UNIT 3
Derivatives and Their Applications
Date
Textbook
Section
Title
Homework
Review of Prerequisite Skills
Pg. 116 # 1-8, 9*GC
3.1
Higher-Order Derivatives, Velocity
& Acceleration
Pg.127 # 1-18
3.2
Maximum and Minimum on an
Interval (Extreme Values)
Pg. 135 # 1-15
Mid-Chapter Review
Pg. 139 # 1-11
Pg. 145 # 1-23
3.3
Optimization Problems
3.4
Optimization Problems in
Economics and Science
Pg. 151 #1-19
Key Concepts Review
Pg. 156 # 1-30
Chapter 3 Practice Test
Pg. 160 # 1-8
Unit 3 Test
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Unit 3: Derivatives and Their Applications
3.1 Higher-Order Derivatives, Velocity and Acceleration
HIGHER-ORDER DERIVATIVES
Consider the function y  f (x) .
The first derivative of
f or “ f prime” is: f ( x)  y  
dy
dx

 d  dy  d y
f or “ f double prime” is: f ( x)   f x   y    y      2
dx  dx  dx
2
The second derivative of

 d  d y  d y

f or “ f triple prime” is: f ( x)   f x   y    y   
dx  dx  dx 3
2
The third derivative of
3

 d d y d y
f or “super 4” is: f 4  x    f  x   y 4    y    3   4
dx  dx  dx

d  d n 1 y  d n y
n 
n 1
n 
 n 1 
x   y  y    n1   n
The n-th derivative of f or “super n” is: f  x    f
dx  dx  dx
3
4
The fourth derivative of
Note that
f is differentiable at x if f n   x  exists.
Ex. 1:
Find the second derivative of f ( x)  3  x  at x  4 .
Ex. 2:
Find f  , f  , f  , f 4 , and f 5 for f x   4 x 3  3x 2  x  1 .
2
 
 
POLYNOMIAL MODELS INVOLVING VELOCITY AND SPEED
In many situations, an object’s position, s, can be represented
as s(t). Recall that average velocity is defined as the rate of
change of displacement over an interval of time.
Instantaneous velocity is the rate of change of
displacement at a specific point in time.
On a displacement-time graph, the slope of a secant
represents average velocity, while the slope of a tangent
represents instantaneous velocity.
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Average velocity =
s
t
Instantaneous velocity
s
t
s (t  t )  s (t )
 lim
t 0
t
s (t  h)  s (h)
 lim
h0
h
 s(t )
 lim
t 0
As a result, the derivative of the position function, s(t ) represents the instantaneous velocity of the object at
time
t . So v(t ) 
ds
 s (t )
dt
a(t )  v (t )  s (t )
Acceleration of the object is:
a(t ) 
dv d 2 s

dt dt 2
Velocity of an object measures how fast it is moving and the direction of movement.
m s 
Speed is the magnitude or absolute value of the velocity, v (t ) , without regard to direction.
m s 
Acceleration is the instantaneous rate of change of velocity with respect to time.
m s 
2
ANALYZING THE MOTION OF A MOVING OBJECT: HORIZONTAL MOTION
If v(t) > 0 or s(t) is increasing, the object is moving to the right/advancing.
If v(t) < 0 or s(t) is decreasing, the object is moving to the left/retreating.
If v(t) = 0, the object is stationary, or at rest.
To determine whether a particle is moving towards a fixed point O or away from a fixed point O, two quantities
need to be considered:  its
and its
.
1. Particles moving toward a fixed point O.

If the particle is located to the right of O, then it must move
other words, if its position is
, its velocity must be
to move toward O. In
.

If the particle is located to the left of O, then it must move
to move toward O.
Thus, if it has a
position, it must have a
velocity.
Therefore, when a particle moves toward O the signs of position s(t) and velocity v(t) are
, or s(t )  v(t )  0 .
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2. Particles moving away from fixed point O.

