Deck 5: Exponential and Logarithmic Functions

Question 1

According to estimates by Paul Kroger Associates, the percent of households that own VCRs is given by where t is measured in years, with t = 0 corresponding to the beginning of 1985. What percent of households owned VCRs at the beginning of 1985? At the beginning of 1994?

A) 64.1%; 67.0%
B) 3.0%; 62.7%
C) 0.9%; 63.4%
D) 0.9%; 64.1%

Answer 1

3.0%; 62.7%

Question 2

According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by where t is measured in years, with corresponding to 2001. What was the total federal debt in 2004? Round answer to three decimal places.

A)
B)
C)
D)
E)

Answer 2

Question 3

A certain piece of machinery was purchased 4 year(s) ago by Garland Mills for $400,000. Its present resale value is $256,000. Assuming that the machine's resale value decreases exponentially, what will it be 2 year(s) from now?

A) The resale value of the machinery will be $204,789
B) The resale value of the machinery will be $204,911
C) The resale value of the machinery will be $204,800
D) The resale value of the machinery will be $203,689

Answer 3

The resale value of the machinery will be $204,800

Question 4

Skeletal remains of the so-called "Pittsburgh Man", unearthed in Pennsylvania, had lost 86% of the carbon 14 they originally contained. Determine the approximate age of the bones. Hint: the decay constant k for carbon 14 is equal to 0.00012.

A) The approximate age of bones is 14,852 year(s).
B) The approximate age of bones is 12,928 year(s).
C) The approximate age of bones is 16,384 year(s).
D) The approximate age of bones is 15,261 year(s).

Question 5

According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by where t is measured in years, with corresponding to 2001. How fast was the total federal debt increasing in 2002? round answer to three decimal places.

A)
B)
C)
D)
E)

Question 6

Universal Instruments found that the monthly demand for its new line of Galaxy Home Computers t months after placing the line on the market was given by . Select the graph of this function. At what level is the demand expected to stabilize?

A) 2,000 computers
B) 1,900 computers
C) 2,200 computers
D) 2,000 computers
E) 1,900 computers

Question 7

The percent of a certain brand of computer chips that will fail after t years of use is estimated to be . What percent of this brand of computer chips are expected to be usable after 3 years?

A) 52.6%
B) 46.9%
C) 33.7%
D) 45.5%

Question 8

Given that a quantity Q(t) is described by the exponential growth function where t is measured in minutes. What is the growth constant? What quantity is present initially? Complete the table of values. Round your answers to the nearest integer.

A) k = 600; Q = 0.03 t 0 10 20 100
Q 600 815 1,104 12,051
B) k = 0.03; Q = 600 t 0 10 20 100
Q 600 810 1,104 12,162
C) k = 0.03; Q = 600 t 0 10 20 100
Q 600 810 1,093 12,051
D) k = 600; Q = 0.03 t 0 10 20 100
Q 600 810 1,093 12,051

Question 9

Wood deposits recovered from an archeological site contain 25% of the carbon 14 they originally contained. How long ago did the tree from which the wood was obtained die? Hint: the decay constant k for carbon 14 is equal to 0.00012.

A) The tree died 10,429 year(s) ago.
B) The tree died 11,552 year(s) ago.
C) The tree died 13,084 year(s) ago.
D) The tree died 8,096 year(s) ago.

Question 10

Phosphorus 32 has a half-life of 14.2 days. If 100 g of this substance are present initially, find the amount present after t days. What amount will be left after 7.1 days? How fast is the phosphorus 32 decaying when t = 7.1?

A)
B)
C)
D)

Question 11

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percent of teachers who are lay teachers is given by where t is measured in decades, with corresponding to the beginning of 1960. Find the year when the percent of lay teachers was increasing most rapidly.

A) 1970
B) 1983
C) 1976
D) 2000
E) 1962

Question 12

During a flu epidemic, the number of children in the Woodbridge Community School System who contracted influenza after t days was given by . How many children were stricken by the flu after the first day? How many children had the flu after 10 days? How many children eventually contracted the disease?

A) 13 children were stricken by the flu after the first day; 1,264 children had the flu after 10 days; 883 children eventually contracted the disease
B) 10 children were stricken by the flu after the first day; 1,264 children had the flu after 10 days; 1,200 children eventually contracted the disease
C) 13 children were stricken by the flu after the first day; 1,125 children had the flu after 10 days; 1,200 children eventually contracted the disease
D) 15 children were stricken by the flu after the first day; 1,125 children had the flu after 10 days; 1,354 children eventually contracted the disease

Question 13

On the basis of data collected during an experiment, a biologist found that the growth of the fruit fly (Drosophila) with a limited food supply could be approximated by the exponential model where t denotes the number of days since the beginning of the experiment. How fast was the population changing on the 20th day?

