Deck 3: Differentiation

Question 1

Let f be the function defined by . Use the differential of f to find the approximate change in y if x changes from 4 to 4.1.

A) 0.02774
B) 0.36056
C) 1.10940
D) 0.05547
E) 0.01387

Answer 1

0.02774

Question 2

Find the differential of the function.

A)
B)
C)
D)
E)

Answer 2

Question 3

Use a differential to approximate . Hint: Let and compute dy with x = 25 and dx = 0.08 .

A) 5.14381
B) 6.18124
C) 5.38115
D) 5.20768

Answer 3

5.20768

Question 4

Find the differential of the function.

A)
B)
C)
D)

Question 5

Use a differential to approximate the quantity to the nearest thousandth.

A)
B)
C)

Question 6

Lupe deposits a sum of $5,000 into an account that pays interest at the rate of r/year compounded monthly. Her investment at the end of 10 yr is given by . Approximately how much more would Lupe's account be worth at the end of the term if her account paid 9.1%/year instead of 9%/year? 9.2%/year instead of 9%/year? 9.3%/year instead of 9%/year?
(Hint: Use the differential of A to answer this questions.)

A) $122.66, $243.31, $367.97
B) $121.66, $263.31, $364.97
C) $111.66, $263.31, $394.97
D) $122.66, $241.31, $367.97
E) $121.66, $243.31, $364.97

Question 7

Let f be the function defined by . Find the approximate change in y if x changes from to . Find the actual change in y if x changes from to .

A)
B)
C)
D)
E)

Question 8

Find the differential of the function.

A)
B)
C)
D)

Question 9

Find the differential of the function.

A)
B)
C)
D)

Question 10

Use a differential to approximate the quantity to the nearest thousandth.

A)
B)
C)
D)

Question 11

Find the differential of the function.

A)
B)
C)
D)

Question 12

Let f be the function defined by . Find the approximate change in y if x changes from 1 to 1.08. Find the actual change in y if x changes from 1 to 1.08.

A)
B)
C)
D)

Question 13

The supply equation for a certain brand of radio is given by , where x is the quantity supplied and p is the unit price in dollars. Use differentials to approximate the change in price when the quantity supplied is increased from 10,000 units to 10,500 units.

A) $0.75
B) $7.00
C) $1.00
D) $0.50
E) $6.50

Question 14

Find the differential of the function.

A)
B)
C)
D)

Question 15

Find the differential of the function.

A)
B)
C)
D)

Question 16

Find the differential of the function.

A)
B)
C)
D)

Question 17

Use a differential to approximate the quantity to the nearest thousandth.

A)
B)
C)

Question 18

The volume of a spherical cancerous tumor is given by . If the radius of a tumor is estimated at 1.2 cm, with a maximum error in measurement of 0.004 cm, determine the error that might occur when the volume of the tumor is calculated.

A)
B)
C)
D)
E)

Question 19

The supply equation for a certain brand of radio is given by , where p is expressed in dollars/bushel and x, the quantity demanded each year, is measured in billions of bushels. The economists are forecasting a harvest of 1.8 billion bushels for the year, with a maximum error of 15% in their forecast. Determine the corresponding maximum error in the predicted price per bushel of soybeans. Round the answer to the nearest percent.

A)
B)
C)
D)
E)

Question 20

Find the differential of the function.

A)
B)
C)
D)
E)

Question 21

A study prepared for the National Association of Realtors estimates that the number of housing starts per year over the next 5 yr will be million units, where r (percent) is the mortgage rate. Use differentials to estimate the decrease in the number of housing starts when the mortgage rate is increased from 10% to 10.6%. Round to the nearest unit.
Give your answer as a number without the units.

Question 22

A study prepared for the National Association of Realtors estimates that the number of housing starts per year over the next 5 yr will be million units, where r (percent) is the mortgage rate. Use differentials to estimate the decrease in the number of housing starts when the mortgage rate is increased from 17% to 17.8%. Round to the nearest unit.

A) 20,119 housing starts
B) 20,133 housing starts
C) 20,856 housing starts
D) 19,172 housing starts

Question 23

A sociologist has found that the number of serious crimes in a certain city each year is described by the function , where x (in cents/dollar deposited) is the level of reinvestment in the area in conventional mortgages by the city's ten largest banks. Use differentials to estimate the change in the number of crimes if the level of reinvestment changes from 16 /dollar deposited to 25 /dollar deposited. Round to the nearest integer.

A) Decrease of 125 crimes/yr.
B) Decrease of 155 crimes/yr.
C) Decrease of 122 crimes/yr.
D) Decrease of 88 crimes/yr.

