate to the nearest cent.
E. On your homework paper, sketch the shape of a graph of the model function, f.
Based on your view of this graph, would you expect tangent line slope estimates
of changes in the unpaid balance to be too large or too small? (Be careful, you are
dealing with negative numbers!)
F. Give an expression for f ′(x). This expression can contain memory cell names.
Also make an entry for Y2 based on this expression.
G. Given function g with image formula g(x) = f ′ (36)⋅ x + b :
MAC Assignment #114
CHMS
-761. Give an expression for g(36) in the form of the sum of two addends.
2. Give an expression for g(36 + I) in the form of the sum of three
addends.
3. If one moves along the graph of g from the point with first coordinate
36 to the points whose first coordinate is represented by “36 + I,” what is
the change in the second coordinate?
H. Give the tangent line slope estimate of the change in the unpaid balance from
the time right after the 36th payment has been made to the time right after the 37th
payment has been made.
IV. Suppose that an object which has been cooled to a temperature of 40˚ Fahrenheit has
been brought into a room with an ambient temperature of 74˚ to warm up. Suppose also
that after 12.5 minutes of warming, the temperature of the object has risen to 61˚.
A. Create an image formula for a vertically scaled, vertically translated
exponential model function, W, for the data points described as (the number of
minutes since the object was brought into the room, the temperature of the object
after that many minutes.) Use “t” as the variable in this image formula. Show the
details of your work, including two different ways to determine the base of the
model function. Give the image formula on your homework paper with the base
and coefficients accurate to the nearest hundredth. Store the most accurate value
possible for the base in a memory cell named “B.”
B. Give an expression for W ′(t) in which the letter “W” appears exactly twice.
This expression should make clear that the rate of temperature change is related to
the object temperature in the way stated in Newton's Law of Cooling.
C. Give an expression for Y1 appropriate for the use of PROGRAM EULER to
approximate points on the graphs of solutions of the differential equation you
created as your answer for “B” right above. Use PROGRAM EULER with an
initial point of (0,40) and an increment size of 0.25 to help get an estimate of the
temperature of the object after 12.5 minutes. Think carefully about how many
steps should be taken to do this. If you make the correct choices, you should find
the estimated temperature as the last entry in list #2. Give this estimate on your
homework paper accurate to the nearest hundredth.
V. Suppose a vertically scaled exponential function, P, is used to model (time in hours,
number of organisms at that time) data points for the early stages in the growth of a
bacteria population cultivated in a nutrient rich environment. Suppose that the number of
bacteria is originally 2400 and after 1.7 hours it is determined to be 2520.
A. Give an expression for Y1 from which a graph of P could be generated. To
ensure the greatest possible accuracy, include memory cell names in this
expression. On your homework paper show the work you did to get this
MAC Assignment #114
CHMS
-77expression. Also, give the expression on your homework paper with the base and
coefficient accurate to the nearest hundredth.
B. Give an expression for P ′(t). Use memory cell names in this expression.
C. Give an equivalent to your answer for “B” which contains the letter “P” exactly
twice.
D. Use your answer to “C” to compute each of these numbers accurate to the
nearest integer.
1. The rate of growth of the bacteria population when the population is
3000.
2. The rate of growth of the bacteria population when the population is
6000.
3. The rate of growth of the bacteria population when the population is
9000.
E. True or False: The pairs described as (the population at any given time, the rate
of growth of the population at that time) lie on a line through the origin. If this is
true, give an exact representation of the slope of that line.
VI. Suppose h(x) = ln( f (x)) is considered to have the form h(x) = g( f (x)) .
B. Give an expression for h ′(x).
A. Give an expression for g(x) .
VII. In each case, give an expression for h ′(x).
2
A. When h(x) = x ⋅ ln(2) .
2
2
3
B. When h(x) = x ⋅ ln(x ) .
x
C. When h(x) = x ⋅ ln(3 ) .
VIII. In each case below there are many, many possible non-equivalent expressions for
h(x) consistent with the given expression for h ′(x). Give any two of them.
5⋅x
A. When h ′(x) = 5⋅ e .
B. When h ′(x) =
1
⋅ P ′(x) and P(x) > 0
P(x)
for all x.
IX. The Ferris wheel in the illustration below has a radius of 50 feet and it is making one
complete rotation every 48 seconds. Reference point R is on a vertical line through the
center of the wheel and it is 7 feet above the ground.
