AP Calculus
Lesson- Curve Fitting
Name:__________________________________
Date:___________________________________
Correlation and Regression
Correlation- When one variable is related to another in some way
Scatterplot- A plot on an x-y plane, where (x, y) are paired data plotted as a single point
Types of plots:
Linear Cases
Perfect +
Strong +
Moderate +
Perfect Strong Moderate _______________________________________________________________________________________
Non-Linear Cases- Only sketch the Perfect Correlations
Exponential
Quadratic
Cubic
Quartic
Natural Logarithmic
None
1
Linear Correlation Coefficient (r)
measures the strength of the linear relationship between the given variables (AKA Pearson’s Product Moment
Correlation Coefficient)
n xy   x  y 
; where n is the number of ordered pairs
r
2
2
2
2
n  x   x  n  y   y 




r 2 (in percent form) is the percent of variation in the y variable that can be explained by variation in the x
variable.
r
Type of correlation
1
Perfect +
.75  r < 1
Strong +
.50  r < .75
Moderate +
-.50 < r < .50
None
-.75 < r  -.50
Moderate -1 < r  -.75
Strong -1
Perfect -
Ex 1: Construct a scatterplot and compute the linear correlation coefficient between x and y.
x 1 2 3 5 9 11
y 3 5 7 10 16 20
Linear Regression
AKA- Least Squares Line, Line of Best Fit, Linear Regression Equation
This is the equation that best fits the data set given.
Calculator uses:
y = ax + b
where a = slope and b = y-intercept
yˆ  b0  b1 x where b1 = slope and b0 = y-intercept
Text uses:
slope  a  b1 
n xy   x  y 


n  x 2   x 
2
 y  x    x  xy
n x    x 
2
y  int ercept  b  bo 
2
2
Ex 2: Using the data from Ex 1, write the equation of the line of best fit. Plot this line on your scatterplot as
verification.
2
Using the Graphing Calculator: Note: You should only need to do steps 2-8, 10 once.
1. Enter x values in L1 and y values in L2
2. Press 2 nd y 
3. Make sure all plots are off, if not, Press 4 (Plotsoff) Enter
4. Press 2 nd y 
5. Select 1
6. Turn this plot on by highlighting On and hitting Enter
7. Press the Down Arrow once and Press Enter
8. Verify that L1 and L2 are listed.
9. Press Zoom 9 (On your graph you should see a scatterplot)
10. 2 nd 0 Press DiagnosticsOn Press Enter twice
11. Press Stat, Right Arrow, 4 (LinReg(ax + b))
12. L1 , L2 , Y1 Enter (Correlation and Regression data is now presented)
13. Zoom 9 (Line should appear on the graph verifying that equation is correct)
Making Predictions
To determine “y” given an “x” you can either:
 Sub the given “x” into the best fit model and solve for “y” or
 Enter the best fit equation into your calculator and hit: 2nd Trace “Value”. Type in the given “x” and hit
enter.
To determine “x” given a “y” you can either:
 Sub the given “y” into the best fit model and solve for “x” or
 Enter the best fit equation into your calculator. Enter the given “y” as an equation. Then find the
intersection (2nd Trace “Intersection” and follow the instructions on the screen).
Other Regressions
There are other regressions that can be determined using the graphing calculator.
Quadratic: y  ax 2  bx  c
Cubic: y  ax 3  bx 2  cx  d
Quartic: y  ax 4  bx 3  cx 2  dx  e
Natural Logarithmic: y  a  b  ln x Exponential: y  a  b x
Power Regression: y  a  x b
Sine Regression: y  A sin( Bx  C )  D
3
Examples:
1.
A study was conducted to determine if there is a relationship between the number of cookies eaten on a
daily basis and an individual male’s weight. The data are given below.
# of cookies daily
12
13
22
31
41
53
65
73
83
99
Weight in lb
120
140
170
195
205
225
250
295
310
325
a. Construct a scatterplot (complete, with appropriate labels)
b. Visually determine the best fit model.
c. Determine, using your calculator, the best regression model. Why is the one you chose the best?
Justify your answer by showing all work.
d. Determine the equation of best fit.
e. What weight would you expect a person who eats 100 cookoes per day to be?
f. How many cookies per day would you expect a person who is 150 lb to eat?
2.
The table below gives the average monthly temperature for Omaha, Nebraska, over a 30 year period.
Jan
Feb
Ma
Apr
Ma
Jun
Jul
Aug Sep
Oct
Nov Dec
21.1 26.9 38.6 51.9 62.4 72.1 76.9 74.1 65.1 53.4 39
25.1
a.
Put the data into your calculator and examine the resulting scatter plot.
b.
Use the sinreg function to regress the data and generate an equation. Plot the equation with the
scatterplot.
c.
Write the equation here:
d.
Amplitude: _______ Period: _______ Frequency: _______
4
AP Calculus
Lesson- Graphing Piecewise/Absolute Value Functions
Name:__________________________________
Date:___________________________________
Objective:
To learn review graphing piecewise and absolute value functions
Do Now:
State the domain in interval notation and determine any asymptotes for the f ( x) 
x2 1
x
______________________________________________________________________________________
Piecewise functions
This is a graph that is exactly what it sounds like. It is a graph that is basically in pieces.
Graph the following:
2 x 2 if x  0
f ( x)  
2  x if x  0
y
The procedure is to graph each part of the function separately.
x
Absolute Value Functions
y
Graph the Following
f ( x)  2 x  3
x
5
(1)
 4  x if x  2
f ( x)  
 x  2 if x  2
(2)
 3  x 2 if x  0
f ( x)  
 2x  3 if x  0
y
y
x
x
(3)
 3x  1
f ( x)  
2
 4x
if x  1
if x  2
(4)
x  1 if x  0
f (x)  
 1 if x  0
y
y
x
x
6
(5)
 1 if x  0

f ( x )   0 if x  0
  1 if x  0

(6)
  x  1 if x  0

f ( x)  
5 if x  0
 x  2 if x  0

y
y
x
x
(7)
 x  2 if x  1

f ( x )   2x if  1  x  1
  x if x  2

(8)
 x 2  1 if x  1
f (x)   3
 x  2 if x  1
y
y
x
x
7
AP Calculus
Lesson- Continuity &
Increasing/decreasing/constant functions
Objectives:


Name:__________________________________
Date:___________________________________
determine whether a function is continuous or discontinuous
determine whether a function is increasing, decreasing, or constant within an
interval
Continuous graphs:
Discontinuous graphs:
Point Discontinuity:
y
x3  8
Example: f ( x) 
x2
x
Jump Discontinuity:
y
 x  1 if x  0
Example: f ( x)  
 x  3 if x  0
x
Infinite Discontinuity:
Example: f ( x) 
y
x
x 1
x
8
AP Calculus
Lesson- Limits and Vertical Asymptotes
Name:__________________________________
Date:___________________________________
Objectives:

determine the limit as the graph approaches a vertical asymptote
Limits:
(1)
Graph f ( x ) 
(a)
lim
x 2
(b)
lim
x 2
(c)
(2)


1
x2
(c)
x
1
x 2 x  2
lim
lim
x  3
(b)
y
1
x2
Graph f ( x ) 
(a)
1
and find the following limits:
x2
lim

x  3
x
and find the following limits:
x3
y
x
x3
x
x3
x
x
x  3 x  3
lim
9
y
(3)
x2
Graph f ( x )  2
and find the following limits:
x 4
(a)
lim
x 2
(b)
(c)
(4)
lim
x  2
x2
x2  4
x2
x2  4
x2
2
x 2 x  4
lim
x 0
(b)

lim
x 0
(c)
x
lim
Graph f ( x ) 
(a)
(5)


1
and find the following limits:
x2
y
1
x2
1
x2
x
1
2
x 0 x
lim
Graph f ( x ) 
x 1
and find the following limits:
x  2x  15
y
2
x 1
x  5 x  2 x  15
(a)
lim
(b)
lim
(c)
lim
2
x 1
x  0 x  2 x  15
2
x
x 1
x  3 x  2 x  15
2
10
AP Calculus
Lesson- Determining limits graphically
Name:__________________________________
Date:___________________________________
Objectives:
(1)

determine the limit of a function graphically
2
and find the following limits:
x3
Graph f ( x ) 
(a)
lim
x  3
(b)

lim
x  3
y
2
x3
2
x3
x
(c)
(2)
2
x  3 x  3
lim
x 3
and find the following limits:
x  2x  3
Graph f ( x ) 
2
(a)
x 3
lim 2
x  1 x  2x  3
(b)
lim
x  1
x 3
x  2x  3
2
(c)
x3
x  1 x  2 x  3
(d)
lim
lim
x 3
(e)
(f)
2

lim
x  3
y
x
x 3
x  2x  3
2
x 3
x  2x  3
2
x 3
x  3 x  2x  3
lim
2
11
Given the value of c, use the graph of f to find each of the following values:
(a)
f (c )
(b) lim f ( x )
x c
(c) lim f ( x )
x c
(1)
(2)
(3)
(4)
(d) lim f ( x )
x c
12
AP Calculus
Lesson - Limits and horizontal asymptotes
Name:__________________________________
Date:___________________________________
Objectives:

