277 Chapter 4: Applications of Differentiation
____ 20. The graph of f is shown below. For which value of x is minimum?
a.
b.
c.
d.
e.
____ 21. A meteorologist measures the atmospheric pressure P (in kilograms per square meter) at
altitude h (in kilometers). The data are shown below. Use the regression capabilities of the graphing
utility to find a linear model for the revised data points obtained by plotting the points
a.
b.
c.
d.
e.
____ 22. A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing
process. The flow rate R (in liters per hour) at time t (in hours) is given by the linear model
Write the linear model in exponential form.
a.
b.
c.
d.
e.
4.6 A Summary of Curve Sketching
278
____ 23.
A meteorologist measures the atmospheric pressure P (in kilograms per square meter)
at altitude h (in kilometers). The data are shown below. Find the rate of change of the pressure with
respect to altitude when
using the relation
. Round your answer to
one decimal place.
a.
b.
c.
d.
e.
279 Chapter 4: Applications of Differentiation
4.6 A Summary of Curve Sketching
4.6 A Summary of Curve Sketching
280
281 Chapter 4: Applications of Differentiation
4.7 Optimization Problems
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find two positive numbers whose product is 181 and whose sum is a minimum.
a.
b.
c.
d.
e.
____ 2. Find two positive numbers such that the sum of the first and twice the second is 56 and
whose product is a maximum.
a.
b. 28 and 14
c.
d.
e.
____ 3. Find the length and width of a rectangle that has perimeter meters and a
maximum area.
12 m; 12 m.
16 m; 9 m.
1m; 23 m.
13 m; 11 m.
6 m; 18 m.
____ 4. Find the length and width of a rectangle that has an area of 968 square feet and whose
perimeter is a minimum.
a.
b.
c.
d.
e.
____ 5. Find the point on the graph of the function that is closest to the point
. Round all numerical values in your answer to four decimal places.
a.
b.
4.7 Optimization Problems
282
c.
d.
e.
____ 6. Find the point on the graph of the function that is closest to the point
.
a.
b.
c.
d.
e.
____ 7. A rectangular page is to contain square inches of print. The margins on each side are 1
inch. Find the dimensions of the page such that the least amount of paper is used.
a.
b.
c.
d.
e.
____ 8. On a given day, the flow rate F (cars per hour) on a congested roadway is given by where v
is the speed of the traffic in miles per hour. What speed will maximize the
flow rate on the road? Round your answer to the nearest mile per hour.
18 miles per hour
16 miles per hour
17 miles per hour
10 miles per hour
21 miles per hour
283
Chapter 4: Applications of Differentiation
____
9.
Determine the dimensions of a rectangular solid (with a square base) with maximum
volume if its surface area is 529 square meters.
a.
Dimensions:
b.
Dimensions:
c.
Dimensions:
d.
Dimensions:
e.
Dimensions:
____ 10. 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 38 feet.
a.
b.
c.
d.
e.
4.7 Optimization Problems
284
____ 11.
A rectangle is bounded by the x– and y-axes and the graph of
(see
figure). What length and width should the rectangle have so that its area is a maximum?
a.
b.
c.
d.
e.
____ 12. A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.
The total volume of the solid is 30 cubic centimeters. Find the radius, r, of the cylinder that
produces the minimum surface area. Round your answer to two decimal places.
a.
b.
c.
d.
e.
285
Chapter 4: Applications of Differentiation
____
13.
The sum of the perimeters of an equilateral triangle and a square is 19. Find the
dimensions of the triangle and the square that produce a minimum total
area. a.
b.
c.
d.
e.
____ 14. A sector with central angle is cut from a circle of radius 10 inches, and the edges of the
sector are brought together to form a cone. Find the magnitude of such that the volume of the
cone is a maximum.
a.
b.
c.
d.
e.
____ 15. Assume that the amount of money deposited in a bank is proportional to the square of the
interest rate the bank pays on this money. Furthermore, the bank can reinvest this money at 36%.
Find the interest rate the bank should pay to maximize profit. (Use the simple interest formula.)
a.
b.
c.
d.
e.
4.7 Optimization Problems
286
4.7 Optimization Problems
Answer Section
between points
287 Chapter 4: Applications of Differentiation
4.8 Differentials
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the equation of the tangent line T to the graph of at the given point
a.
b.
c.
d.
e.
____
2.
Find the equation of the tangent line T to the graph of
at the given
point
a.
b.
c.
d.
e.
____
3.
Compare dy and
for
at x = –1 with
Give your
answers to four decimal places.
a.
b.
c.
d.
e.
4.8 Differentials
288
____
4.
Compare dy and
for
at x = 0 with
Give your
answers to four decimal places.
a.
b.
c.
d.
e.
____
5.
Find the differential dy of the function
a.
b.
c.
d.
e.
____
6.
Find the differential dy of the function
.
a.
b.
c.
d.
e.
____
7.
Find the differential dy of the function
.
a.
b.
c.
d.
e.
289
Chapter 4: Applications of Differentiation
____
8.
The measurement of the side of a square floor tile is 14 inches, with a possible error
of
inch. Use differentials to approximate the possible propagated error in computing the area of
the square.
a.
b.
c.
d.
e.
____ 9. The measurements of the base and altitude of a triangle are found to be 54 and 33
centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate
the propagated error in computing the area of the triangle. Round your answer to four decimal places.
a.
b.
c.
d.
e.
____ 10.
The measurement of the radius of the end of a log is found to be 28 inches, with a
possible error of
inch. Use differentials to approximate the possible propagated error in computing
the area of the end of the log.
a.
b.
c.
d.
e.
____ 11. The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.03
inch. Use differentials to approximate the maximum possible error in calculating the volume of the
sphere. Round your answer to two decimal places.
a.
b.
c.
d.
e.
4.8 Differentials
290
____ 12.
The radius of a spherical balloon is measured to be 12 inches, with a possible error of
0.06 inch. Use differentials to approximate the maximum possible error in calculating the surface
area of the sphere. Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 13. The total stopping distance T of a vehicle is where T is in feet and x is the
speed in miles per hour. Use differentials to approximate the percent change in total stopping
distance as speed changes from
to
miles per hour. Round your answer to one decimal
place.
a.
37.4%
b.
16.5%
c.
32.2%
d.
168.0%
e.
116.7%
____
14.
The range R of a projectile is
where
is the initial velocity in feet
per second and
is the angle of elevation. If
feet per second and
is changed from
to
use differentials to approximate the change in the range. Round your answer to the nearest
integer.
2,593 ft
4,568 ft
5,623 ft
2,811 ft
5,186 ft
____
15.
Use differentials to approximate the value of
. Round your answer to four
decimal places.
a.
5.0000
b.
5.0100
c.
5.0400
d.
5.0200
e.
5.0300
____
16.
Use differentials to approximate the value of
. Round your answer to four
decimal places.
a.
1.9683
b.
1.9483
c.
1.9383
d.
1.9583
e.
1.9783
291 Chapter 4: Applications of Differentiation
4.8 Differentials
Answer Section