If the particle is located to the right of O, then it must move
In other words, if its position is

, its velocity must be
If the particle is located to the left of O, then it must move
to move away from O.
.
to move away from O.
Thus, if it has a
position, it must have a
velocity. Therefore, when a
particle moves away from O the signs of position s(t) and velocity v(t) are
, or
s(t )  v(t )  0 .
ACCELERATION
1. Acceleration to the right or upwards is positive.
2. Acceleration to the left or downwards is negative.
3. When acceleration is in the same direction as velocity, an object will speed up.
ie.
v(t )  a (t )  0
4. When acceleration and velocity are in opposite directions, an object will slow down.
ie.
v(t )  a(t )  0
5. Velocity is not changing if
Ex. 3:
a(t )  v(t )  s (t )  0
The position of an object moving along a straight line can be modeled by the function
s(t )  3t 3  40.5t 2  162t , where s is the position in metres at t seconds, and t  0 .
a) Determine the initial position of the object.
b) Determine the velocity at 2 s, 5 s and 7 s.
c) When is the object stationary?
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d) When is the object moving in a positive direction? Negative direction?
e) Determine the total distance traveled during the first eight seconds of motion.
f)
Determine the acceleration at 2 s, 5s and at 7 s.
g) Determine when the velocity is not changing.
Ex. 3: a) Discuss the motion of an object moving on a horizontal line if its position is given by
st   t 2  1t  16, 0  t  10 , where s in meters and t is in seconds. Include the initial velocity,
final velocity and any acceleration in your discussion.
b) Is the object moving away or towards origin at 6
seconds?
ANALYZING THE MOTION OF A FALLING OBJECT: VERTICAL MOTION
When an object is thrown upward, the height of the object (metres) after t (seconds) is given by the formula:
1
h   gt 2  v0 t  h0
2
where g represents the acceleration due to gravity in metres per second squared, t seconds represents the
time, v0 metres per second represents the initial velocity, and h0 metres is the initial height of the object.
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Ex. 4:
A rock is thrown into the air from a bridge 15 m above the water. Its height above the water is a function
of the time since it was thrown. The height of the rock, in metres, above the water can be modeled by
the function h(t )  4.9t 2  12t  15 , where h(t) represents the height in metres at t seconds.
a) Determine the instantaneous velocity at 1 s and at 2 s .
b) Determine the initial velocity of the rock.
c) When is the rock at its maximum height? What is the maximum height?
Time,
(t sec)
Height,
h(t)
Velocity,
h’(t)
0.0
15.0
12.0
0.5
19.8
7.1
1.0
22.1
2.2
1.5
22.0
-2.7
2.0
19.4
-7.6
2.5
14.4
-12.5
3.0
6.9
-17.4
d) What is the velocity of the rock when it enters the water?
e) Is the rock slowing down or speeding up when it enters the water? Consider the graph of the velocity
function.
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v ( t )
(first differences)
t(s)
v(t)=h’(t)
(m/s)
0
12.0
1
2.2
-9.8
2
-7.6
-9.8
3
-17.4
-9.8
4
-27.2
-9.8
3-5
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Unit 3: Derivatives and Their Applications
3.2 Minimum and Maximum on an Interval, Extreme Values
Illustration:
The maximum value of a function that has a derivative at all points in an interval occurs at a “peak”
(ie. f ′(c ) = 0 ) or at an endpoint of the interval. The minimum value occurs at a “valley” (ie. f ′(c ) = 0 )
or at an endpoint. This is true no matter how many peaks and valleys the graph has in the interval.
GLOBAL MAXIMUM
A function f has a global (absolute) maximum at x = c
f ( x) ≤ f (c) for all x ∈ D f .
f (c) is called the global (absolute) maximum value.
(c, f (c)) is called the global (absolute) maximum point.
if
GLOBAL MINIMUM
A function f has a global (absolute) minimum at
x = c if f ( x) ≥ f (c) for all x ∈ D f .
f (c) is called the global (absolute) minimum value.
(c, f (c)) is called the global (absolute) minimum point.
EXTREMUM AND EXTREMA
An extremum is either a minimum or a maximum (value, point, local or global). Extrema is the plural of
extremum.
GLOBAL (ABSOLUTE) EXTREMA ALGORITHM
To find the global (absolute) extrema for a continuous function
1. Identify the critical numbers over (a, b )
f over a closed interval [a, b] :
f (c) at each critical number c in (a, b )
3. Find the values f (a ) and f (b)
2. Find the values of the function
4. from the values obtained at part 2) and 3):
⇒ the largest represents the global (absolute) maximum value
⇒ the least represents the global (absolute) minimum value (s)
Note: c is a critical number if either f ′(c ) = 0 or f ′(c ) = DNE
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Ex. 1:
State, with reason, why the maximum/minimum algorithm can or cannot be used to determine the