determine the limit as the graph approaches a horizontal asymptote
Limits where x approaches infinity:
Limits where x approaches a constant
Find each of the following limits:
(1)
(3)
lim
5x  4
x
x  
(2)
2x
x   x  4
lim sin X
(4)
x 1
2
x   x  3
(6)
3x
2
x   x  2 x  1
x 
(5)
x2  2
lim
x   x  1
lim
lim
lim
4
13
Find the limit for each of the following algebraically:
(1)
lim x  4 
(2)
x  3
(3)
(7)
x 5
(4)
lim
x 2  2x  4
x2
x 2
(6)
lim
x2  1
lim
x 1 x  1
(8)
x  2
(5)
 
lim 2x 2
lim 2x  1
lim
x 0
sin 3x
2x
x  22  4
x
x 0
x2
x 2 x  x  6
lim
2
14
AP Calculus
Lesson- Continuity
Name:____________________________________
Date:_____________________________________
A function is continuous at a point c in its domain if and only if the limit of the function as x approaches c is
equal to the value of the function at c, or mathematically: f(x) is continuous at c iff lim f ( x)  f (c)
x c
A function is continuous at an endpoint if and only if the one sided limit at that endpoint equals the value of the
function there, or mathematically:
f(x) is continuous at left end point a iff lim f ( x )  f ( a )
xa
f(x) is continuous at right end point b iff lim f ( x)  f (b)
x b
* Remember * A limit only exists if the left hand limit is equal to the right hand limit, or mathematically:
lim f ( x)  L iff
x c
lim f ( x)  L  lim f ( x)
x c 
x c
There are various types of discontinuities:
a. Removable Discontinuities- These are discontinuities where the limit exists at a point, but it doesn’t
equal the value of the function at that point. Figures (b) and (c) are examples of removable
discontinuities at x = 0.
Let’s see how to remove a discontinuity:
f (3) 
0
 undefined, but it’s an indeterminate form
0
(x  3)(x 2  3x  2)
(x 2  3x  2) 20 10
x 3  7x  6
lim
 lim
 lim


x 3
x 3
x 3
(x  3)(x  3)
(x  3)
6
3
x2  9
So the limit exists, but does not equal the value of the function, making it a removable discontinuity.
15
So you can write an extended function, which is simply a function that fills in the hole (removable discount).
Method 1: Write a piecewise function
Method 2: Rewrite as a new, simplified, function
(if possible)
 x 3  7x  6
,x  3

g (x )   x 2  9
10

,x  3
3

g (x ) 
x 2  3x  2
x 3
b. Jump Discontinuities- These are discontinuities where the left and right hand limit at a point both exist
but are not equal to each other. Figure (d) is an example of a jump discontinuity at x = 0.
The mathematical expression for a jump discontinuity is:
lim f ( x)  L1 lim f ( x) L2 L1  L2
x c 
Questions:
x c
Is f(x) = int x continuous on [ 0 , 2 ]?
How about the interval ( 0 , 1 )?
c. Infinite Discontinuities- These are discontinuities where the left and right hand limits at a point go to
positive or negative infinity. Figures (e) and 2.22 are examples of infinite discontinuities.
The Mathematical expression for an infinite discontinuity is:
lim f ( x)   lim f ( x) 
x c 
x c
d. Oscillating Discontinuities- These are the most difficult to define as they are simply functions which
oscillate around a point too much to have a limit at that point. Figure (f) is an example of an oscillating
discontinuity.
16
Epsilon-Delta Definition of a Limit
lim f (x )  L
if, for every 
x c
 0 , then there exists a   0 such that for all x,
if 0  x  c   then f (x )  L  
What does this mean? It basically means that no matter what interval (with width ε on each side) around your
desired limit (y-value) is picked, you can pick an interval around the given x-value (with width δ on each side)
in which all x-values will have a corresponding y-value in the chosen ε-interval.
Whoa… that’s quite wordy. Let’s have a look at it graphically:
Say we’re given y = 2x – 1,
and we want to prove that
the lim(2x  1)  7 .
x 4
Given an ε-range of 2,
we can determine that
a δ-range of 1 will result
in all corresponding
y-values being in the
ε-range.
Now, since this can be shown
for any ε, the limit must exist.
Now, take the function y = x , an c-value of 1,
a limit of 1, and an ε of 0.25. If you find the
corresponding x-values to each end of the
ε-range ( 5 and 3 ), you get 25 and 9 . These
4
4
16
16
are not the same distance from 1. To determine
the δ value, you want to take the smaller value
of 7 to create a viable δ-interval so that every
16
x-value in that interval has a y-value in the ε range.
1- 9
16
=7
16
-1= 9
25
16
16
Now what happens in the case
where limits don’t exist?
If a limit doesn’t exist, then an ε
can be chosen that will cause
some of the x’s in the δ-range to
not have a corresponding
y-value in the ε-range.
17
So, again for a limit to exist, it basically means that no matter what interval around your desired limit (with
width ε on each side) is picked, you can pick an interval around the given x-value (with width δ on each side) in
which all x-values will have a corresponding y-value in the chosen ε-interval.
6+ε
6
6-ε
c
5- δ
5+ δ
5+V
Now, what if you simply want to prove a limit exists algebraically? Then you need to solve the ε –δ definition
to get the corresponding x-values for each end of the ε-range, determine how far each of those x-values is from
the c-value, and use that info to find a general δ in terms of ε.
Ex 1. Prove lim(9  x )  5
x 4
Let’s use f (x )  L   to help us find a δ for
0  x c  
(9  x )  5  
4x  
  4 x  
   4  x    4
  4  x    4
  4  x    4
Now how far are each of these x-values from the given c-value of 4?
(  4)  4  
4  (  4)  
(Upper bound – c-value)
(c-value – Lower bound)
So, letting δ = ε will allow every x-value in the δ-interval to have a corresponding y-value in the ε-range,
proving the limit always exists.
18
1
 1
x 1 x 
Ex 2. lim
Ex 3. lim x  5  2
x 9
Ex 4. lim x  1  0
2
x  1
19
Name_______________________
Date____________
The Sandwich Theorem
AP Calc
g ( x )  lim h ( x )  L
If g(x) < f(x) < h(x) for all x ≠ c in some interval about c, and lim
x c
x c
f (x )  L
then: lim
x c
f ( x ) where f(x) = x2 sin  1 
Let’s take a look at xlim
0
x 
Now let’s look at lim
x 
sin x
:
x
Practice:
20
1. Find the limit of x2cos(1/x) as x approaches 0.
2. We know that
x-1 < int x < x
Use the sandwich theorem to determine:
int x
x 
x
a. lim
int x
x -
x
b. lim
21
AP Calculus
Lesson- Logistic Functions
Name:____________________________________
Date:_____________________________________
Key fact of todays lesson: Logistic functions curve UP in the beginning and curve BACK towards the end. We
use specialized vocabulary to discuss curving. CONCAVE UP means the slope of the curve is getting higher
and higher. CONCAVE DOWN means the slope of the curve is getting lower and lower.
The parabola y = x2 is always
The curve at right is concave
while and then concave up:
concave up.
down for a
10
.
1  2e 0.25 x
The curve is concave up until approximately x = 1.386 and is
concave down from then on.
The graph at right is the logistic function y 
The traditional format of the logistic curve is:
y
L
1  ae  bx
I want you to play around with this on your graphing calculators and tell me what effect each variable has on
the curve.
 1st:Can you successfully graph the equation
y
10
? Make sure that you can get this on your
1  2e 0.25 x
screen.
 2nd:
What happens when you change the numerator, L? Try a few different numbers and then
describe the effect:
22


3rd:
What happens if you change a, the coefficient of the “e” term? Try a few numbers. Does it
matter if it’s negative? A fraction? Describe the effect.
4th:
What about the “b” value. What is the effect of changing this? Can it be positive and still
logistic? Can the absolute value of x be larger than 1?
Great. Put it back to the original: y 
10
1  2e 0.25 x
How big does x have to be before your calculator thinks that y is exactly 10? Try larger and larger x values
until you see it switch. In reality, the function never actually equals 10 but at some point it is indistinguishable.
Work backwards. What does x have to be to yield a y value of exactly 8? You can estimate this by using the
tables and continually changing the tableset values until you see the “Y” you want. You can also graph it and
use TRACE, but your answer will be pretty far off.
BEST OF ALL: Solve the equation 8 
10
for x.
1  2e 0.25 x
23
AP Calculus
Lesson- Limits and Rates of Change
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to use limits to find rates of change.
Average Rate of Change-
Slope of the secant line
What is a secant?
y

