Deck 3: Differentiation

Question 1

Find the differential of the function. ​ ​

A)
B)
C)
D)

Answer 1

Question 2

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) 14%
B) 7%
C) 61%
D) 12%
E) 24%

Answer 2

24%

Question 3

A hemisphere-shaped dome of radius 70 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 0.03 in. thick. ​
Hint: The volume of a hemisphere of radius r is

A) 76.97 ft3
B) 76.54 ft3
C) 77.54 ft3
D) 76.78 ft3

Answer 3

76.97 ft3

Question 4

Find the differential of the function. ​ ​

A)
B)
C)
D)

Question 5

Find the differential of the function. ​ ​

A)
B) ​
C)
D)

Question 6

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 21%. How will this affect the resistance R? Assume that l is constant. ​

A) It will increase by 105%.
B) It will drop by 84%.
C) It will increase by 84%.
D) It will drop by 105%.

Question 7

Let f be the function defined by . Find the approximate change in y if x changes from 2 to 2.05. Find the actual change in y if x changes from 2 to 2.05.
​

A) ,
B) ,
C) ​ ,
D) ​ ,

Question 8

Find the differential of the function. ​ ​

A)
B)
C)
D)

Question 9

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

A)
B)
C)
D)

Question 10

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

A)
B)
C)

Question 11

Find the differential of the function. ​
​ ​

A)
B) ​
C)
D)

Question 12

The length of each edge of a cube is 17 cm, with a possible error in measurement of 0.02 cm. Use differentials to estimate the error that might occur when the volume of the cube is calculated. ​

A) An error is approximately ±16.68 cm³.
B) An error is approximately ±17.34 cm³.
C) An error is approximately ±16.95 cm³.
D) An error is approximately ±17.65 cm³.

Question 13

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) 0.06963
B) 0.13926
C) 2.29783
D) 2.78524
E) 0.00870

Question 14

The volume of a spherical cancerous tumor is given by . ​
If the radius of a tumor is estimated at 1.7 cm, with a maximum error in measurement of 0.003 cm, determine the error that might occur when the volume of the tumor is calculated.

A) ±0.003 cm³
B) ±0.009 cm³
C) ±0.036 cm³
D) ±0.185 cm³
E) ±0.109 cm³

Question 15

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

Question 16

Find the differential of the function. ​ ​

A)
B)
C)
D)

Question 17

Use a differential to approximate . Hint: Let ​ and compute dy with x = 4 and dx = 0.02 .

A) 2.83477
B) 2.57789
C) 1.83793
D) 2.50375

Question 18

Find the differential of the function. ​ ​

A)
B)
C)
D)

Question 19

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

A)
B)
C)

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 17% to 17.8%. Round to the nearest unit. Give your answer as a number without the units.

Question 22

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 20%. How will this affect the resistance R? Assume that l is constant. It will drop by __________%.

Question 23

The volume of a spherical cancerous tumor is given by . If the radius of a tumor is estimated at 0.8 cm, with a maximum error in measurement of 0.006 cm, determine the error that might occur when the volume of the tumor is calculated. Round the answer to the nearest thousandth, if necessary. ± __________ cm³

Question 24

Find the differential of the function.

Question 25

Find the differential of the function.

Question 26

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 0.04 in. thick. Round the result to the nearest hundredth, if necessary.

Question 27

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 28

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 29

Use a differential to approximate . Round the result to five decimal places, if necessary.
​

Question 30

Find the differential of the function.

Question 31

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 17.2%. Round to the nearest unit.

A) 12,567 housing starts
B) 12,819 housing starts
C) 12,362 housing starts
D) 11,956 housing starts

Question 32

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

Question 33

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

Question 34

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 18 /dollar deposited to 26 /dollar deposited. Round to the nearest integer

A)Decrease of 60 crimes/yr.
B)Decrease of 24 crimes/yr.
C)Decrease of 56 crimes/yr.
D)Decrease of 57 crimes/yr.

Question 35

The length of each edge of a cube is 16 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 ± __________ cm³.

Question 36

Find the differential of the function.

Question 37

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

Question 38

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 21 /dollar deposited to 25 /dollar deposited. Round to the nearest integer. Decrease of __________ crimes/yr.

Question 39

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 70 to 72 items. Round to the nearest second. __________ sec

Question 40

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 42 to 45 items. Round to the nearest second.

A) 187 sec
B) 169 sec
C) 193 sec
D) 171 sec

Question 41

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

A)
B)
C)
D)

Question 42

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

A)
B)
C)
D)
E)

Question 43

Find the differential of the function.