A) 19 flies/day
B) 17 flies/day
C) 16 flies/day
D) 20 flies/day
E) 18 flies/day

Question 14

The radioactive element polonium decays according to the law where is the initial amount and the time t is measured in days. If the amount of polonium left after 560 days is 40 mg, what was the initial amount present?

A) The initial amount was 642 mg
B) The initial amount was 640 mg
C) The initial amount was 635 mg
D) The initial amount was 629 mg

Question 15

Given that a quantity Q(t) exhibiting exponential decay is described by the function where t is measured in years. What is the decay constant? What quantity is present initially? Complete the table of values. Round your answer to the nearest integer.

A) k = - 0.01; Q = 2,500 t 0 5 10 20 100
Q 2,500 2,378 2,262 2,047 920
B) k = - 0.01; Q = 2,500 t 0 5 10 20 100
Q 2,500 2,378 2,207 2,047 925
C) k = 2,500; Q = - 2,500 t 0 5 10 20 100
Q 2,500 2,489 2,207 2,058 925
D) k = 2,500; Q = - 2,500 t 0 5 10 20 100
Q 2,500 2,489 2,262 2,058 920

Question 16

The American Court Reporting Institute finds that the average student taking Advanced Machine Shorthand, an intensive 20-week course, progresses according to the function , , where Q(t) measures the number of words (per minute) of dictation that the student can take in machine shorthand after t weeks in the course. Sketch the graph of the function Q and answer the following questions:
A.What is the beginning shorthand speed for the average student in this course?
B.What shorthand speed does the average student attain halfway through the course?
C.How many words per minute can the average student take after completing this course?

A) a.75 words/min b.126 words/min c.157 words/min
B) a.65 words/min b.116 words/min c.147 words/min
C) a.55 words/min b.106 words/min c.137 words/min

Question 17

The length (in centimeters) of a typical Pacific halibut t years old is approximately . What is the length of a typical 6-year-old Pacific halibut? How fast is the length of a typical 6-year-old Pacific halibut increasing? What is the maximum length a typical Pacific halibut can attain?

A) 151.2 cm; 0.957 cm/yr; 226 cm;
B) 138.1 cm; 11.6 cm/yr; 199 cm;
C) 199.0 cm; 13.7 cm/yr; 253 cm;
D) 108.3 cm; 12.3 cm/yr; 199 cm;

Question 18

Strontium 90, a radioactive isotope of strontium, is present in the fallout resulting from nuclear explosions. It is especially hazardous to animal life, including humans, because, upon ingestion of contaminated food, it is absorbed into the bone structure. Its half-life is 27 years. If the amount of strontium 90 in a certain area is found to be four times the "safe" level, find how much time must elapse before an "acceptable level" is reached.

A) 30.3 year(s) must elapse before an "acceptable level" is reached.
B) 54 year(s) must elapse before an "acceptable level" is reached.
C) 57 year(s) must elapse before an "acceptable level" is reached.
D) 39 year(s) must elapse before an "acceptable level" is reached.

Question 19

If the temperature is constant, then the atmospheric pressure P (in pounds/square inch) varies with the altitude above sea level h in accordance with the law where is the atmospheric pressure at sea level and k is a constant. If the atmospheric pressure is at sea level and at 5,000 ft, how fast is the atmospheric pressure changing with respect to altitude at an altitude of 15,000 ft? Give the answer to four decimal places.

A)
B)
C)
D)
E)

Question 20

The growth rate of the bacterium, a common bacterium found in the human intestine, is proportional to its size. Under ideal laboratory conditions, when this bacterium is grown in a nutrient broth medium, the number of cells in a culture doubles approximately every 20 min. If the initial cell population is 100, determine the function Q(t) that expresses the exponential growth of the number of cells of this bacterium as a function of time t (in minutes). How long will it take for a colony of 100 cells to increase to a population of 1 million? If the initial cell population were 1,000, how would this alter the model?

A)
B)
C)
D)

Question 21

The concentration of a drug in grams/cubic centimeter (g/cm3) t minutes after it has been injected into the bloodstream is given by where u, l, and k are positive constants, with l > u. At what time is the concentration of the drug the greatest? What will be the concentration of the drug in the long run?

