Chapter 4 Curve
Part 1
20. Sketch a graph of the function with the following
properties:
1. What is the second derivative test?
2. Restate the following using calculus: “Our prices
are rising slower than any place in town.”
f ’(x) > 0 when x < -1
f ’(x) > 0 when x > 3
f ’(x) < 0 when x < 3
f ’’(x) < 0 when x < 2
f ’’(x) > 0 when x > 2
For numbers 3-19, find all critical points of the given
21. Sketch a graph of the function with the following
function. Determine where the graph is rising or
properties:
falling, and find where the graph is concave up or
f ’(x) > 0 when x < 2 and when 2 < x < 5
f ’(x) < 0 when x > 5
f ’(x) = 0
f ’’(x) < 0 when x < 2 and when 4 < x < 7
f ’’(x) > 0 when 2 < x < 4 and when x > 7
down.
2
3. y = 2(x + 20) – 8(x + 20) + 7
4. y =
1 3
x – 9x + 2
3
22. Sketch a graph of the function with the following
properties:
18
5. y = 1 + 2x +
x
4
3
f ’(x) > 0 when x < 1
f ’(x) < 0 when x > 1
f ’’(x) > 0 when x < 1
f ’’(x) > 0 when x > 1
2
6. y = u + 6u – 24u + 26
3
7. y = (t + t)
3
2
23. Sketch the graph of a function with the following
properties: There are relative extrema at (-1,7)
and (3,2). There is an inflection point at (1,4).
The graph is concave down only when x < 1.
The x-intercept is (-4,0), and the y-intercept is
(0,5).
4
8. y = t – 3t
6
9. y = 5t – 6t
10. y =
3
x
x 1
2
2
24. Examine the graphs :
1/3
-3t
a. f(x) = (x + 1)
2/3
b. f(x) = (x + 1)
4/3
c. f(x) = (x + 1)
5/3
d. f(x) = (x + 1)
11. y = t e
12. y =
e x  ex
e x  e x
Generalize to make a statement about the effect of a
n/3
positive integer n on the graph of f(x) = (x + 1) .
2
13. y =ln(x )
14. y = x
4/3
5
(x - 27)
3
25. Find constants A, B, and C that guarantee that
3
2
the function f(x) = Ax + Bx + C will have a
relative extremum at (2,11) and an inflection
point at (1,5). Sketch the graph of f.
2
15. y = x + 2x - x + 11
16. y = θ + cos 2θ for 0 ≤ θ ≤ 2π
17. y =
1
sin 2t + cos t for -π ≤ t ≤ π
4
-1
18. y = 2x – sin x for -1 ≤ x ≤ 1
2
19. y = tan x – x + 3 on (
 3
,
2 2
)
x
-x
x
-x
26. Let S(x) = .5(e – e ) and C(x) = .5(e + e ).
These functions are known as the hyperbolic
sine and hyperbolic cosine, respectively. These
functions are examined in Section 7.8.
a. Show that S’(x) = C(x) and C’(x) =
S(x).
b. Sketch the graphs of S and C.
27. Set up an appropriate model to answer the given
question. Be sure to state your assumptions.
At noon on a certain day, Frank sets out to
assemble five stereo sets. His rate of assembly
increases steadily throughout the afternoon until
4 PM at which time he has completed three sets.
After that he assembles sets at a slower and
slower rate until he finally completes the fifth set
at 8 PM. Sketch a rough graph of a function that
represents the number of sets Frank has
completed after t hours of work.
largest lateral surface area that can be inscribed in
the sphere. Hint: the lateral surface area is S =
2πr2h.
38. Find the dimensions of the right circular cylinder
of largest volume that can be inscribed in a right
circular cone of radius R and altitude H.
28. Describe an optimization procedure.
29. A woman plans to fence off a rectangular garden
2
whose area is 64 ft . What should be the dimensions
of the garden if she wants to minimize the amount of
fencing used?
30. Pull out a sheet of 8.5-in. by 11-in. binder paper.
Cut squares from the corners and fold the sides up
to form a container. Show that the maximum volume
of such a container is about 1 liter.
31. Why is it important to check endpoints when
finding an optimum value?
32. The highway department is planning to build a
rectangular picnic area for motorists along a major
2
highway. It is to have an area of 5,000 yd and is to
be fenced off on the three sides not adjacent to the
highway. What is the least amount of fencing that
will be needed to complete the job?
33. Farmer Jones has to build a fence to enclose a
2
1,200 m rectangular area ABCD. Fencing costs $3
per meter, but Farmer Smith has agreed to pay half
the cost of fencing CD , which borders the property.