Question 24

Use a differential to approximate .
Round the result to five decimal places, if necessary.
Hint: Let and compute dy with x = 25 and dx = 0.02 . __________

Question 25

The supply equation for a certain brand of radio is given by , where x is the quantity supplied and p is the unit price in dollars. Use differentials to approximate the change in price when the quantity supplied is increased from 10,000 units to 10,500 units.
$__________

Question 26

Use a differential to approximate the quantity. Round the result to the nearest thousandth, if necessary. __________

Question 27

Use a differential to approximate the quantity. Round the result to the nearest thousandth, if necessary. __________

Question 28

The length of time (in seconds) a certain individual takes to learn a list of n items is approximated by .
Use differentials to approximate the additional time it takes the individual to learn the items on a list when n is increased from 72 to 78 items. Round to the nearest second.
__________ sec

Question 29

Research done in the 1930s by the French physiologist Jean Poiseuille showed that the resistance R of a blood vessel of length l and radius r is where k is a constant. Suppose a dose of the drug TPA increases r by 22%. How will this affect the resistance R? Assume that l is constant.

A) It will increase by 110%.
B) It will drop by 110%.
C) It will increase by 88%.
D) It will drop by 88%.

Question 30

The length of each edge of a cube is 18 cm, with a possible error in measurement of 0.01 cm. Use differentials to estimate the error that might occur when the volume of the cube is calculated.
Round the result to the nearest hundredth, if necessary.
An error is approximately __________ .

Question 31

The length of each edge of a cube is 19 cm, with a possible error in measurement of 0.01 cm. Use differentials to estimate the error that might occur when the volume of the cube is calculated.

A) An error is approximately
B) An error is approximately
C) An error is approximately
D) An error is approximately

Question 32

Research done in the 1930s by the French physiologist Jean Poiseuille showed that the resistance R of a blood vessel of length l and radius r is , where k is a constant. Suppose a dose of the drug TPA increases r by 14%. How will this affect the resistance R? Assume that l is constant.
It will drop by __________%.

Question 33

A hemisphere-shaped dome of radius 40 ft is to be coated with a layer of rust-proofer before painting. Use differentials to estimate the amount of rust-proofer needed if the coat is to be 0.01 in. thick.
Round the result to the nearest hundredth, if necessary.
Hint: The volume of a hemisphere of radius r is .
__________

Question 34

A hemisphere-shaped dome of radius 30 ft is to be coated with a layer of rust-proofer before painting. Use differentials to estimate the amount of rust-proofer needed if the coat is to be 0.05 in. thick. Hint: The volume of a hemisphere of radius r is .

A)
B)
C)
D)

Question 35

A sociologist has found that the number of serious crimes in a certain city each year is described by the function where x (in cents/dollar deposited) is the level of reinvestment in the area in conventional mortgages by the city's ten largest banks. Use differentials to estimate the change in the number of crimes if the level of reinvestment changes from 27 /dollar deposited to 34 /dollar deposited. Round to the nearest integer.
Decrease of __________ crimes/yr.

Question 36

The length of time (in seconds) a certain individual takes to learn a list of n items is approximated by . Use differentials to approximate the additional time it takes the individual to learn the items on a list when n is increased from 68 to 76 items. Round to the nearest second.

A) 348 sec
B) 385 sec
C) 436 sec
D) 392 sec

Question 37

The volume of a spherical cancerous tumor is given by .
If the radius of a tumor is estimated at 0.7 cm, with a maximum error in measurement of 0.007 cm, determine the error that might occur when the volume of the tumor is calculated. Round the answer to the nearest thousandth, if necessary. __________

Question 38

Use a differential to approximate the quantity. Round the result to the nearest thousandth, if necessary. __________

Question 39

Use a differential to approximate the quantity to the nearest thousandth.

A)
B)
C)
D)
E)

Question 40

A certain country's government economists have determined that the demand equation for soybeans in that country is given by , where p is expressed in dollars/bushel and x, the quantity demanded each year, is measured in billions of bushels. The economists are forecasting a harvest of 1.8 billion bushels for the year, with a maximum error of 15% in their forecast.
Determine the corresponding maximum error in the predicted price per bushel of soybeans. Round the answer to the nearest percent.
__________%

Question 41

Find by implicit differentiation.

A)
B)
C)
D)
E)

Question 42

Find an equation of the tangent line to the graph of the function f defined by the equation at the indicated point.

A)
B)
C)
D)
E)

Question 43

Find the differential of the function.

Question 44

Let f be the function defined by .
Find the differential of f. __________
Find the approximate change in y if x changes from 3 to 3.07. __________
Find the actual change in y if x changes from 3 to 3.07. __________

Question 45

Find the differential of the function.

Question 46

Find by implicit differentiation.

A)
B)
C)
D)

Question 47

Find by implicit differentiation.

A)
B)
C)
D)
E)

Question 48

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

A)
B)
C)
D)

Question 49

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 15 mph. How fast is the distance between them changing at 1 p.m.?

A)
B)
C)
D)
E)

Question 50

Find the second derivative of the function defined implicitly by the equation.

A)
B)
C)
D)
E)

Question 51

Find the differential of the function.

Question 52

Find the differential of the function.

Question 53

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

A)
B)
C)
D)

Question 54

Find the differential of the function.

Question 55

Find the differential of the function.

Question 56

Find the differential of the function.

Question 57

Find the differential of the function.