maximum and minimum values for
Ex. 2:
y=
3x
,
x−2
−1 ≤ x ≤ 3.
The amount of current, in amperes (A), in an electrical system is given by the function
C (t ) = −t 3 + t 2 + 21t , where t is the time in seconds and 0 ≤ t ≤ 5 . Determine the times at which
the current is at its maximum and minimum, and determine the amount of current in the system at these
times.
Ex. 3:
The concentration C (t ) , in milligrams per cubic centimeter, of a certain medicine in a patient’s
bloodstream is given by
C (t ) =
0.1t
, where t is the number of hours after the medicine is taken.
(t + 3) 2
Determine the maximum and minimum concentrations between the first and sixth hours after the
medicine is taken.
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Unit 3: Derivatives and Their Applications
3.3 Optimization Problems
ALGORITHM FOR SOLVING OPTIMIZATION PROBLEMS
1. Read and understand the problem’s text.
2. Draw a diagram (if necessary).
3. Assign variables to the quantities involved and state restrictions according to the situation.
4. Write relations between these variables.
5. Identify the variable that is minimized or maximized. This is the dependant variable.
6. Use the other relations (called constraints) to express the dependent variable (the one which is minimized or
maximized) as a function of one single variable (the independent variable).
7. Find extrema (maximum or minimum) for the dependant variable (using global extrema algorithm, first
derivative test or the second derivative test).
8. Check if extrema satisfy the conditions of the application.
9. Find the value of other variables at extrema (If necessary).
10. Write the conclusion statement.
DOUBLE DERIVATIVE TEST
If f’’(x)>0, then f(x) has a relative minimum value.
If f’’(x)<0, then f(x) has a relative maximum value.
Ex. 1:
OPTIMIZATION PROBLEMS INVOLVING NUMBERS
Find two positive numbers with a product equal to 200 such that the sum of one number and twice the
other number is as small as possible. What is the minimum value of the sum?
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Ex. 2:
MAXIMIZE THE VOLUME GIVEN THE SHAPE AND AREA
If 2700 cm2 of material is available to make a box with a square base and an open top, find the
dimensions (length, width, and height) of the box that give the largest volume of the box. What is the
maximum volume of the box?
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Ex. 3:
MINIMIZE THE AREA GIVEN THE SHAPE AND VOLUME
A cylindrical can is to be made to hold 1000 cm3 (one litre) of oil. Find the dimensions (radius and
height) of the can that will minimize the cost of the metal to make the can.
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Ex. 4:
MINIMIZE THE COST GIVEN THE SHAPE AND VOLUME
A closed box with a square base is to contain 252 cm3 . The bottom costs $0.05/ cm2, the top costs
$0.02/ cm2 and the sides costs $0.03/ cm2 Find the dimensions (base side and height) that will
minimize the cost.
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Ex. 5:
MINIMIZE THE COST FOR AN UNDERGROUND/UNDERWATER CABLE
A cable television company is laying cable in an area with underground utilities. Two subdivisions are
located on opposite sides of a 200m wide river. The company has to connect points P and Q with cable,
where P is on the South bank and Q is on the North bank 2000m East of P. It cost $8/m to lay cable
underground and $10 /m to lay cable underwater. What is the least expensive way to lay the cable and
what is the minimum cost?
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Ex. 6:
GEOMETRY
How close does the curve y 
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x come to the point (3/ 2,0) ?
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Ex. 7:
GEOMETRY
Find the dimensions of the largest right-cylinder that can be inscribed in a cone of radius R = 3m and
height H = 6m
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MCV 4U Optimization Questions
1.
Find two numbers whose difference is 150 and whose product is a minimum.
2.
Find two positive numbers with product 200 such that the sum of one number and twice the second
number is as small as possible.
3.
A rectangle has a perimeter of 100 cm. What length and width should it have so that its area is a
maximum?
4.
Show that a rectangle with a given area has a minimum perimeter when it is a square.
5.
A box with a square base and open top must have a volume of 400 cm3 . Find the dimensions of the
box that minimizes the amount of material used.
6.
A box with an open top is to be constructed from a square piece of cardboard that is 3 m wide, by
cutting out a square from each from each of the four corners and bending up the sides. Find the
largest volume that such a box can have.
7.
A farmer wants to fence an area of 750 000 m 2 in a rectangular field and divide it in half with a fence