Given x is the length of side CD , what is the
minimum amount (to the nearest cent) Jones has to
pay?
34. Find the rectangle of largest area that can be
inscribed in a semicircle of radius R, assuming that
one side of the rectangle lied on the diameter of the
semicircle.
35. A tinsmith wants to make an open-topped box
out of a rectangular sheet of tin 24 in. wide and 45
in. long. The tinsmith plans to cut congruent squares
out of each corner of the sheet and then bend the
edges of the sheet upward to form the sides of the
box. What are the dimensions of the largest box that
can be made in this fashion?
36. Find the dimensions of the right circular cylinder
of largest volume that can be inscribed in a sphere
of radius R.
37. Given a sphere of radius R, find the radius r and
altitude 2h of the right circular cylinder with the
39. A truck is 250 mi due east of a sports car and is
traveling west at a constant speed of 60 mi/h.
Meanwhile, the sports car is going north at 80 mi/h.
When will the truck and the car be closest to each
other? What is the minimum distance between
them? Hint: Minimize the square of the distance.
40. Show that of all rectangles with a given
perimeter, the square has the largest area.
41. Show that of all rectangles with a given area, the
square has the smallest perimeter.
42. A closed box with a square base is to be built to
house an ant colony. The bottom of the box and all
2
four sides are to be made of material costing $1/ft ,
and the top is to be constructed of glass costing
2
$5/ft . What are the dimensions of the box of
greatest volume that can be constructed for $72?
43. According to postal regulations, the girth plus the
length of a parcel sent by fourth-class mail may not
exceed 108 in. What is the largest possible volume
of a rectangular parcel with two square sides that
can be sent by fourth-class mail?
44. A man 6 feet tall walks at a rate of 5 feet per
second away from a light that is 15 feet above the
ground. When he is 10 feet from the base of the
light,
a) at what rate is the tip of his shadow moving?
b) at what rate is the length of his shadow
changing?
48. A trough is 12 feet long and 3 feet across the
top. Its ends are isosceles triangles with altitudes of
3 feet. If water is being pumped into the trough at 2
cubic feet per minute, how fast is the water level
rising when the water is 1 foot deep?
49. An airplane is flying at an altitude of 6 miles and
passes directly over a radar antenna. When the
plane is 10 miles away (s = 10), the radar detects
the distance s is changing at a rate of 240 miles per
hour. What is the speed of the plane?
45. As a spherical raindrop falls, it reaches a layer of
dry air and begins to evaporate at a rate that is
2
proportional to its surface area (S = 4 πr ). Show
that the radius of the raindrop decreases at a
constant rate.
46. When a certain polyatomic gas undergoes
adiabatic expansion, its pressure p and volume v
satisfy the equation
1.3
pv = k
where k is a constant. Find the relationship between
the related rates dp/dt and dv/dt.
47. A fish is reeled in at a rate of 1 foot per second
from a point 15 feet above the water. At what rate is
the angle between the line and the water changing
when there are 25 feet of line out?
50. Missy Smith is at a point A on the north bank of
a long, straight river 6 mi. wide. Directly across from
her on the south bank is a point B, and she wishes
to reach a cabin C located s miles down the river
from B. Given that Missy can row at 6 mi/h (including
the effect of the current) and run at 10 mi/h, what is
the minimum time (to the nearest minute) required
for her to travel from A to C in each case?
a) s = 4
b) s = 6
2
52. A poster is to contain 108 cm of printed matter,
with margins of 6 cm each at top and bottom and 2
cm on the sides. What is the minimum cost of the
poster if it is to be made of material costing
2
20¢/cm ?
53. An isosceles trapezoid has a base of 14 cm and
slant sides of 6 cm. What is the largest area of such
a trapezoid?
x.
dx
dy
a) Given
= 3, find
when x = 4
dt
dt
58. y =
b) Given
dy
dx
= 2, find
when x = 25
dt
dt
59. A point is moving along the graph of y =
1
.
1 x2
dx
dy
= 2 cm/sec. Find
for the given values of x.
dt
dt
a) x = -2
55. A cylindrical container with no top is to be
constructed to hold a fixed volume of liquid. The cost
2
of the material used for the bottom is 50¢/in , and
the cost of the material used for the curved face is
2
30¢/in . Use calculus to find the radius of the least
expensive container.
3
56. Use the fact that 12 oz ≈ 355 mL = 255 cm to
find the dimensions of the 12-oz Coke® can that can
be constructed using the least amount of metal.