Question 58

A 30-ft ladder leaning against a wall begins to slide. How fast is the top of the ladder sliding down the wall at the instant of time when the bottom of the ladder is 18 ft from the wall and sliding away from the wall at the rate of 6 ft/sec? Hint: Refer to the accompanying figure. By the Pythagorean theorem, . Find when and .

A)
B)
C)
D)
E)

Question 59

Find the derivative by solving the given implicit equation for y explicitly in terms of x.

A)
B)
C)
D)

Question 60

Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation . If 23,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 800 cartons/week, at what rate is the wholesale price changing? (Hint: To find the value of p when , solve the supply equation for p when ).

A) Increasing at the rate of
B) Increasing at the rate of
C) Dropping at the rate of
D) Dropping at the rate of
E) Dropping at the rate of

Question 61

Find by implicit differentiation.

A)
B)
C)
D)
E)

Question 62

The demand function for a certain brand of compact disc is where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Compute the elasticity of demand and determine whether the demand is inelastic, unitary, or elastic when x = 13.

A) unitary
B) elastic
C) inelastic

Question 63

A spectator watches a rowing race from the edge of a river bank. The lead boat is moving in a straight line that is 90 ft from the river bank. If the boat is moving at a constant speed of 20 ft/sec, how fast is the boat moving away from the spectator when it is 400 ft past her?

A) 26.94 ft/sec
B) 17.28 ft/sec
C) 61.85 ft/sec
D) 19.51 ft/sec

Question 64

Find by implicit differentiation.

A)
B)
C)
D)

Question 65

The demand equation for a certain brand of metal alloy audiocassette tape is , where x represents the number (in thousands) of ten-packs demanded each week when the unit price is $p. How fast is the quantity demanded increasing when the unit price/ten-pack is $16 and the selling price is dropping at the rate of $.15/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 16, solve the equation for x when p = 16.

A) Increasing at the rate of 44 ten packs/wk.
B) Increasing at the rate of 71 ten packs/wk.
C) Increasing at the rate of 60 ten packs/wk.
D) Increasing at the rate of 69 ten packs/wk.

Question 66

Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation .
If 23,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 800 cartons/week, at what rate is the wholesale price changing? Round the answer to the nearest tenth. (Hint: To find the value of p when , solve the supply equation for p when ).
Round the answer to the nearest tenth, if necessary.
__________

Question 67

A spectator watches a rowing race from the edge of a river bank. The lead boat is moving in a straight line that is 330 ft from the river bank. If the boat is moving at a constant speed of 20 ft/sec, how fast is the boat moving away from the spectator when it is 560 ft past her?
Round the answer to the nearest hundredth, if necessary.
__________ ft/sec

Question 68

Find by implicit differentiation.

A)
B)
C)
D)
E)

Question 69

Find by implicit differentiation.

A)
B)
C)
D)

Question 70

Find by implicit differentiation.

A)
B)
C)
D)

Question 71

A 5-ft tall man is walking away from a street light 10 ft high at a speed of 5 ft/sec. How fast is the tip of his shadow moving along the ground?

A) 4 ft/sec
B) 8 ft/sec
C) 3 ft/sec
D) 10 ft/sec

Question 72

Find by implicit differentiation.

A)
B)
C)
D)

Question 73

The base of a 25-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 24 ft from the wall, the base is moving at the rate of 7 ft/sec. How fast is the top of the ladder sliding down the wall at that instant of time?
Round the answer to the nearest tenth, if necessary.
__________ ft/sec

Question 74

A 5-ft tall man is walking away from a street light 10 ft high at a speed of 4 ft/sec . How fast is the tip of his shadow moving along the ground?
__________ ft/sec

Question 75

Find an equation of the tangent line to the graph of the function f defined by the given equation at the point .

A)
B)
C)
D)

Question 76

Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded changing when x = 13, p = 73, and the price/tire is increasing at the rate of $6/week?

A) Dropping at the rate of 278 tires/wk.
B) Dropping at the rate of 231 tires/wk.
C) Dropping at the rate of 226 tires/wk.
D) Dropping at the rate of 268 tires/wk.

Question 77

The volume of a right-circular cylinder of radius r and height h is . Suppose the radius and height of the cylinder are changing with respect to time t. At a certain instant of time, the radius and height of the cylinder are 8 and 2 in. and are increasing at the rate of 0.5 and 0.2 in./sec, respectively. How fast is the volume of the cylinder increasing?

A)
B)
C)
D)

Question 78

The base of a 5-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 4 ft from the wall, the base is moving at the rate of 6 ft/sec. How fast is the top of the ladder sliding down the wall at that instant of time?

A) 5.6 ft/sec
B) 6.5 ft/sec
C) 8 ft/sec
D) 9.3 ft/sec

Question 79

Find by implicit differentiation.

A)
B)
C)
D)

Question 80

In calm waters oil spilling from the ruptured hull of a grounded tanker spreads in all directions. If the area polluted is a circle and its radius is increasing at a rate of 3 ft/sec, determine how fast the area is increasing when the radius of the circle is 70 ft.

A)
B)
C)
D)