MAC Assignment #114
CHMS
-78-
A
R
The work that has been done in recent assignments is preparatory to simulating the
motion of a ball that is released to the side of the Ferris wheel at a time when point A is
rising through a position 80 feet above a point on the ground right beneath it. Here is a
summary of some key results:
 2 ⋅π 
The function f with image formula f (x) = 50 ⋅sin
⋅ x  is a model for the data
 48 
points described as (the time since point A passed reference point R, the
horizontal displacement of point A relative to a vertical line through the center of
the wheel).
 2 ⋅π 
The function g with image formula g(x)= − 50 ⋅cos
⋅ x  + 57 is a model for
 48

the data points described as (the time since point A passed reference point R, the
displacement of point A relative to a point on the ground directly beneath it).
 2 ⋅π 
The smallest positive solution of the equation 80= − 50 ⋅cos
⋅ x  + 57 has
 48

been stored in your calculators memory as “Q.”
A. If the intersection of the vertical reference line through the center of the wheel
and the ground is used as the origin, give exact representations of the coordinates
of the point from which the ball is released. When you do this, do not be
concerned about the width of a seat of the Ferris wheel, or the distance the ball is
held out to the side of the wheel when it is released.
B.
1. Give an expression for f ′(x).
2. Give a representation of the initial horizontal component of the velocity
of the ball. Use the letter “Q” in this representation.
C.
1. Give an expression for g ′(x).
2. Give a representation of the initial vertical component of the velocity of
the ball. Use the letter “Q” in this representation.
MAC Assignment #114
CHMS
-79 2 ⋅π 
 2 ⋅π 
 ⋅cos
X. “Climb the little ladder” from s ′(t) = v(t) = 50 ⋅ 
⋅ Q to an expression
 48 
 48

for s(t) which meets the condition s(0) = F . Assume that “Q” and “F” represent real
constants.
MAC Assignment #114
CHMS
-80-
ASSIGNMENT #115
I.
(7, 3)
a portion of the
graph of function f
(4,¯ 1)
(1, ¯ 5)
A. Consistent with the graph of function f in the illustration, sketch that portion of
the graph of f ′ which has first coordinates in the interval [0.5, 7.5].
B. Consistent with the graph you sketched for “A” right above, sketch that portion
of the graph of f ′′ which has first coordinates in the interval [0.5, 7.5]. Keep in
mind that f ′′ is the derivative of f ′ .
C. Consistent with the illustration given and with the sketches you made, give the
right column of each table. Notice that one of the right columns is to contain
images in function f while the other is to contain images in the derivative of
function f. Assume that the point with first coordinate 4 is “special.”
pre-images
for function f
1
2
3
4
5
6
7
images for
function f
pre-images
for function f ′
images for
function f ′
1
2
3
4
5
6
7
II. Suppose a family wishes to save a total of $40,000 to meet college expenses when
their young child is expected to enroll in college fifteen years from now. They have
found an investment which will pay dividends at an annual rate of 8.6%. They plan to
make an initial deposit of $3000 and equal monthly deposits subsequently.
A. Create a difference equation with an initial condition which can be used to
compute the month by month change in the balance in the college account.
MAC Assignment #115
CHMS
-81B. Suppose function f is a “perfect” model for the data points described as (the
number of monthly deposits which have been made, the college account balance
after that many payments).
1. Give the exact coordinates of at least two points which must be on the
graph of f.
2. Give an expression for Y2 from which a graph of f could be generated.
Use memory cell names in this expression so that the graph will come as
close as the accuracy of your calculator will allow to passing through all
the data points. Also give an image formula for function f on your
homework paper. The base in this image formula should be exact and the
coefficients should be accurate to the nearest hundredth.
3. By making use of both the Y2 expression and the difference equation,
calculate, accurate to the nearest hundredth, the monthly deposit the family
must make to meet its savings goal.
C. Give an expression for Y1 from which a graph of the tangent line slope
function for f could be generated. For the sake of accuracy, use memory cell
names in this expression. Also, on your homework paper, make a sketch showing
the shape of the graph of this tangent line slope function.
D. Give, accurate to the nearest hundredth, the tangent line slope estimate of the
change in the account balance from just after the 100th deposit is made until just
after the 101st deposit is made.
III.
A. Write a differential equation which expresses the fact that the rates of growth
of a population whose growth is modeled by a function, P1, varies directly as the
changing sizes of that population. Use 0.1 as the constant of variation.