Compare these dimensions with a Coke from your
refrigerator. What do you think accounts for the
difference?
57. A stained glass window in the form of an
equilateral triangle is built on top of a rectangular
window. The rectangular part of the window is one of
clear glass and transmits twice as much light per
square foot as the triangular part, which is made of
stained glass. If the entire window has a perimeter of
20 ft, find the dimensions (to the nearest ft) of the
window that will admit the most light.
b) x = 2 c) x = 0 d) x = 10
60. Using the graph below, determine whether:
dy
increases or decreases for increasing x and
dt
dx
constant
dt
dx
b)
increases or decreases for increasing y and
dt
dy
constant
.
dt
a)
61. Find the rate of change of the distance between
the origin and a moving point on the graph of y = sin
(d)
Use a graphing utility to graph the
function in part (c) and estimate the solution
from the graph.
dx
x if
= 2 centimeters per second.
dt
(e)
Use the calculus to find the critical
number of the function in part (c). Then find the
two numbers.
65. The included angle of the two sides of constant
equal length s of an isosceles triangle is θ.
a) Show that the area of the triangle is given by A =
2
.5s sin θ.
In Exercises 2-6, find two positive numbers that
satisfy the given requirements.
b) If θ is increasing at the rate of .5 radians per
minute, find the rate of change of the area when θ =
2.
The sum is S and the product is a maximum.
3.
The product is 192 and the sum is a minimum.

6
and θ =

.
3
c) Explain why the rate of change of the area of the
area of the triangle is not constant even though
d
is constant.
dt
5.
The second number is the reciprocal of the first
and the sum is a minimum.
62. All edges of a cube are expanding at a rate of 3
centimeters per second. How fast is the volume
changing when each edge is:
a) 1 centimeter?
centimeters?
b) 10
63. At a sand and gravel plant, sand is falling off a
conveyor and onto a conical pile at a rate of 10 cubic
feet per minute. The diameter of the base of the
cone is approximately three times the altitude. At
what rate is the height of the pile changing when the
pile is 15 feet high?
Part 2
1.
Numerical, Graphical, and Analytic Analysis
Find two positive numbers whose sum is 110 and whose
product is a maximum.
(a)
Analytically complete six rows of a table
such as the one below. (The first two
rows are shown.)
First Number
x
10
Second Number
Product P
110 – 10
10(110 – 10) = 1000
20
110 – 20
20(110 – 20) = 1800
(b) Use a graphing utility to generate additional
rows of the table. Use the table to estimate the
solution. (Hint: Use the table feature of the
graphing utility.)
(c)
4.
The product is 192 and the sum of the first plus
three times the second is a
minimum.
Write the Product P as a function of x.
6.
The sum of the first and twice the second is 100
and the product is a maximum.
In Exercise 7 and 8, find the length and width of a
rectangle that has the given perimeter and a
maximum area.
7.
Perimeter: 100 meters
8.
Perimeter: P units
In Exercise 9 and 10, find the length and width of a
rectangle that has the given area and a minimum
perimeter.
9.
Area: 64 square feet
10.
Area: A square centimeters
In Exercises 11 and 12, find the point on the graph of
the function that is closest to the given point.
Function
x
11.
f(x) =
12.
f(x) = x (2, ½)
Point
(4, 0)
2
13.
Chemical Reaction
In an autocatalytic
chemical reaction the product formed is a catalyst for
the reaction. If Q0 is the amount of the original
substance and x is
the amount of catalyst formed,
the rate of chemical reaction is
dQ
 kx(Q0  x).
dx
For what value of x will the rate of chemical
reaction be greatest?
(b)
14.
Traffic Control On a given day, the flow rate F
(cars per hour) on a
congested roadway is
F
v
22  0.02v 2
(c)
Use calculus to find the critical number
of the function in part (b) and find
the
maximum value.
(d)
where v is the speed of the traffic in miles per
hour. What speed will maximize
the flow rate on
the road?
Write the volume V as a function of x.
Use a graphing utility to graph the
function in part (b) and verify the
maximum volume from the graph.
15.
Area A farmer plans to fence a rectangular
pasture adjacent to a river. The pasture must contain
180,000 square meters in order to provide enough grass
for the herd. What dimensions would require the least
amount of fencing if no fencing is needed along the
river?
17.
Volume
(a)
Verify that each of the rectangular solids
shown in the figure has a surface area of 150
square inches.
18.