x
Find the slope of the secant line for each.
Ex 1: f ( x)  x 3  x for [1,3]
Ex 3:
f ( x)  2e 2 x for [0,  ]
Instantaneous Rate of Change-
Ex 2:
f ( x)  2 x  1  3 for [0,2]
Ex 4:
f ( x) 
x 1
for [1.5,1.5]
x2  4
Slope of the tangent line at a point
What is a tangent?
Derivation
Using limits, find the slope of the line tangent to the function y  f (x) at the point (a, f(a)).
24
Ex 5: Find the slope of the line tangent to f ( x)  x 2  4 when x = 1. What is the equation of this tangent line?
Normal to a Curve-
Equation of the line perpendicular to the tangent to a curve at a given point
Ex 6: Find the normal to f ( x)  9  3x  x 2 at x = 0.
What about Piecewise Functions?
Process:
Find the tangent of each piece at the given endpoint. Compare. If =, it is what it is. If not =,
then the function is not differentiable at that point.
Ex 7: Determine if the following piecewise function is differentiable at x = 0. If so, what is the slope of the
tangent line.
2  2 x  x 2 if x  0
f ( x)  
2 x  2 if x  0
25
AP Calculus
Lesson- Definition of a Derivative
Name:_________________________________
Date:__________________________________
For a circle:
For the graph of a function?
secant
tangent
f(x) = x(x-2)(x-4)
A tangent
-touches but does not cross?
-touches at only one point?
How will we define a tangent line to the graph of y=f(x) at the point (c, f(c))?
Step 1:
Slope of the secant line through the point (c, f(c)) and a second point of the curve:
Step 2:
Slope of the tangent line at the point (c, f(c))
26
DEFINITION: If f is defined on an open interval containing c and if the limit exists, then the line passing
through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)).
Example: Find the slope of the tangent line to the graph of y = x² at the point (3, 9).
Note: If lim
x 0
f (c  h )  f (c )
  then the tangent line at (c, f(c)) is vertical.
h
Example: Find the slope of the tangent line to the graph y  3 x at x = 0.
y3 x
27
Test for Differentiability
If a function is not continuous at a certain point, then it is not differentiable at that point.
~Check to see if the function is continuous
f ( x )  f (c )
~If so, use the alternate derivative formula f ' ( x)  lim
to determine if the function has a left- and
x c
xc
right-handed limit that are the same.
~If so, the function is differentiable. If not, it’s not.
Test for Continuity
If a function is not differentiable at a certain point, then it is not continuous at that point.
Textbook definition: “f is differentiable on the closed interval [a,b] if
If f is not continuous at x=c then f cannot be differentiable at x=c.
But f can be continuous at c and still not be differentiable at x=c.”
Example: Find the derivative with respect to x of the function y = 2/x.
28
Name________________________________
Date_________________
AP Calculus
Differentiability
A function is differentiable at a point c in its domain if and only if the function is continuous at that point and
the limit of the derivative function as x approaches c is equal to the value of the derivative of the function at c,
or mathematically
f(x) is differentiable at c iff
lim f ' ( x)  f ' (c)
x c
A function is differentiable at an endpoint if and only if the one sided limit at that endpoint equals the
value of the function there, or mathematically
f(x) is differentiable at left end point a iff
lim f ' ( x)  f ' (a)
x a
f(x) is differentiable at right end point b iff
lim f ' ( x)  f ' (b)
x b
Another way to think about differentiability is the idea of “smoothness”. If a curve is differentiable at a point, it
is smooth at that point. It’s also said that differentiability implies local linearity. This means if you were to
zoom in at a differentiable point, the more you zoom in the more the graph looks like a line.
Let’s look at the function f(x) = x cos(3x). Now if we were to zoom in at the point (2, 1.9) you will see how the
graph begins to look more and more linear. The curve is “smooth” at that point.
There are various types of points of non-differentiability:
a. Discontinuous- These are points where the function is discontinuous. If the function is
discontinuous, then it can’t be differentiable.
To justify this, just use the limit tests from continuity for the various types of discontinuities.
The rest of the types of points of non-differentiability are when f(x) is continuous, but the derivative is not.
29
b. Corner- a corner is a when the function is continuous at a point c, but
lim f ' ( x )  L1 lim f ' ( x ) L2 and L1  L2
x c 
x c
The absolute value function is an excellent example of a corner. It is continuous for all real
numbers, but at its turning point the slope from the left is not equal to the slope from the right.
 x , x  0
f ( x)  x  
 x ,x  0
lim f ( x)  0, lim f ( x)  0, lim f ( x)  0
x 0 
x 0 
x 0
 1 , x  0
f ' ( x)  
 1 ,x  0
lim f ' ( x )  1, lim f ' ( x )  1
x 0 
x 0 
To justify a corner, show that the function is continuous at c (if necessary) and that the left and
right hand limits of the derivative at that point are not equal.
c. Cusp- a cusp is when the function is continuous at c, but
lim f ' ( x)   lim f ' ( x)  
x c 
x c 
Notice that they approach different infinities.
The equation y = x2/3 has an example of a cusp. Notice the limits as you approach 0 from each side
approaches different infinities. The graph gets steeper as you zoom in.
f (x ) = x2/3
2 1
f ' ( x) = x 3
3
f ' ( x) 
2
3x
1
3
lim f ' ( x )  , lim f ' ( x )  
x 0 
x 0 
To justify a cusp, show that the function is continuous at c (if necessary) and that the left and
right hand limits of the derivatives at c equal different infinities.
30
d. Vertical Tangent- a vertical tangent is when the function is continuous at c, but
lim f ' ( x )   lim f ' ( x ) 
x c 
x c
Notice that they approach the same infinity.
The equation y = x1/3 has an example of a vertical tangent. Notice the limits as you approach 0 from
each side approaches the same infinities. The graph gets more vertically linear as you zoom in.
f (x ) = x1/3
1 2
f ' ( x) = x 3
3
f ' ( x) 
1
3x
2
3
lim f ' ( x )  , lim f ' ( x )  
x 0 
x 0 
To justify a vertical tangent, show that the function is continuous at c (if necessary) and that the
left and right hand limits of the derivative at c equal the same infinity.
31
AP Calculus
Lesson- Derivative Shortcut Rules
Name:_________________________________
Date:__________________________________
Objective:
To learn the most common derivative rules
Constant Rule:
The derivative of a constant is 0.
Power Rule:
d n
x  nx n 1
dx
 
Examples:
f ( x) 
f ( x)  x10
f ( x)  x 5
Constant Multiple Rule:
1
x2
f ( x)  3 x
d
d
[cf ( x)]  c [ f ( x)]
dx
dx
Examples:
k ( x) 
f ( x) 
f ( x)  5 x
f ( x)  4 x 3
2
3x 2
m( x ) 
2
x
f ( x)  0.47 x100
5
 3x 
2
Sum and Difference Rules: The derivative of the sum (or difference) equals the sum (or difference) of the
derivatives.
d
[ f ( x)  g ( x)]
dx
f ( x)  x 3  4 x 2  9 x  46
2
g ( x)  5 x 2  5
x
Example:
Find
32
Derivative of the Sine/Cosine Functions:
d (sin x)
 cos x
dx
Special Trig Rules:
sin x
lim
x 0
x
Why do these work? Take a look at the graphs of each…
and
d (cos x)
  sin x
dx
1  cos x
0
x 0
x
lim
Proofs of Sine/Cosine Derivative Rules:
Examples:
g ( x)   cos x
h( x)  x  sin x
j ( x) 
4
2
 4 cos x
x
More Derivative Practice:
Check each using 2nd CALC 6 with an x-value of  .
1. Find the slope of the tangent to the graph of g (t )  2  3cos t at the point ( , 1)
2. Find the equation of the tangent line to the graph of y  3(5  x)2 at the point (5, 0).
3. Determine the points, if any, at which the function has a horizontal tangent line.
1
(a) y  x3  x
(b) y  2
x
33
AP Calculus
Lesson- Position Function and Velocity
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to use derivatives as a means to solve projectile motion problems.
Velocity:
Derivative of the Position Function = Instantaneous Velocity Function
Position Function s(t) =
Example 1 A ball is dropped from a height of 100 feet. It’s height, in feet, t seconds later is given by
s(t) = -16t² + 100
What is the average velocity of the ball between 1 and 2 seconds after it is dropped?
change in dis tan ce
Note: Average velocity is
also Negative velocity means the object’s speed is in a
change in time
downward direction
What is the average velocity of the ball between 1 and 1.5 seconds after it is dropped?
What is the average velocity of the ball between 1 and 1.1 seconds after it is dropped?
What is the instantaneous velocity at exactly 1 second after it is dropped?
Note: Instantaneous velocity is found by finding the first derivative then subbing in a given value of t.
v(t) =
34
Example 2
A ball is thrown straight down from the top of a 220-foot building with an initial velocity of
-22ft/sec. What is its velocity after 3 seconds?
s(t)=
v(t)=
v(3)=.
What is its velocity after falling 108 feet?
Find the position (height above ground) after falling 108 feet.
Find the time t at which the ball is at this position.
Find the velocity at this particular t.
Example3
A ball is thrown straight down from the top of a 220-foot building with an initial velocity of
-22ft/sec. What is its velocity after 3 seconds?
35
More Examples:
1.
A bolt is dropped from a bridge under construction, falling 89 m to the valley below the bridge. (a) How
much time does it take to pass through the last 27 % of its fall? What is its speed (b) when it begins that
last 27 % of its fall and (c) just before it reaches the ground?
2.
A ball is thrown vertically downward from the top of a 36.2-m-tall building. The ball passes the top of a
window that is 11.8 m above the ground 2.00 s after being thrown. What is the speed of the ball as it
passes the top of the window?
3.
In the figure below, a stuntman drives a car (without negative lift) over the top of a hill, the cross section
of which can be approximated by a circle of radius R = 250 m. What is the greatest speed at which he
can drive without the car leaving the road at the top of the hill?
36
AP Calculus
Lesson – Marginal Analysis
Name:____________________________________
Date:_____________________________________
Objective:
To learn about cost, revenue, profit, marginal revenue, and marginal productivity.
Terms:
Productivity- ratio of output units to input units
Marginal Productivity- additional output resulting from adding one more unit of input
Cost (C(x))- may be fixed or variable.
 Fixed- not reliant upon number of units
 Variable- depends on number of units produced
Total cost- Fixed cost plus variable cost
Marginal Cost (C’(x))- additional cost for producing one more input unit
Revenue (R(x))- amount received in sale of units produced.
Marginal Revenue (R’(x))- additional amount received when producing one more input unit
Profit (P(x))- revenue minus cost: R(x) – C(x)
Marginal Profit- P’(x) = R’(x) – C’(x)
Example 1:
The cost and revenue functions for production of table saws are: C(x) = 7200 + 6x & R(x)= 20x – 0.0033x2,
where x is the number of saws produced.
a.
find the marginal cost
b.
find the marginal revenue
c.
evaluate R’(1500) and R’(4500) and interpret your answers
d.
graph R(x) and C(x) on the same set of axes. Determine the break-even points and the regions of
gain and loss.
e.
Find the profit function, P(x)
f.
Find the marginal profit.
g.
evaluate P’(1500) and P’(4500) and interpret your answers
37
Example 2: Union membership as a percentage of the labor force can be modeled by
M ( x)  0.0003x 3  0.066 x 2  4.64 x  81 , where x is the number of years after 1900 and M is membership as a
percentage of the labor force.
a.
Find the rate at which membership is changing in 1960.
b.
Find the rate at which membership is changing in 1985.
c.
Find the average rate of change between 1960 and 1985.
Example 3:
A cell phone manufacturing company finds that the cost of producing x phones is modeled by
c( x)  250  40 x and the profit from producing x phones is modeled by p( x)  0.2 x  0.25x 2 dollars.
a.
find the marginal cost function
b.
find the marginal cost when 100 phones are produced
c.
find the marginal profit function
d.
find the marginal profit when 100 phones are produced
e.
find the marginal revenue function
f.
find the marginal revenue when 100 phones are produced
38
AP Calculus
Lesson – Product and Quotient Rules
Name:____________________________________
Date:_____________________________________
The Product Rule
If f and g are two differentiable functions, then their product fg is also a differentiable function and
d
 f ( x) g ( x)  f ( x) g ( x)  g ( x) f ( x) .
dx
In words,
The derivative of a product is
the 1st function times the derivative of the 2nd plus the 2nd function times the derivative of the 1st.
Example 1
Differentiate: h( x)   6 x  5   x 3  3
Example 2
If g ( x)  x sin x , find g ( x) .
Example 3
y  sin x cos x
Find
dy
.
dx
39
The Quotient Rule
If f and g are two differentiable functions, then their quotient f/g is also a differentiable function (wherever
g ( x )  0 ) and
d  f ( x)  g ( x) f ( x)  f ( x) g ( x)
.