parallel to one of the sides of the rectangle. How can this be done so as to minimize the cost of the
fence?
8.
Find the point on the line y = 5 x + 4 , that is closest to the origin.
9.
Find the point on the parabola 2 y = x 2 that is closest to the point (− 4,1) .
10.
A can is to be made to hold a litre of oil. Find the radius of the can that will minimize the cost of the
metal to make the can. ( 1L = 1000 cm3 )
11.
A piece of wire 40 cm long is cut into two pieces. One piece is bent into the shape of a square and the
other is bent into the shape of a circle. How should the wire be cut so that the total area enclosed is a)
a maximum ? b) a minimum?
12.
A rectangle is inscribed in a semicircle of radius 2 cm. Find the largest area of such a rectangle.
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13.
A Norman window has the shape of a rectangle capped by a semicircular region. If the perimeter of
the window is 8 m, find the width of the window that will admit the greatest amount of light.
14.
A boat leaves a dock at noon and heads West at a speed of 25
20
km
. Another boat heads North at
h
km
and reaches the same dock at 1:00 pm. When were the two boats closest to each other?
h
15.
Find the largest possible volume of a right circular cylinder that is inscribed in a sphere of radius “r”.
16.
A 1 km racetrack is to be built with two straight sides and semicircles at the ends. Find the dimensions
of the track that encloses the maximum area.
17.
Find the lengths of the sides of an isosceles triangle which has a perimeter of 12 m and a maximum
area.
18.
A Gable window has the form of a rectangle topped by an equilateral triangle, the sides of which are
equal to the width of the rectangle. Find the maximum area of the window if the perimeter is 600 m.
19.
Find the dimensions and area of the largest rectangle having its two lower corners on the x-axis and its
two upper corners on the parabola y = 16 − x 2 .
20.
Find the volume of the largest open topped rectangular box that has a square base and a total surface
area of 3600 cm 2 .
21.
A right circular cone is inscribed in a sphere of radius 15 cm. Find the dimensions of the cone that has
the maximum volume.
22.
A cylinder of radius ‘r’ is inscribed in a cone of height H and base radius R. Show that the maximum
4
volume of the cylinder is the volume of the cone.
9
23.
At 1 pm a Ship sailing South at 18 knots is 40 miles due East of a Boat travelling East at 24 knots.
When will the two vessels be closest to each other?
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24.
cm
. At the same moment, a
s
cm
toy sail ship starts from a point 8 2 m Northeast of the tugboat and travels West at 7
. How
s
closely do the two toys approach each other?
25.
Two isolated farms are situated 12 km apart on a straight country road that runs parallel to the main
highway which is 20 km away. The power company decides to run a wire from the highway to a
junction box, and from there, wires of equal length to the two farms. Where should the junction box
be placed to minimize the length of wire needed?
26.
A sailor in a rowboat that is 8 km off a straight coastline wants to reach a point on shore that is 10 km
from the point directly opposite her present position. (In the shortest possible time!!) Toward what
point on shore should she steer and how long does it take her to reach her destination if she can row at
km
km
and she can run (on the beach) at 6
?
4
h
h
27.
A real estate firm owns 250 apartments that can be rented at $460 per month each. For each $5 per
month increase in rent, there are two vacancies created that cannot be filled. What should the
monthly rent be to maximize the total revenue? What is the maximum revenue?
28.
Electric power for a home on one bank of a straight river (that is 200 m wide) must come from a power
station that is 500 m downstream on the opposite bank of the river. If it costs twice as much to lay
cable underwater as it does to lay cable on land, what path should be chosen to minimize the cost of
the cable?
29.
A wooden chest is rectangular in shape with its length along the front twice as long as its width. The
top, front, and two sides of the chest are made of Oak. The back and bottom are made of pine. The
chest is to have a volume of 0.25 m3 . Oak costs three times as much as pine. Find the dimensions of
the chest that minimize the cost.
A toy tugboat is launched from the side of a pond and travels North at 5
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Optimization Question Answers
1.
a = 75, b = -75
2.
a = 20, b = 10
3.
l = 25, w = 25
256 3
32
8
, y=
, Area =
units 2
9
3
3
19.
x=
20.
base = 20 3 , h = 10 3 ,
Volume = 12000 3 cm3
21.
h = 20, r = 200
4.
5.
base = 20, height = 10
6.
Max. Volume = .5 x 2 x 2 = 2 m3
7.
length = 1060.6, width = 707.1
22.
23.
time = 64 minutes + 1pm = 2:04 pm
8.
 − 10 2 
, 