(b)
Find the volume of each.
(c)
Determine the dimensions of a
rectangular solid (with a square base) of
maximum volume if its surface area is
150 square inches.
Numerical, Graphical, and Analytic Analysis
An open box of maximum volume is to
be made from a square piece of material, 24
inches on a side, by cutting equal squares from
the corners and turning up the sides (see figure).
(a)
Analytically complete six rows of a table
such as the one below. (The first two rows are
shown.) Use the table to guess the maximum
volume.
Height
Length and Width
Volume
1
24 – 2(1)
1[24 – 2(1)] = 484
2
24 – 2(2)
2[24 – 2(2)] = 800
2
2
19.
(a)
Solve Exercise 18 given that the square
piece of material is s meters on a side.
(b)
If the dimensions of the square piece of
material are doubled, how does the volume change?
20.
Numerical, Graphical, and Analytic Analysis
A physical fitness room consists of a
rectangle with a semicircle on each end. A
200-meter running track runs around the
outside of the room.
(a)
Draw a figure to represent the problem.
Let x and y represent the length and
width of the rectangle.
(b)
Analytically complete six rows of a table
such as the one below. (The first two
rows are shown.) Use the table to
guess the maximum area of the
rectangular region.
Length
x
10
Width y
2

20
2

21.
23.
Area
(100  10)
(10)
(100  20)
(20)
2

2

(100  10)  573
Length
A right triangle is formed in the
first quadrant by the x- and y-axes and a line
through the point (1, 2) (see figure).
(a)
Write the length L of the hypotenuse as
a function of x.
(b)
Use a graphing utility to graphically
approximate x such that the length of
the hypotenuse is a minimum.
(c)
Find the vertices of the triangle such
that its area is a minimum.
(100  20)  1019
(c)
Write the area A as a function of x.
(d)
Use calculus to find the critical number
of the function in part (c) and find the
maximum value.
(e)
Use a graphing utility to graph the
function in part (c) and verify the
maximum area from the graph.
24.
Area Find the dimensions of the largest
isosceles triangle that can be inscribed in a
circle of radius 4 (see figure).
25.
Area
Area A Norman window is constructed by
adjoining a semicircle to the top of an ordinary
rectangular window (see figure). Find the
dimensions of a Norman window of maximum
area if the total perimeter is 16 feet.
A rectangle is bounded by the x-axis
and the semicircle y  25  x (see figure).
What length and width should the rectangle
have so that its area is a maximum?
2
22.
Area A rectangle is bounded by the x- and yaxes and the graph of y = (6-x)/2 (see figure).
What length and width should the rectangle
have so that its area is a maximum?
Use an appropriate local linear
approximation to estimate the value of the
given quantity.
26. (3.02)2
27. (1.97)3
28.
29.
30.
31.
65
24
80.9
Part 3
1-8 . Produce graphs of f that reveal all the
important aspects of the curve. In particular,
you should use graphs of f' and f" to estimate
the intervals of increase and decrease,
extreme values, intervals of concavity, and
inflection points.
1. f(x) = 4x4- 7x2 + 4x + 6
2. f(x) = 8x5 + 45x4 + 80x3 + 90x2 + 200x
3 2
x - 3x - 5
4
4. f(x) =
5. f(x) =
x  x 3 - 2x  2
x2  x - 2
x
x 3 - x 2 - 4x  1
6. f(x) = tan x + 5 cos x
7. f(x) = x2 sin x, -7 ≤x ≤7
8. f(x) =
13. f(x) =
14. f(x) =
36.03
3. f(x) =
13-14. Sketch the graph by hand using
asymptotes and intercepts, but not derivatives.
Then use your sketch as a guide to producing
graphs (with a graphing device) that display the
major features of the curve. Use these graphs
to estimate the maximum and minimum values.
ex
x2 - 9
9-10 . Produce graphs of f that reveal all the
important aspects of the curve. Estimate the
intervals of increase and decrease, extreme
values, intervals of concavity, and inflection
points, and use calculus to find these quantities
exactly.
9. f(x) = 8x3 – 3x2 - 10
10.f(x) = x 9 - x 2
11-12 . Produce a graph of f that shows all the
important aspects of the curve. Estimate the
local maximum and minimum values and then
use calculus to find these values exactly. Use a
graph of f" to estimate the inflection points.
11. f(x) = ex^3-x
12. f(x) =ecosx
( x  4)( x - 3)2
x 4 ( x - 1)
10x( x - 1)4
(x - 2)3 ( x  1)2