Question 44

Find by implicit differentiation. ​ ​

A)
B)
C)
D)

Question 45

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 46

Find by implicit differentiation. ​ ​

A) ​
B)
C)
D)

Question 47

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

A)
B)
C) 2x
D) 2

Question 48

Two ships leave the same port at noon. Ship A sails north at 21 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.? ​

A) mph
B) mph
C) mph
D) 32 mph
E) 24 mph

Question 49

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 = 10. ​

A) elastic
B) inelastic
C) unitary

Question 50

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 3 ft/sec? ​
Hint: Refer to the accompanying figure. By the Pythagorean theorem, . Find when x=18 and

a = 30; b = 3

A) 4ft/sec
B) ft/sec
C) ft/sec
D) 3 ft/ssec

Question 51

Find by implicit differentiation. ​ ​

A)
B)
C)
D)
E)

Question 52

Let f be the function defined by . Find the differential of f. ​
dy = __________ ​
Find the approximate change in y if x changes from 3 to 3.01. ​
dy = __________ ​
Find the actual change in y if x changes from 3 to 3.01.
__________

Question 53

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 54

Find by implicit differentiation. ​ ​

A)
B)
C)
D)

Question 55

Find the differential of the function.

Question 56

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

A)
B)
C)
D)

Question 57

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 25,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 1,000 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 3.6 dollars/carton/wk
B) Increasing at the rate of 3.7 dollars/carton/wk
C) Dropping at the rate of 3.7 dollars/carton/wk
D) Dropping at the rate of 3.6 dollars/carton/wk

Question 58

Find the differential of the function.

Question 59

Find the differential of the function.

Question 60

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 5 ft/sec, determine how fast the area is increasing when the radius of the circle is 40 ft. ​

A) 200 ft²/sec
B) 400π ft²/sec
C) 400 ft²/sec

Question 61

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 (to the nearest hundredth) when x = 15. __________
​Determine whether the demand is inelastic, unitary, or elastic when x = 15. __________

Question 62

Two ships leave the same port at noon. Ship A sails north at 20 mph, and ship B sails east at 12 mph. How fast is the distance between them changing at 1 p.m.?

Question 63

The base of a 52-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 48 ft from the wall, the base is moving at the rate of 9 ft/sec. ​
How fast is the top of the ladder sliding down the wall at that instant of time?

A) 20.9 ft/sec
B) 21.6 ft/sec
C) 26.5 ft/sec
D) 17.2 ft/sec

Question 64

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 = 72, and the price/tire is increasing at the rate of $7/week? Round the answer to the nearest integer. Dropping at the rate of __________ tires/wk.

Question 65

Find by implicit differentiation. ​ ​

A)
B)
C)
D)

Question 66

A 25-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 15 ft from the wall and sliding away from the wall at the rate of 6 ft/sec?
​

Question 67

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 6 and 3 in. and are increasing at the rate of 0.4 and 0.1 in./sec, respectively. How fast is the volume of the cylinder increasing?

A) 18 in³/sec
B) 10.7 in³/sec
C) 22.7 in³/sec
D) 9.4 in³/sec

Question 68

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 27,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 1,000 cartons/week, at what rate is the wholesale price changing? Round the answer to the nearest tenth.

Question 69

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 $10 and the selling price is dropping at the rate of $.13/ten-pack/week? Round the answer to the nearest integer.

Question 70

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) 3 ft/sec
B) 16 ft/sec
C) 5 ft/sec
D) 10 ft/sec

Question 71

The base of a 80-ft ladder leaning against a wall begins to slide away from the wall. At the instant of time when the base is 64 ft from the wall, the base is moving at the rate of 9 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 72

Find by implicit differentiation. ​ ​

A)
B)
C)
D)

Question 73

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 $20 and the selling price is dropping at the rate of $.11/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 20, solve the equation for x when p = 20.

A) Increasing at the rate of 93 ten packs/wk.
B) Increasing at the rate of 82 ten packs/wk.
C) Increasing at the rate of 66 ten packs/wk.
D) Increasing at the rate of 31 ten packs/wk.

​

Question 74

A spectator watches a rowing race from the edge of a river bank. The lead boat is moving in a straight line that is 110 ft from the river bank. If the boat is moving at a constant speed of 10 ft/sec, how fast is the boat moving away from the spectator when it is 600 ft past her? Round the answer to the nearest hundredth, if necessary.
__________ ft/sec

Question 75

Find by implicit differentiation.

Question 76

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

Question 77

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 = 10, p = 67, and the price/tire is increasing at the rate of $4/week?

A) Dropping at the rate of 189 tires/wk.
B) Dropping at the rate of 166 tires/wk.
C) Dropping at the rate of 237 tires/wk.
D) Dropping at the rate of 200 tires/wk.

Question 78

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 50 ft/sec, how fast is the boat moving away from the spectator when it is 560 ft past her? ​

A) 18.63 ft/sec
B) 43.08 ft/sec
C) 61.95 ft/sec
D) 73.01 ft/sec

Question 79

A 6-ft tall man is walking away from a street light 12 ft high at a speed of 6 ft/sec . How fast is the tip of his shadow moving along the ground?
​
__________ ft/sec

Question 80

Find by implicit differentiation. ​ ​

A)
B)
C)
D)