A)
B)
C)
D)

Question 22

Skeletal remains of the so-called "Pittsburgh Man", unearthed in Pennsylvania, had lost 86% of the carbon 14 they originally contained. Determine the approximate age of the bones. Round your answer to the nearest year. Hint: the decay constant k for carbon 14 is equal to 0.00012.
__________ years

Question 23

The U.S. population is approximated by the function where P(t) is measured in millions of people and t is measured in 30-year intervals, with t = 0 corresponding to 1930. What is the expected population of the United States in 2050 ( t = 4 )?

A) 616.6 millions
B) 398.6 millions
C) 401.6 millions
D) 404.0 millions

Question 24

Wood deposits recovered from an archeological site contain 22% of the carbon 14 they originally contained. How long ago did the tree from which the wood was obtained die? Round your answer to the nearest year. Hint: the decay constant k for carbon 14 is equal to 0.00012.
__________ years

Question 25

The length (in centimeters) of a typical Pacific halibut t years old is approximately . What is the length of a typical 6-year-old Pacific halibut? How fast is the length of a typical 6-year-old Pacific halibut increasing? What is the maximum length a typical Pacific halibut can attain? Round your answers to the nearest tenth.

Question 26

If the temperature is constant, then the atmospheric pressure P (in pounds/square inch) varies with the altitude above sea level h in accordance with the law , where is the atmospheric pressure at sea level and k is a constant.
If the atmospheric pressure is 15 at sea level and 13.5 at 5,000 ft, find the atmospheric pressure at an altitude of 15,000 ft. Please round the answer to one decimal place.
__________ How fast is the atmospheric pressure changing with respect to altitude at an altitude of 15,000 ft? Please round the answer to four decimal places.
__________

Question 27

The length (in centimeters) of a common commercial fish is approximated by the von-Bertalanffy growth function where a, b, and k are positive constants. Select the graph of f.

A)
B)
C)
D)
E)

Question 28

The radioactive element polonium decays according to the law , where is the initial amount and the time t is measured in days. If the amount of polonium left after 420 days is 80 mg, what was the initial amount present?
__________ mg

Question 29

Suppose a radioactive substance decays according to the formula . How long will it take for the substance to decay to half the original amount?

A) It will take 40 years for the substance to decay to half the original amount.
B) It will take 5,170 years for the substance to decay to half the original amount.
C) It will take 5,281 years for the substance to decay to half the original amount.
D) It will take 5,626 years for the substance to decay to half the original amount.

Question 30

On the basis of data collected during an experiment, a biologist found that the growth of the fruit fly (Drosophila) with a limited food supply could be approximated by the exponential model , where t denotes the number of days since the beginning of the experiment.
What was the initial fruit fly population in the experiment? __________
What was the maximum fruit fly population that could be expected under this laboratory condition? __________
What was the population of the fruit fly colony on the 20th day? Please give the answer as an integer. __________
How fast was the population changing on the 20th day? Please give the answer as an integer. __________

Question 31

A certain piece of machinery was purchased 4 years ago by Garland Mills for $300,000. Its present resale value is $192,000. Assuming that the machine's resale value decreases exponentially, what will it be 3 years from now? Round your answer to the nearest dollar.
$__________

Question 32

The change from religious to lay teachers at Roman Catholic schools has been partly attributed to the decline in the number of women and men entering religious orders. The percent of teachers who are lay teachers is given by where t measured in decades, with corresponding to the beginning of 1960.
What percent of teachers were lay teachers at the beginning of 1990? Please round the answer to one decimal place.
__________ %
How fast was the percent of lay teachers changing at the beginning of 1990? Please round the answer to two decimal places.
__________ % / year
Find the year when the percent of lay teachers was increasing most rapidly.

Question 33

Strontium 90, a radioactive isotope of strontium, is present in the fallout resulting from nuclear explosions. It is especially hazardous to animal life, including humans, because, upon ingestion of contaminated food, it is absorbed into the bone structure. Its half-life is 27 years. If the amount of strontium 90 in a certain area is found to be four times the "safe" level, find how much time must elapse before an "acceptable level" is reached.
__________ years

Question 34

Find the derivative of the function.

A)
B)
C)
D)

Question 35

The percent of a certain brand of computer chips that will fail after t years of use is estimated to be .
What percent of this brand of computer chips are expected to be usable after 5 years? Round your answer to one decimal place.
__________%

Question 36

The U.S. population is approximated by the function , where P(t) is measured in millions of people and t is measured in 30-year intervals, with t = 0 corresponding to 1930. What is the expected population of the United States in 2050 (t = 4)? Round your answer to one decimal place.
__________ million

Question 37

Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The number of students who learned of the new program t hours later is given by the function . If 600 students on campus had heard about the new program 2 hours after the ceremony, how many students had heard about the policy after 4 hours?