2
dx  g ( x) 
 g ( x)
In words,
The derivative of a quotient is
the denominator times the derivative of the numerator minus the numerator times the derivative of the
denominator, all “over’ the denominator –squared.
x2  2
, find f ( x ) .
2x  7
Example 4
If f ( x) 
Example 5
Differentiate t ( x) 
Example 6
x3  3x  2
dy
y
. Find
.
2
dx
x 1
sin x
.
x
40
x3  3x  2
dy
. Find
.
2
dx
x
Example 7
y
Example 8
Find f ( x ) if f ( x) 
Example 9
Differentiate
Example 10
5
.
4 x4
1 
2
y  x2  
.
 x x 1
 x2  x  3  2
f ( x)   2
  x  x  1
 x 1 
41
Derivatives of Trig Functions (Section 2.3)
If y  sin x,
dy
 cos x
dx
If y  cos x,
dy
  sin x
dx
If y  tan x,
dy

dx
If y  cot x,
dy

dx
If y  sec x,
dy

dx
If y  csc x,
dy

dx
Higher Order Derivatives
(Section 2.3)
The derivative of the first derivative of f is called the second derivative of f.
The derivative of the second derivative of f is called the third derivative of f.
And so on… for the nth derivative of f.
Notation:
Function
y
f(x)
First Derivative
y
dy
dx
Dx  y 
f ( x )
d
 f ( x)
dx
Second derivative
y 
d2y
dx 2
Dx 2  y 
f ( x)
d2
 f ( x) 
dx 2
Third derivative
y
d3y
dx 3
Dx3  y 
f   x 
d3
 f ( x) 
dx3
Fourth derivative
y (4)
d4y
dx 4
Dx 4  y 
f (4)  x 
d4
 f ( x) 
dx 4
 x
dn
 f ( x) 
dx n
.
.
.
n th derivative
.
.
.
y
(n)
dny
dx n
Dx  y 
n
f
( n)
42
Example 1
Find the second derivative of f ( x)  12 3 x .
Example 2
Find the second derivative of h(t )  4sin t  5cos t .
When our function is a position function , y = s(t), then
the first derivative is the
the second derivative is the
function = s’(t)
function =s”(t)
90t
where v is measured in feet per second.
4t  10
Find the vehicle’s velocity and acceleration at each of the following times:
Example 3
An auto’s velocity starting from rest is v(t ) 
(a)
1 second
(b)
5 seconds
(c)
10 seconds
43
Example 4
The graphs of f , f  and f  are shown on the same set of coordinate axes. Which is which?
Explain your reasoning.
Example 5
A car moves along an x axis through a distance of 970 m, starting at rest (at x = 0) and ending at rest (at x = 970
m). Through the first 1/4 of that distance, its acceleration is +3.25 m/s2. Through the next 3/4 of that distance,
its acceleration is -1.08 m/s2. What are (a) its travel time through the 970 m and (b) its maximum speed?
44
AP Calculus
Lesson- Understanding Motion Graphs
Name:____________________________________
Date:_____________________________________
Motion Graphs
Constant acceleration motion can be characterized by motion equations and by motion graphs. The graphs of
distance, velocity and acceleration as functions of time below were calculated for one-dimensional motion using
the motion equations in a spreadsheet. The acceleration does change, but it is constant within a given time
segment so that the constant acceleration equations can be used. For variable acceleration (i.e., continuously
changing), then calculus methods must be used to calculate the motion graphs.
A considerable amount of information about the motion can be obtained by examining the slope of the various
graphs. The slope of the graph of position as a function of time is equal to the velocity at that time, and the
slope of the graph of velocity as a function of time is equal to the acceleration.
45
The Slopes of Motion Graphs
In this example where the initial position and velocity were zero, the height of the position curve is a measure of
the area under the velocity curve. The height of the position curve will increase so long as the velocity is
constant. As the velocity becomes negative, the position curve drops as the net positive area under the velocity
curve decreases. Likewise the height of the velocity curve is a measure of the area under the acceleration curve.
Example 1:
Analyze the position, velocity, and acceleration functions for a rock that is thrown off of a 50
foot platform with an initial velocity of 10 ft/sec.
Example 2:
A particle travels along a horizontal path. Its motion is modeled by x(t )  2 x 3  8 x 2  3x  2 .
Analyze the position, velocity, and acceleration functions.
46
1.
1 2
gt  v0 t  y 0
2
s(t )  16t 2  10t  50
s ' (t )  32t  10
s" (t )  32
s (t ) 
Position:
height is increasing to vertex (t=0.3125 sec) then begins decreasing until hits the ground (t=2.108
sec)
Velocity:
decreasing velocity to the max height, then increasing velocity in negative direction until impact.
Acceleration: constant at -32 ft/sec2
--------------------------------------------------------------------------------------------------------------------------------------x(t )  2t 3  8t 2  3t  2
2.
x' (t )  6t 2  16t  3
x" (t )  12t  16
Important points for x(t)
Important points for x’(t)
y-intercept: 2
Zeroes: 0.849 and 3.488
Max @ (0.203, 2.296)
Min @ (2.464, -9.259)
y-intercept: 3
Zeroes: 0.203 and 2.464
Min @ (1.333, -7.667)
Only Position and Velocity Graphs Plotted
47
All Three Graphs Plotted
Observations:
1.
max/min x(t) = zeroes of x’(t)
This tells us that we can find the max /min by finding the first
derivative and setting =0 to solve for t.
2.
If x(t) has a (-) slope then the object is moving backward.
If x(t) has a (+) slope then the object is moving forward.
If x(t) has a (0) slope then the object is stationary (no speed). This also indicates the object’s turning pt.
3.
If x’(t) is below the horizontal axis then the object is moving backward and if x(t) is above the
horizontal axis then the object is moving forward but it’s the steepness of the slope that gives the
magnitude of the velocity (or speed).
*Note: The steeper the slope, the greater the velocity.
4.
In this case, the graph can be interpreted as a particle decelerating in a forward direction for the most
minute of instances before it hits its first turning point. Then it moves backwards with increasing (to the
max) then decreasing velocity until it comes to a stop at which point it starts to move forward with
increasing velocity.
Particularly: The particle starts off (time=0) at 3 units/time moving forward and decreases its speed to 0
units/time after time= 0.203. At this point it begins going backward picking up speed until it hits its
maximum speed after time=1.333. At this point, the particle’s maximum speed is 7.667 units/time. It
then begins to slow down until its speed is 0 units/time. This occurs at time=2.464. At that time, the
particle turns again and begins accelerating without bound.
5.
To get a sense of the horizontal movement, turn your page sideways so that the original vertical axis is at
the top of the page and look at the path created by x(t).
6.
The acceleration graph tells us that a(t) or v”(t) is continuously changing over time. The zero of the
acceleration graph is the inflection point which helps us to determine when there is a change in the
direction of the velocity function. In this instance, it tells us that the direction of the velocity is changing
at t = 1.333. That indicates also, the moment when acceleration is equal to 0.
48
AP Calculus
Lesson- Chain Rule
Name:____________________________________
Date:_____________________________________
Chain Rule
Let y be a differentiable function of u, y = f(u),
and u be a differentiable function of x, u = g(x).
d  f  g ( x)  d
d
Then,
  f (u )   g ( x) .
dx
du
dx
Think: (derivative of the outside) times (derivative of the inside)
dy dy du