 13 13 
24.
t=
(− 2,2)
25.
9.
d= 3.46 km
(from midpoint between two farms)
r=3
26.
x = 7.15 km down shore
10.
27.
x = 16.5 increases in rent.
Rent = (217) ( $542.50)
28.
x = 115.5 m. ( downstream)
29.
length = .956
Width = .478
Height = .546
11.
500
π
= 5.4 cm
All Square: Area = 100cm 2 ,
All Circle: Area = 128.6 cm 2
Square & Circle.: sq: 22.41 cm,
circle: 17.59 cm
12.
x = 2 , y = 2 , Area = 4cm 2
13.
r=
8
16
, w=
π +4
π +4
14.
t=
16
hours = 23.4 minutes
41
15.
h=
4 3πR 3
2R
, Vol. =
9
3
16.
4800
, Minimum distance = 186 cm
37
1
km, l = 0.
π
2π
(totally circular track)
r=
500
m. =
17.
s = 4 . ( Equilateral Triangle)
18.
Max. Area = 21087.4 cm 2
x = 70.29, y= 89.13
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Unit 3: Derivatives and Their Applications
3.4 Optimization Problems In Economics and Science
Ex. 1:
MAXIMIZING REVENUE
The bus company Carsec Travels carries about 20 000 riders per day for a fare of $0.90. A survey
indicates that if the fare is decreased by $0.05, the number of riders will increase by 2000. What ticket
price would result in the greatest revenue?
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Ex. 2:
OPTIMIZING COST TO MAXIMIZE PROFIT
Through market research, a computer manufacturer found that x thousand units of its new laptop will
sell at a price of 2000 − 5 x dollars per unit. The cost, C, in dollars to produce this many units is
C ( x) = 15 000 000 + 1 800 000 x + 75 x 2 . Determine the level of sales that will maximize profit.
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Ex. 3:
OPTIMIZING DIMENSIONS TO MINIMIZE COST
A 20 000 m3 rectangular cistern is to be made from a reinforced concrete so that the interior length will
be twice the height. If the cost is $40/ m2 for the base, $100/ m2 for the side walls, and $200/ m2 for the
roof, then find the interior dimensions that will keep the cost to a minimum. To protect the water table,
the building code specifies that no excavation can be more than 22 m deep. It also specifies that all
cisterns must be at least 1 m in depth.
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Ex. 4:
AMOUNT OF ACID IN STOMACH OVER TIME
A researcher, Anita Brane, found that the level of antacid in a person’s stomach, t minutes after a
certain brand of antacid tablet is taken, is
L(t ) =
a. Determine the value of t for which L ′(t ) = 0 .
6t
.
t + 2t + 1
2
b. Determine L(t ) for the value you found in part a.
From the graph of L(t ) given below, what can you predict about the level of antacid in a person’s
stomach after 1 min?
d. What is happening to the level of antacid in a person’s stomach from 2 ≤ t ≤ 8 ?
c.
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Ex. 5:
OPTIMIZING COSTS WITH VARIABLE RATES
A power house, P, is on the bank of a straight river 200 m wide, and a factory, F, is on the other bank
400 m downstream from P. The cable has to be taken across the river under water at a cost of $12/m.
On land, the cost is $6/m. What path should be chosen to minimize the cost?
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