B. Use PROGRAM EULER with an initial point of (0,5) and 20 increments of
size 1 to generate a set of points which approximate a set of 21 points on the
graph of the solution of your differential equation from “A” which passes through
(0,5). Recall that when you use PROGRAM EULER you need to enter as Y1 an
expression from which your calculator can compute tangent line slopes from
second coordinates that are stored in the memory cell named “S.” Based on an
observation of a scatter plot of these points, what might be wrong with using the
solution of this differential equation as a model over an extended period of time
for (time, population) data points?
C. Write a differential equation which expresses the fact that the rates of growth
of a population whose growth is modeled by a function P2, varies directly as the
difference between a carrying capacity of 100 and the changing populations.
Again, use 0.1 as the constant of variation.
MAC Assignment #115
CHMS
-82D. Use PROGRAM EULER with an initial point of (0,5) and 40 increments of
size 1 to generate a set of points which approximate a set of 41 points on the
graph of the solution of your differential equation from “C” which passes through
(0,5). Based on an observation of a scatter plot of these points, what might be
wrong with using the solution of this differential equation as a model for the early
stages in the growth of a population?
E. Create a new differential equation which has a solution whose graph passes
through (0,5). This solution should combine the desirable features of the model
functions P1 and P2 from “A” and “C” above while avoiding any undesirable
features you might have noted in your answers for “B” and “D.” Think carefully
about the constant you use in this differential equation. Choose it so that the
portion of the graph near the vertical coordinate axis would be approximated by
the graph of P1 while the portion far to the right of the vertical coordinate axis
would be approximated by the graph of P2.
IV. This problem is about describing the form of image formulas for members of a family
of functions with the “generic” name f which meet the condition f ′(x) = 3 ⋅ f (x) :
A. Give an equivalent to the given equation which has the letter “f” on only one
side of the equal sign. Assume that no point on the graph of any of the family of
functions has second coordinate 0.
B. Assume that f (x) > 0 for all x in the domain of f and then create from your
answer for “A” right above an equation which relates ln( f (x)) to expressions
describing a large family of parallel lines.
C. Create from your answer for “B” an equation which does not contain “ln.”
D. Create an equivalent to your answer for “C” which contains neither a plus sign
nor a minus sign.
E. True or False: If for some function, g, all the pairs of the form (g(x), g′(x)) lie
on a line through the origin with slope 3, that function must be a vertically scaled
exponential function.
V. In each case, give an expression for h ′(x).
sin(x )
A. When h(x) = 4
.
4
B. When h(x) = (sin(x)) .
sin( x)
C. When h(x) = 5
MAC Assignment #115
⋅ (cos(x)) 5 .
CHMS
-83VI. In each case below there are many, many possible non-equivalent expressions for
h(x) consistent with the given expression for h ′(x). Give any two of them.
A. When h ′(x) = ln(3) ⋅ 3cos( x ) ⋅(− sin(x)) .
B. When h ′(x) =
MAC Assignment #115
1
⋅ P ′(x) and P(x) > 30 for all x.
P(x) − 30
CHMS
-84-
ASSIGNMENT #116
I. A doctor has prescribed a strong medicine for a serious medical condition. She has
recommended a dosage that contains 1.2 milligrams of the active ingredient to be taken
every six hours. Experience has shown that because of chemical changes and the action
of the kidneys 25 percent of the active ingredient in a patient's body will not be present
six hours later.
A. In each case, give the amount of active ingredient one would expect to be
present in the patient's body:
1. Right after the first dose has been taken.
2. Right after the second dose has been taken.
3. Right after the third dose has been taken.
B. Create a difference equation with an initial condition which can be used to
compute the number of milligrams of active ingredient in the patients body from
the time one dose is taken to right after each new dose is taken.
C. Put your calculator in sequence MODE and make an entry for u(n) based on
your answer for “B.” Also push WINDOW and make appropriate entries for
nMin and u(nMin) (Does it make more sense to set nMin to “0” or “1”?) Then
use the command “seq(X,X,1, 20, 1)” to generate list #1 and the command
“u(L1)” to generate list #2. Set your calculator in function MODE. Then, after
viewing a scatter plot of the list #1, list #2 data points, give the form of an image
formula for a model function A, for the data points described as (the number of
doses taken, the amount of active ingredient in the patient's body).