In other symbols:
dx du dx
Ok so that didn’t make much sense…
Here are three explanations at varying levels of abstraction. The book is also excellent.
1) The chain rule is used with composed functions. The goal is to end up with a function (or the derivative of a
function) that uses one function in terms of the other. The derivative of the composed function is equal to the
product of the derivative of each function.
2) Let’s try to illustrate this: Assume two functions, g(f) and f(t). g(f) uses f as the input and outputs a value,
g, for any given f. Likewise, f(t) uses t as the input and outputs f for any given value of t. This is a perfect
chain: in goes t, out comes f, which goes into g(f). The end result is an output, g, from an input of t. The
derivative of this chain is the derivative of the outside times the derivative of the inside.
3) We’ll use g(f) = f2 and f(t) = 3t. We can rewrite g(f) by using the definition of f. The new function g(f(t)) is
[f(t)]2, or [3t]2, which simplifies to 9t2. The derivative of this (using the rule we already have) is 2∙9∙t, or 18t.
We can also find this using the chain rule:
 The derivative of [f(t)]2 is 2[f(t)]
 The derivative of 3t is 3
 The product of the derivatives is 2[f(t)] ∙ 3, or 6[f(t)]
 Substituting for f(t), this gives 6 ∙ 3t, which is 18t.
Directions
Try the three problems below. They can all be solved by composing first and taking the derivative of the
composed function but don’t do it! Try the chain rule, then check yourselves by composing and taking the
derivative.
1. Let f(h) = 2h2 and h(x) = 3x
Find df/dx using both the chain rule and
taking the derivative of the composition.
Check: the r-o-c at x = 2 is 72
2. Let g(k) = k2 – 2k and k(x) = 2x
Find dg/dx using both methods.
Check: x = 3 is the first integer value at
which the rate-of-change exceeds 20.
3. Let’s examine the speed of a cyclist downhill. Simplifying wildly, the bike “falls” down a
50
hill at S(t) = 9.8t2. Wind resistance increases as a function of speed, given by R ( s ) 
.
s
Find the rate-of-change of resistance as a function of time. Use both methods. At what time
does resistance equal 15 mph?
49
More Practice:
dy du
dy
, , and
. This is the equivalent to asking for the
du dx
dx
derivative of each one separately and then of the composed function.
1. Given: y  2  u and u  x 3  1 , find
80000
and that
p
price p is a function of time given by p = 1.6t + 9, where t is in days. To summarize, this says that the price
of an object increases with time and that the demand for that object is inversely proportional to its price.
From this, it is a short jump to ask how the demand for the object varies with time.
i.
Find a function, f(x), that expresses the demand for an item in terms of time, t, in days.
ii.
Find the derivative, dD/dt. This will give you the rate at which demand is changing for
any given time, t, in days.
iii.
How quickly is demand changing after 100 days.
iv.
Is this rate of change rising or falling?
2. Marginal demand. Suppose that the demand function for a product is given by D( p) 
50
Find the first derivative of each:
1.
y  sin  x 4  5 
1
2.
y  sin 2 x  (sin x)2
5.
y  cos3 4t 2
8.
g ( x)  3x  5 tan(2 x)7
 
4.
y
7.
y
10.
Find the derivative of the function f ( x) 
x
2
 2x
x

3
x4  4
3.
y  x2  3
6.
f ( x)  x  3x  9 
9.
g ( x) 
3
cos x
csc x
1
 2 
 cos x at the point  ,  .
x
2 
51
11.

Find an equation of the tangent line to the graph of f ( x)  9  x 2

2
3
at the point (1,4).
Graph the function and its tangent.
x
has a horizontal tangent.
2x 1
12.
Determine the points at which the graph of f ( x) 
13.
The sum of $8000 is deposited into an annuity paying r% interest compounded quarterly. After four
years, the value of the annuity is V (r )  8000(1  0.0025r )16 . Evaluate V(5) and V’(5), and interpret
your answers.
52
AP Calculus
WKST- Chain Rule
Name:____________________________________
Date:_____________________________________
Find the derivative of each. Use the chain rule even if it could be done another way. In fact, if it can be done
another way do so and use that to check your work!
1.
f ( x)  2 x  1
2.
g ( x)  2 sin( x 2  4)
3.
y  4( x  2) 4
4.
f ( x) 
5.
g ( x) 
6.
 1  2
y
 x  2x
 x 1
7.
y  4 sec 2 (sin x)
8.
 f  g ' ( x) if
x 1
x2 1
2
3
3x  1


3
f ( x )  3 x  2 & g ( x)  x 3  8
9.
y
sec( x  2)
csc( 2 x)
10.
f ( x)  ax  h   k ;
2
where all but x are constants.
53
AP Calculus
Multiple Choice Chain Rule Practice
Name:____________________________________
Date:_____________________________________
#1.
#2.
#3.
#4.
54
#5.
#6.
#7.
#8.
55
#9.
#10.
56
EXPONENTIAL AND LOG FUNCTIONS
57
PROPERTIES OF EXPONENTIAL FUNCTIONS AND LOGS
58
AP Calculus
Review- Exponential and Logarithmic Functions part 1
Name:____________________________________
Date:_____________________________________
Write each expression in terms of simpler logarithmic forms:
4
s5
(2) log b 7
u
(1) log b x 5 y
(5) log3 42
(3) log b
1
c8
(4) logb
m 5n 3
p
Given loga n, evaluate each logarithm to four decimal places:
(7) log6 0.00098
(6) log 1 5
2
Solve each equation and round answers to four decimal places where necessary:
(8) log2 x  3
(9) log5 4  log5 x  log5 36
(10) 1000  75e0.5 x
(12) log 7
1
x
49
(11) log 6 x  2
(13) log x 4 
1
2
(14) 10 x  27.5
(15) log x  log 5  log 2  log( x  3)
(16) log x  log 2  1
(17) log4 x  3
(18) log9 (5  x )  3 log9 2
(19) log 20  log x  1
(20) 2  1.002 4 x
(21) e 25 x  1.25
(22) log( x  10)  log( x  5)  2
(23) log 6 216 
1
log 6 36  log 6 x
2
59
AP Calculus
Review- Exponential and Logarithmic Functions part 2
Name:____________________________________
Date:_____________________________________
SHOW ALL WORK:
(1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22
million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he
need for this investment?
(2) The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially:
a.
Express the number of guppies as a function of the time t.
b.
Use your answer from part (a) to find how many guppies are present after 1 week?
c.
Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies?
(3) The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by
i
log
 0.00235 x . What is the intensity, to the nearest tenth, at a depth of 40 feet?
12
(4) Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula
i
D  10 log
, where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per
i0
square meter is a standardized sound level. Use this information and formula to find the intensity of the
sound at the concert.
60
(5) How many years, to the nearest year, will it take the world population to double if it grows continuously at
an annual rate of 2%.
(6) Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If
you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two
banks?
(7) Determine how much time, to the nearest year, is required for an investment to double in value if interest is
earned at the rate of 5.75% compounded quarterly.
61
AP Calculus
Lesson- Derivatives of Exponentials and Logs
Name:____________________________________
Date:_____________________________________
Objective:
to learn how to find the derivative of an exponential or logarithmic function
Finding the Derivative of f ( x)  e x
Using limit process:
RULE:
Examples:
Find f’(x) for each:
1.
f ( x)  3e
4.
f ( x)  x 3 e x
7.
x
f ( x)  e 3 x  2
3.
ex
f ( x) 
x
f ( x)  (e x  4)3
6.
f ( x )  (e x ) 3
f ( x)  e x
9.
f ( x)  x 7 e
2.
f ( x)  e  3x
5.
8.
x
6
6
62
Finding the Derivative of f ( x)  ln x
Derivation:
Examples:
1.
f(x) = 4 ln x
4.
7.
f ( x)  ln( 5 x)
2.
f ( x)  x 2  ln x
3.
f ( x)  (8x  3 ln x) 3
5.
f ( x)  3x 2 ln( 4 x)
6.
f ( x)  (ln x) 4
For what values of x does f ( x)  x 2 ln( x) have a horizontal tangent line?
Putting it all together….
8.
Find f’(x) for f ( x)  63 x5
9.
Find f’(x) for f ( x)  log 8 (3x  2)
63
Application Problems:
1.
The percent of information retained t months after being tested on that information is given by:
f(t) = 82 – 14 ln (t + 1). Evaluate f(2) and f’(2) and interpret your answers.
2.
The cost and revenue functions for producing x units of a certain product are C(x) = 6000 + 15x and
R(x) = 1811 ln(1.2x + 1).
a.
Find the marginal cost if 200 units are produced.
b.
Find the marginal revenue if 200 units are produced.
c.
Find the marginal profit if 200 units are produced.
d.
Graph the two functions and use your graphing calculator to find the break-even points.
e.
To make a profit, how many units should be produced.
64
AP Calculus
Lesson- Implicit Differentiation
Name:____________________________________
Date:_____________________________________
So far we have been working with functions in explicit form (equations solved for y in terms of x). Now we
will learn how to work with implicit forms of equations (equations not solved for y or not easily solved for y).
Explicit Form of y = function of x;
y written explicitly in terms of x
y
Implicit Form of y = function of x;
an equation that relates y to x but where y cannot
necessarily be isolated
1
x
xy  1
Implicit Differentiation
Step 1
Step 2
Step 3
Step 4
Note:
Differentiate both sides of the equation with respect to x.
dy
Collect all terms which contain
on one side of the equation and everything else on the other side.
dx
dy
Factor
out of all the terms on the one side.
dx
dy
Solve for
dx
dy
will be in terms of x and y.
dx
Example 1
Graph x 2  y 2  16 and find
dy
implicitly.
dx
Find all points where the graph has a horizontal tangent.
dy
Using implicit form of
:
dx
Using explicit form of
dy
:
dx
65
Find all points where the graph has a vertical tangent.
Using implicit form of
dy
:
dx
Using explicit form of
dy
:
dx
Example 2
a.
Find the first derivative of x 2 y  y 2 x  2 by implicit differentiation.
b.
Find f’(2)
66
Example 3
a.
x2 y 2