A) 975
B) 970
C) 980
D) 960
E) 965

Question 38

During a flu epidemic, the number of children in the Woodbridge Community School System who contracted influenza after t days was given by .
How many children were stricken by the flu after the first day? Round your answer to the nearest integer.
__________ children
How many children had the flu after 10 days? Round your answer to the nearest integer.
__________ children
How many children eventually contracted the disease? Round your answer to the nearest integer.
__________ children

Question 39

Three hundred students attended the dedication ceremony of a new building on a college campus. The president of the traditionally female college announced a new expansion program, which included plans to make the college coeducational. The number of students who learned of the new program t hours later is given by the function .
If 600 students on campus had heard about the new program 2 hours after the ceremony, how many students had heard about the policy after 4 hours? __________
How fast was the news spreading 4 hours after the ceremony? Round the answer to the nearest integer. __________

Question 40

According to estimates by Paul Kroger Associates, the percent of households that own VCRs is given by , where t is measured in years, with t = 0 corresponding to the beginning of 1985.
What percent of households owned VCRs at the beginning of 1985? Round your answer to the nearest tenth.
__________%
At the beginning of 1993?. Round your answer to the nearest tenth.
__________%

Question 41

Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)
E)

Question 42

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 43

Find the derivative of the function.

A)
B)
C)
D)

Question 44

Find the derivative of the function.

A)
B)
C)
D)

Question 45

Find the derivative of the function.

A)
B)
C)
D)

Question 46

Find the derivative of the function.

A)
B)
C)
D)

Question 47

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 48

Find the second derivative of the function.

A)
B)
C)
D)

Question 49

Find the second derivative of the function.

A)
B)
C)
D)

Question 50

Find the derivative of the function.

A)
B)
C)
D)
E)

Question 51

Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)

Question 52

Find the derivative of the function.

A)
B)
C)
D)

Question 53

Find an equation of the tangent line to the graph of at the point .

A)
B)
C)
D)

Question 54

Find the derivative of the function.

A)
B)
C)
D)

Question 55

Find the derivative of the function.

A)
B)
C)
D)

Question 56

Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)

Question 57

Find the second derivative of the function.

A)
B)
C)
D)
E)

Question 58

Find the derivative of the function.

A)
B)
C)
D)

Question 59

Find the derivative of the function.

A)
B)
C)
D)

Question 60

Use logarithmic differentiation to find the derivative of the function.

A)
B)
C)
D)
E)

Question 61

For children between the ages of 5 and 13 years old, the Ehrenberg equation gives the relationship between the weight W (in kilograms) and the height h (in meters) of a child. Use differentials to estimate the change in the weight of a child who grows from 0.8 m to 1 m. Round the answer to the nearest tenth.
__________ kg

Question 62

Find the derivative of the function.

Question 63

Find the derivative of the function.

Question 64

Find the inflection points of the function .

A)
B)
C)
D)
E)

Question 65

Find an equation of the tangent line to the graph of at its inflection point.

A)
B)
C)
D)
E)

Question 66

Find the inflection points of the function .

A)
B)
C)
D) There are no inflection points.

Question 67

The strain (percent of compression) on the lumbar vertebral disks in an adult human as a function of the load x (in kilograms) is given by . What is the rate of change of the strain with respect to the load when the load is 300 kg? When the load is 450 kg? Give the answers to four decimal places.

A)
B)
C)
D)

Question 68

For children between the ages of 5 and 13 years old, the Ehrenberg equation gives the relationship between the weight W (in kilograms) and the height h (in meters) of a child. Use differentials to estimate the change in the weight of a child who grows from 1.1 m to 1.4 m.

A) kg
B) kg
C) kg
D) kg
E) kg

Question 69

Find the derivative of the function.

Question 70

Find the derivative of the function.

Question 71

Find the derivative of the function.

Question 72

Find the derivative of the function.

Question 73

Find the derivative of the function.

Question 74

Find the derivative of the function.

Question 75

Find the derivative of the function.

Question 76

Use the curve-sketching guideline, to select the graph of the function.

A)
B)
C)

Question 77

On the Richter scale, the magnitude R of an earthquake is given by the formula where I is the intensity of the earthquake being measured and is the standard reference intensity. Suppose an earthquake is measured with a magnitude of 6 on the Richter scale with an error of at most 2%. Use differentials to find the error in the intensity of the earthquake. Round your answer to the nearest integer.

A)
B)
C)
D)
E)

Question 78

Find the derivative of the function.

Question 79

Find the derivative of the function.

Question 80

Find the derivative of the function.