D. Create an image formula for the function, A, from “C” right above. Use exact
numbers in this image formula. Also, on your homework paper, indicate clearly
how you determined any of these numbers which were not immediately known.
E. Experience has shown that a concentration of more than 4.0 milligrams of the
active ingredient in this medicine in a patient's body could have severe negative
effects. Create an equation whose solution will help answer the question, “What
is the maximum number of doses of the medicine the doctor should have the
patient take to avoid these severe negative consequences?”
F. Use your answer for “E” and the intersection command to determine the
number of doses the doctor should tell the patient to take so that the concentration
of the medicine will not pass the danger level.
II. Shown at the top of the next page is the graph of the derivative of each member of a
large family of anti-derivatives.
MAC Assignment #116
CHMS
-85graph of f, ′ the derivative
of each member of a family of
possibleanti-derivatives
(0, f ′(0))
(4, f ′(4))
(0, 0)
(2, 0)
(4, 0)
(5.5, 0)
A. Make a number line diagram for f ′ which overlaps slightly the interval [¯ 1,6].
B. Consistent with your answer for “A” and consistent with the sketch, make a
number line diagram for f ′′ .
C. Sketch a curve which shows all the features which your diagrams indicate
should be shared by the graph of each member of the family of anti-derivatives of
f ′ . Use “f” as a generic name for members of this family, and label “key” points
on the curve you sketch with notation of the form (a, f (a)) . Use numerical values
for “a.”
III.
graph of a model function
for (time (in minutes), temperature (in
degrees Fahrenheit)) data points
(0, 120)
(50,75)
A. There are many different vertically scaled, vertically translated exponential
functions whose graphs would have the shape shown in the illustration above.
Assume that the horizontal line through (50,75) is a horizontal asymptote for these
graphs. Give image formulas for any two such functions, f1 and f2 .
B. Give an image formula for a function, f, whose graph has the shape shown in
the illustration if it is known that the graph of f passes through the point (28,90).
Give this image formula with exact coefficients and the base accurate to the
nearest hundredth. Also, store in a memory cell named “B” the best estimate
possible for the base. Enter as Y2 an expression from which a graph of f can be
generated.
C. Based on the shape of the graph of f, would tangent line slope estimates of
changes in the temperature be too small or too large?
MAC Assignment #116
CHMS
-86-
D. According to the model, accurate to the nearest thousandth of a degree, what
would be the change of temperature during the time interval [28, 29]?
E. Give on your homework paper an expression for f ′(x) with the coefficients
accurate to the nearest thousandth.
F. Calculate, accurate to the nearest thousandth of a degree the tangent line slope
estimate of the change of temperature over the time interval [28,29].
IV. This problem is about describing the form of image formulas for a family of functions
with the “generic” name “f ” which meet the conditions f ′(x)= − 0.5 ⋅ ( f (x) − 80) and
f (x) > 80 for all x:
A. Give an equivalent to the given equation which has the letter “f” on only one
side of the equal sign. Assume that no point on the graph of any of the family of
functions has second coordinate 80.
B. Create from your answer for “A” right above an equation which relates
ln( f (x) − 80) to expressions describing a large family of parallel lines.
C. Create from your answer for “B” an equation which does not contain “ln.”
D. Create an equivalent to your answer for “C” which contains no more than one
plus sign or minus sign.
E. True or False: If for some function, g, all the pairs of the form (g(x) − 80, g′(x))
lie on a line through the origin with slope ¯ 0.5, that function must be a vertically
scaled and vertically translated exponential function.
V. In each case, give an expression for h ′(x).
A. When h(x) = (x 2 + sin(x))
−
2
.
B. When h(x) =
1
.
(x + sin(x))2
2
x
C. When h(x) = ln(0.8 ) ⋅sin(2⋅ x) .
VI. In each case below there are many, many possible non-equivalent expressions for
h(x) consistent with the given expression for h ′(x). Give any two of them.
A. When h ′(x) = 3⋅ e
B. When h ′(x) =
MAC Assignment #116
3⋅ x
+ 3 ⋅sin(3 ⋅ x) + 3 ⋅ x .
2 ⋅π  −  2 ⋅π  
⋅  sin
⋅ x  .
 40  
40 
CHMS
-87-
 2 ⋅π 
C. When h ′(x) = cos
⋅ x .
 30

MAC Assignment #116
CHMS