1
Graph the hyperbola:
6 8
b.
Find an equation of the tangent line to the graph of the hyperbola
Example 4
Find an equation of the tangent line to the astroid x
Example 5
Differentiate implicitly:
2sin x cos y  1
a.
b.
2
3
y
x2 y 2

 1 at the point (3,-2).
6 8
2
3
 5 at the point (8,1).
cot y  x  y
67
AP Calculus
Lesson- Related Rates
Name:____________________________________
Date:_____________________________________
Objectives:
Identify a mathematical relationship between quantities that are each changing.
Use one or more rates to determine another rate.
Process:
Step 1: Draw a diagram.
Step 2: Determine which quantities and rates are given, and which to be found.
Step 3: Identify the primary function to use. (Often this is a formula from geometry.)
Step 4: Differentiate with respect to the independent variable
Step 5: Write a related rates equation
Step 6: Substitute known quantities and solve for desired rate.
[NOTE: Do not substitute known quantities before this last step!]
Type 1: Explicit Function of One Variable
Examples
1.
Air is being blown into a sphere at the rate of 6 cubic inches per minute. How fast is the radius
changing when the radius of the sphere is 2 inches?
2.
The edge of a cube is increasing at a rate of 2 inches per minute. At the instant the edge is 3 inches, how
fast is the volume increasing?
68
3.
A point moves along the curve y   x  3 such that its x-coordinate is increasing at 4 units per second.
(a)
At the moment x = 1, how fast is the y-coordinate changing? Interpret your answer based on the
shape of the graph and the location of the point.
2
(b)
At the moment x = 1, how fast is the point’s distance from the origin changing?
Type 2 - Implicit Function of One Variable
4.
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at
a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is
6 ft from the wall?
Type 3: Functions of Two Variables—2 rates given
5.
The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at
a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the
area is 100 cm2?
Type 4: Functions of Two Variables—1 rate given--secondary equation needed
6.
A water tank has the shape of an inverted circular cone with base radius 2m and height 4 m. If water is
being pumped into the tank at a rate of 2m3/min, find the rate at which the water level is rising when the
water is 3 m deep.
69
Mixed Problem Set- Related Rates
1.
A conical tank is being filled with water. The tank has height 4 ft and radius 3 ft. If water is being
pumped in at a constant rate of 2 cubic inches per minute, find the rate at which the height of the cone
changes when the height is 26 inches. Note the difference in units.
2.
A searchlight is positioned 10 meters from a sidewalk. A person is walking along the sidewalk at a
constant speed of 2 meters per second. The searchlight rotates so that it shines on the person. Find the
rate at which the searchlight rotates when the person is 25 meters from the searchlight.
3.
A person 5 feet tall is walking toward an18 foot pole. A light is positioned at the top of the pole. Find
the rate at which the length of the person’s shadow is changing when the person is 30 feet from the pole
and walking at a constant speed of 6 feet per second.
4.
The length of a rectangle increases by 3 feet per minute while the width decreases by 2 feet per minute.
When the length is 15 feet and the width is 40 feet, what is the rate at which the following changes:
a. area
b. perimeter
c. diagonal
70
5.
6.
7.
1
C 2 h where C is the circumference of the tree in meters at
12
ground level and h is the height of the tree in meters. Both C and h are functions of time t in years.
The volume of a tree is given by V 
dV
. What does it represent in practical terms?
dt
a.
Find a formula for
b.
Suppose the circumference grows at a rate of 0.2 meters/year and the height grows at a rate of 4
meters/year. How fast is the volume of the tree growing when the circumference is 5 meters and
the height is 22 meters?
a.
When the radius of a spherical balloon is 10 cm, how fast is the volume of the balloon changing
with respect to change in its radius?
b.
If the radius of the balloon is increasing by 0.5 cm/sec, at what rate is the air being blown into
the balloon when the radius is 6 cm?
c.
When the volume of the balloon is 50 cubic cm, at what rate is the radius of the balloon
changing?
When hyperventilating, a person breathes in and out very rapidly. A spirogram is a machine that
draws a graph of the volume of air in a person’s lungs as a function of time. During hyperventilation, the
person’s spirogram trace might be represented by V  3  0.05 cos200t  where V is the volume of air
in liters in the lungs at time t minutes.
a.
Sketch a graph of one period of this function.
b.
What is the rate of flow of air in liters/minute? Sketch a graph of this function.
c.
Mark the following on each of the graphs above.
i) the interval when the person is breathing in
ii) the interval when the person is breathing out
iii) the time when the rate of flow of air is a maximum when the person is breathing in
71
AP Calculus
Lesson- Extrema and Critical Number
Name:____________________________________
Date:_____________________________________
Extreme Value Theorem
If f is continuous on a closed interval I = [a,b] then f has both a minimum and a maximum value on I.
f has a relative maximum at c or f(c) is a relative maximum of f if there is some open interval I containing c
on which f(c) is the maximum.
f has a relative minimum at c or f(c) is a relative minimum of f if there is some open interval I containing c on
which f(c) is the minimum.
Relative Extrema
Let I be any interval (closed or open) containing the x-value c.
The extreme values, or extrema, of a function f on I are defined by:
The absolute maximum of f on I is f(c) if f (c)  f ( x) for all x in I.
The absolute minimum of f on I is f(c) if f (c)  f ( x) for all x in I.
NOTE: A function need not have a maximum value or a minimum value on a given interval I.
Process
 Find the first derivative and plug the x-value of the coordinate of the relative min/max & solve for f’(x)
Or

Find the left- and right-hand side derivatives using lim
x c
f ( x )  f (c )
xc
Examples:
For each: determine any relative extrema and then determine the derivative at each relative extremum.
1.
f ( x)  x on the interval [-2,2].
2.
f ( x) 
1
on the interval [-2, 2].
x
3.
f ( x)  x 2 on the interval (-1, 2).
72
4.
f ( x)  x 3  3 x 2
5.
f ( x)  x
6.
f ( x)  sin x
7.
f ( x) 
2
3
8
x 4
2
Critical Values
A critical value is a number c in the domain of f for which f (c)  0 or f (c ) does not exist.
*Relative Extrema Occur Only at Critical Numbers
* One result of this theorem is that, on a closed interval, absolute extrema must occur at local extrema or at the
endpoints.
Long story short: Find the derivative, set = 0, solve for x, plug x back into original function.
Examples:
Find the critical value(s)
8.
f ( x)  x
10.
f ( x) 
2
3
8
x 4
2
9.
f ( x)  sin x
11.
y  2x 3
73
Method for Finding Absolute Extrema on a Closed Interval [a,b]
(1)
(2)
(3)
(4)
Find the critical numbers of f in (a,b).
Evaluate f at each critical number in (a,b).
Evaluate f(a) and f(b).
Compare: the least of these y-values is the minimum; the greatest is the maximum.
2x  5
on the interval [0, 5].
3
12.
Find the maximum and minimum values of f ( x) 
13.
Find the maximum and minimum values of f ( x)  x 2  2 x  4 on the interval [-2, 1].
14.
Find the maximum and minimum values of f ( x)  x3  12 x on the interval [0, 4].
15.
Find the maximum and minimum values of g ( x)  3 x on the interval [-1, 1].
16.
   
,
Find the maximum and minimum values of h( x)  sec x on the interval 
.
 6 3 
74
AP Calculus
Lesson- First Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
Learn about Increasing and Decreasing Functions & the First Derivative Test for Extrema
Graphically
f increasing
f decreasing
Numerically
When x’s go up, y’s go up.
When x’s go up, y’s go down
Sign of the Derivative
f ( x )  0
f ( x )  0
First Derivative Test
If f  changes from – to + at c, f has a relative minimum at (c, f(c)).
If f  changes from + to – at c, f has a relative maximum at (c, f(c)).
If f  doesn’t change sign at c, then f(c) is not a relative extremum for f.
Where “c” is a critical value.
Process:
Step 1 Find all critical numbers of f in the given interval; break the interval into smaller intervals using these
critical values, points of discontinuity, and the endpoints.
Step 2 Create a sign chart, by picking an x-value in each interval and finding the sign of f  there.
Step 3 For each sub-interval state whether f is increasing or decreasing there.
Example
1.
Tell where f ( x)  x3  2 is increasing, decreasing, and identify any relative extrema.
x3
is increasing, decreasing, and identify any relative extrema.
x2
2.
Tell where f ( x) 
3.
Use the graph of f  to tell where f is increasing, decreasing, and identify any relative extrema.
75
AP Calculus
Lesson- Concavity, Inflection, 2nd Derivative Test
Name:____________________________________
Date:_____________________________________
Objective:
To use derivatives to determine concavity and points of inflection
Concavity-
Function has a hill (concave down) or a valley (concave up)
Points of Inflection- Location at which a function goes from being concave down to concave up (or vice versa)
Concave Up
Concave Down
Inflection
Second Derivative Test for Concavity
Process:
1. Find the 1st derivative
2. Set = 0 and solve
3. Find the 2nd derivative
4. Set = 0 and solve (result could be a point of inflection)
5. Create sign chart
Guided Example
1 3 1 2
x  x
3
2
2
f ' ( x)  x  x
f ( x) 
f " ( x)  2 x  1
0  2x  1
x  1 / 2
0x x
2
x  0,1
Intervals
Test Values
Sign of f’(x)
Sign of f”(x)
Conclusion
 ,1
-2
+
Increasing
Concave
down
-1
(-1, -1/2)
-1
-3/4
0
REL Decreasing
MAX Concave
down
-1/2
-1/2
X
0
PT of
INF
(-1/2, 0)
-1/4
+
Decreasing
Concave up
0
0
0
REL
MIN
0, 
2
+
+
Increasing
Concave up
Since the function went from concave down to concave up at x = -1/2, that must be the inflection point.
76
Examples
1.
Find the intervals where f ( x ) 
6
is concave upward and concave downward.
x 3
2
Determine relative extrema, concavity, points of inflection.
2.
g ( x)  x 3  2 .
3.
  
h( x)  2 x  tan x ;   ,  .
 2 2
Let f be a function with f (c)  0 and the second derivative f  exists on an open interval containing c.
Then
if f”(c)>0, then f has a relative minimum at (c, f(c)),
if f”(c)<0, then f has a relative maximum at (c, f(c)),
if f”(c)=0, then test fails; must use the 1st Derivative Test for Extrema (could be max, min, or neither)
Find all the relative extrema
x
.
x 1
4.
f ( x)  x3  9 x 2  27 x.
5.
f ( x) 
6.
f ( x)  x 2  1.
7.
f ( x)  sin x  cos x.
77
AP Calculus
Lesson- Curve Sketching
Name:____________________________________
Date:_____________________________________
Objective:
To learn how to sketch a graph using calculus and without using a graphing calculator
Analyzing and Sketching the Graph of a Function
1. Decide domain and (where possible) range.
2. Determine x- and y-intercepts.
3. Note symmetry where applicable. (even function: f(-x) = f(x) -- symmetric across y-axis
odd function: f(-x) = -f(x) – symmetric across origin)
4. Find any points of discontinuity.
5. Find any vertical and horizontal asymptotes.
6. Find x-values where f and f are 0 or non-existent. Use these to get relative extrema and pts. of inflection.
7. Find end behaviors that are infinite.
Example 1 Polynomial Function
f(x) = x4 - 12x3 + 48x2 - 64x
Example 2
Rational Function
2
2( x  9)
f ( x)  2
x 4


78
AP Calculus
Lesson- Optimization
Name:____________________________________
Date:_____________________________________
Optimization Problems—Applied Minimum and Maximum Problems
Method for Solving:
Step 1 Define variables needed to describe the quantity to be optimized.
Step 2 Write a primary equation for the quantity to be optimized.
Step 3 Use other info given in the problem to write a secondary equation that relates the variables used in the
primary equation. Solve this equation to get one variable in terms of the other. Use this equality to rewrite the primary equation in terms of one variable.
Step 4 Find a feasible domain for the function you wrote in Step 3, i.e., upper and lower bounds on the variable
that make sense for your problem.
Step 5 Find absolute or relative max or min of your function on the feasible domain.
Step 6 Write your answer in an English sentence.
Example 1
Sr. Karlien wants to create a rectangular garden. She has available 100 feet of fencing to make
the border. What are the dimensions of the plot with the largest area that can be enclosed with
this fence?
Example 2
Find 2 positive numbers whose product is 192 and whose sum is a minimum.
Example 3
A manufacturer wants to design an open box having a square base and a surface area of 108
square inches. What dimensions will produce a box of maximum volume?
Example 4
A box with a square base and open top must have a volume of 32,000 cubic cm.
Find the dimensions of the box that minimize the amount of material used
79
Mixed Problem Set- Optimization
1.
A box with an open top and a square base is to have a surface area of 108 square inches. Determine the
dimensions that will maximize the volume of the box.
2.
Two posts, one 12 feet high and the other 28 feet high, are 30 feet apart. They are anchored by two wires
running from the top of each pole to a single stake in the ground at a point between the two posts. Where
should the stake be placed so that the minimum amount of wire is used?
3.
A bus stop shelter has two square sides, a back, and a roof. The volume is 256 cubic feet. What
dimension will allow for the least amount of material to be used?
4.
Scrumptious Soup Company makes a soup can with a volume of 250 cm3. What dimensions will allow
for the minimum amount of metal to produce the can?
5.
A jumbo-size can of baked beans has a volume of 600 cm3. What dimensions will allow for the
minimum amount of metal to produce the can?
6.
The surface area of a can of chunked chicken requires 60 square inches of material. What dimensions
allow for maximum volume?
7.
A large can of tuna requires a surface area of 100 square inches. What dimensions provide the maximum
volume?
8.
A wire of length 12 inches can be bent into a circle, a square, or cut to make both a circle and a
square. How much wire should be used for the circle if the total area enclosed by the figure(s) is
to be a minimum? A maximum?
|
circle
9.
square
A window consisting of a rectangle topped by a semicircle is to have an outer perimeter P. Find the
radius of the semicircle if the area of the window is to be a maximum.
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10.
A rectangular field as shown is to be bounded by a fence. Find the dimensions of the field with
maximum area that can be enclosed with 1000 feet of fencing. You can assume that fencing is not
needed along the river and building.
11.
A company manufactures cylindrical barrels to store nuclear waste. The top and bottom of the barrels
are to be made with material that costs $10 per square foot and the rest is made with material that costs
$8 per square foot. If each barrel is to hold 5 cubic feet, find the dimensions of the barrel that will
minimize the total cost.
12.
The operating cost of a truck is 12 
13.
A furniture business rents chairs for conferences. A contract is drawn to rent and deliver up to 400 chairs
for a particular meeting. The exact number would be determined by the customer later. The price will be
$90 per chair up to 300 chairs. If the order goes above 300 chairs, the price would be reduced by $0.25
per chair for every additional chair ordered above 300. This reduced price would be applied to the entire
order. Determine the largest and smallest revenues this business can make under this contract.
14.
The speed of traffic through the Lincoln Tunnel depends on the density of the traffic. Let S be the speed
in miles per hour and D be the density in vehicles per mile. The relationship between S and D is
D
approximately S  42 
for D  100 . Find the density that will maximize the hourly flow.
3
15.
A commercial cattle company currently allows 20 steer per acre of grazing land. On average a steer
weighs 2000 pounds at the market. Estimates by the Department of Agriculture indicate that the average
weight per steer will be reduced by 50 pounds for each additional steer added per acre of grazing land.
How many steer per acre should be allowed in order to optimize the total market weight of the cattle?
x
cents per mile when the truck travels x miles per hour. If the
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driver earns $6 per hour, what is the most economical speed to operate the truck on a 400 mile turnpike?
Due to construction, the truck can only travel between 35 and 60 miles per hour.
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16.
Catching Rainwater. A 1125-ft3 open-top rectangular tank with a square base x ft on a side and
y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with
the tank involve not only the material from which the tank is made but also an excavation charge
proportional to the product xy. If the total cost is c = 5(x2 + 4xy) + 10xy, what values of x and y will
minimize it?
17.
Designing a Poster. You are designing a rectangular poster to contain 50 in2 of printing with a 4-in.
margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the
amount of paper used?
18.
Vertical Motion. The height of an object moving vertically is given by s = -16t2 + 96t + 112,
with s in ft and t in sec. Find (a) the object's velocity when t = 0, (b) its maximum height and when it
occurs, and (c) its velocity when s = 0.
19.
Finding an Angle. Two sides of a triangle have lengths a and b, and the angle between them is  .
What value of  will maximize the triangle's area? (Hint: A = ½ ab sin  .)
20.
Designing a Can.
What are the dimensions of the lightest open-top right circular cylindrical can that
will hold a volume of 1000 cm3?
21.
Designing a Can.
You are designing a 1000-cm3 right circular cylindrical can whose manufacture
will take waste into account. There is no waste in cutting the aluminum for the side, but the top and
bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of
aluminum used up by the can will therefore be A = 8r2 + 27 rh the ratio of h to r for the most
economical can is?
22.
Designing a Box with Lid A piece of cardboard measures 10- by 15-in. Two equal squares are
removed from the comers of a 10-in. side as shown in the figure. Two equal rectangles are removed
from the other corners so that the tabs can be folded to form a rectangular box with lid.
(a)
Write a formula V(x) for the volume of the box.
(b)
Find the domain of V for the problem situation and graph V over this domain.
(c)
Use a graphical method to find the maximum volume and the value of x that gives it.
(d)
Confirm your result in (c) analytically.
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23.
Designing a Suitcase.
A 24- by 36-in. sheet of cardboard is folded in half to form a 24- by 18-in.
rectangle as shown in the figure. Then four congruent squares of side length x are cut from the corners
of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and
a lid.
(a)
Write a formula V(x) for the volume of the box.
(b)
Find the domain of V for the problem situation and graph V over this domain.
(c)
Use a graphical method to find the maximum volume and the value of x that gives it.
(d)
Confirm your result in (c) analytically.
(e)
Find a value of x that yields a volume of 1120 in3.
24.
Quickest Route.
Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a
straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where
should she land her boat to reach the village in the least amount of time?
25.
Inscribing Rectangles.
A rectangle is to be inscribed under the arch of the curve y = 4cos(0.5x)
from x = -  to x =  . What are the dimensions of the rectangle with largest area, and what is the largest
area?
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AP Calculus
Lesson- Business Applications
Name:____________________________________
Date:_____________________________________
Objective-
To learn about maximizing yield and revenue while minimizing costs of inventory control.
Cobb-Douglas productivity models
Recall:
Revenue (R(x))- amount received in sale of units produced.
Maximizing Revenue involves determining the price at which a quantity should be sold in order to
obtain maximum revenue.
Profit (P(x))- revenue minus cost: R(x) – C(x); where R is revenue and C is cost
Maximizing Profit involves determining the price at which a quantity should be sold in order to obtain
maximum profit.
Example 1:
Rosie’s Discount Mart sells paperback books. At price $p, Rosie can sell q( p)  5 p 2  55 p  60 books. What
price would give Rosie the greatest revenue?
Example 2:
Mark’s restaurant can produce one chicken sandwich for $2. The sandwiches sell for $5 each and at this price,
his customers buy 1200 sandwiches each month. Because of rising costs from suppliers, the restaurant is
planning on raising the price of the sandwich. Based on the results of the previous price increases, Mark
estimates that he will sell 120 fewer sandwiches each month for every $1 he increases the price. At what price
should the sandwiches be sold to maximize Mark’s profit? What is the maximum profit?
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Maximizing yield involves optimization situations in which an increase in one variable causes a decrease in
another related variable.
Example 3:
Taylor’s Orchard has always planted 40 trees per acre, with a yield of 300 apples per tree. For each additional
tree planted per acre, the yield drops by 5 apples per tree. How many trees should be planted per acre for
maximum yield?
Inventory Control:
deciding the most appropriate time to produce x quantity of an item.
Possibilities:
1.
Produce all of a given item at the beginning of a year.
Advantages: All items are on hand for immediate sale. Cheaper to produce at bulk rate.
Disadvantage: Need to store unsold items
2.
Produce items throughout the year as needed.
Advantage: No cost to store items
Disadvantage: Will cost more to produce in smaller quantities
Example:
MAC Boats anticipates a demand for 12,000 fishing boats over the next year. The start-up costs
of each production run are $5000, and it costs the company $40 to store each boat during the year. How many
boats should be made during each production run to minimize total costs? How many production runs should
there be?
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Cobb-Douglas Productivity Model
The productivity of a plant or factory is given by: q  Kx a y1a where q is the number of units produced, x
is the number of employees, and y is the operating budget or capital. K and a are constants that are determined
by each individual factory or plant, with 0 < a < 1.
Example:
Brian’s Beach Shop manufactures surfboards. Daily operating costs are $80 per employee and
$25 per machine. The number of surfboards produced each day is given by q  4.5x 0.8 y 0.2 , where x is the
number of employees and y is the number of machines. If Brian wants to produce 90 surfboards each day at
minimum cost, how many employees and how many machines should he use?
Point of Diminishing Returns
the point at which the rate of growth of the profit function begins to decline.
The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars) on
1 3
s  6 s 2  400 . Find the amount of money the company should spend on advertising in
advertising is P ( s ) 
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order to yield the maximum profit. Find the point of diminishing returns.
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Mixed Problem Set- Business Applications
1.
Marge is planning a casino bus trip. If 100 people sign up, the cost is $300 per person. For each
additional person above 100, the cost per person is reduced by $2 per person. To maximize Marge's
revenue, how many people should go on the trip? What is the cost per person?
2.
A charter dinner cruise boat holds 50 people. The company will charter the boat for 35 or more people.
If 40 people are on board, the cost per person is $150. For each additional person, the cost per person is
reduced by $3. How large a group should be on the cruise to maximize the revenue? What is the cost per
person?
3.
A peach orchard has an average yield of 90 bushels per tree if there are 20 trees per acre. For each
additional tree per acre, the yield decreases by 3 bushels per tree. How many trees should the orchard
plant per acre to maximize the yield? What is the total yield?
4.
An orange grove plants 25 trees per acre and gets a yield of 116 bushels of oranges per tree. For each
additional tree planted per acre, the yield decreases by 4 bushels per tree. How many trees should be
planted per acre to maximize the yield? What is the total yield?
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5.
Matt's Top 40 rents movies. If the rental fee is $4 each, Matt knows he can rent 100 movies per week.
For each additional $1 increase in the rental fee, Matt loses 10 rentals per week. What rental fee should
Matt charge for a movie to maximize his revenue? If each movie costs Matt $2, what should his rental
price be to maximize profit?
6.
Barb's Babysitting charges $8 per hour and, at that rate, averages 20 jobs each week. For each additional
$1 charge per hour, the number of jobs per week declines by two. What should Barb charge per hour to
maximize revenue? If Barb spends $2 per job on supplies, what should she charge per hour to
maximize her profits?
7.
When Jerry's Jalopies charges $20 to do an oil change, there are 80 customers per month. For each
additional $1 charge, the number of customers per month drops by four. If it costs Jerry $5 per customer
for the supplies, what should he charge for an oil change to maximize profits? How many customers
will there be each month?
8.
Missy's Tutoring charges $35 per hour for a tutoring session and has 60 clients each week. For each
additional $1 charge, there are two fewer clients each week. It costs $12 per client for supplies. What
should Missy charge per hour to maximize profits? How many clients will there be each week?
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9.
A bicycle plant assembles 2000 bicycles per month. Each production run costs $1200, and it costs $20
to store a bicycle for a month. How many production runs should the plant use to minimize inventory
costs? How many bicycles are assembled in each production run?
10.
A soda bottling company bottles 20,000 cases of lime soda each year. Each production run costs $1400,
plus a storage cost of $18 per case. How many production runs should the company use to minimize
inventory costs? How many cases are bottled in each production run?
11.
A textbook publisher estimates that the demand for a new calculus book will be 6000 copies. Each book
costs $12 to print, and set-up costs are $1800 for each printing run. Storage costs $3 per book per year.
How many books should be printed per printing run and how many printings should there be to
minimize inventory costs?
12.
A car dealer expects to sell 500 new convertibles tllis year. Each convertible costs the dealer $16,000
plus a fixed $5000 delivery fee per order. It costs $500 to store each car for a year. How many orders
should be made and how many cars should there be in each order to minimize inventory costs?
13.
A golf club manufacturer finds that production of golf clubs follows the model q. = 25xo.25l.75, where
x is the number of employees and y is the number of machines. If the manufacturer must produce 2000
clubs and it costs $70 per employee and $20 per machine, how many employees and how many
machines will minimize costs? If production is increased to 3000 clubs and costs increase to $30 per
machine, how many employees and how many machines will minimize costs?
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The following is a lesson from “Paul’s Online Math Notes”
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AP Calculus
Activity- Logistic Functions and their Derivatives
Name:____________________________________
Date:_____________________________________
Objective:
To determine the derivative of various logistic functions
To verify the rules for derivatives of logistic functions
p (t ) 
1)
Find the derivative of:
2)
Find the second derivative of:
3)
Find the derivative of:
Rule for logistic functions:
1
1  e t
p (t ) 
p(t ) 
1
1  e t
K
for any constant K
K  e t
d
p(t )  p (t )  1  p (t ) 
dt
Verify your answer to numbers 1 and 2 using the above formula.
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AP Calculus
Activity- Visualizing Derivative of Sine
Name:____________________________________
Date:_____________________________________
Objective:
To SEE where the derivative of sine comes from.
This is a “lab” to do with your partner. Essentially, you are asked to create a model sine curve and then look for
places at which you already the value of the derivative. You are then asked to build up a second curve that
represents the plot of the points you are finding. This will become your derivative graph.
1. do you remember how to graph a sine curve? Take a moment to sketch a sine curve on -2, 2. Label any
points you are certain of.
2. Next memory question: How do you find maxima and minima? How do you locate a point of inflection?
Maxima/Minima:
Inflections:
3. On your sketch, highlight any points where the derivative of the sine function is zero.
4. On your sketch, highlight the points of inflection. What will they represent on your derivative graph?
5. Take all the points that you’ve found so far and graph them on a new set of axes. To the left and right of
these points, determine whether the slope is rising or falling.
6. What does your graph suggest? What is the derivative of sin (x)?
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