Deck 9: Exponential and Logarithmic Functions and Calculus

Question 1

The table lists the approximate value V of a mid-sized sedan for the years 2003 through 2009. The variable t represents the time in years, with corresponding to 2003. Use the regression capabilities of a graphing utility to fit an exponential model to the data. Round all numerical values in your answer to four decimal places. Your answer may be slightly different depending on the number of digits your graphing utility uses in computations.

A)
B)
C)
D)
E)

Answer 1

Question 2

Use implicit differentiation to find .

A)
B)
C)
D)
E)

Answer 2

Question 3

Use implicit differentiation to find .

A)
B)
C)
D)
E)

Answer 3

Question 4

The value V (in dollars) of an item t years after it is purchased is . Find the rate of change of V with respect to t when . Round your answer to two decimal places.

A)-1839.21 dollars/year
B)245.40 dollars/year
C)-1324.10 dollars/year
D)1839.21 dollars/year
E)-245.40 dollars/year

Question 5

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 6

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 7

Differentiate the function .

A)
B)
C)
D)
E)

Question 8

Suppose the table below shows the temperature at which a given liquid boils at selected pressures p (pounds per square inch). A model that approximates the data is . Find the rate of change of T with respect to p when . Round your answer to two decimal places.p51014.696(1 atm)2030406080100T327.43404.29448.84485.71536.18573.59629.02670.61704.38

A)5.34
B)4.25
C)152.22
D)153.31
E)1.09

Question 9

Find the following limit if it exists: . Use when appropriate.

A)
B)does not exist
C)4
D)
E)1

Question 10

Find an equation of the tangent line to the graph of at the point (0,1) .

A)
B)
C)
D)
E)

Question 11

Use logarithmic differentiation to find the derivative of .

A)
B)
C)
D)
E)

Question 12

Suppose the term of a mortgage t (in years) can be approximated by where x is the monthly payment in dollars. Use the model to approximate the term of a home mortgage for which the monthly payment is . What is the total amount paid? Round your answer to the nearest cent. Your answer may be slightly different depending on the number of digits your technology uses in computations.

A)$1284.55
B)$308,293.13
C)$15,414.66
D)$51,382.19
E)$616,586.26

Question 13

Use implicit differentiation to find .

A)
B)
C)
D)
E)

Question 14

Find the derivative of the function .

A)
B)
C)
D)
E)

Question 15

Find the derivative of the function .

A)
B)
C)
D)
E)

Question 16

Find the derivative of the function .

A)
B)
C)
D)
E)

Question 17

Evaluate the definite integral.

A)
B)
C)
D)
E)

Question 18

Find the derivative of the function .

A)
B)
C)
D)
E)

Question 19

Locate any relative extrema and inflection points of the function . Use a graphing utility to confirm your results.

A)relative maximum value: ; inflection point: (0, 0)
B)relative minimum value: ; inflection point: (0, 0)
C)relative minimum value: ; no inflection points
D)relative minimum value: ; no inflection points
E)relative maximum value: ; no inflection points

Question 20

Find the derivative of the function .

A)
B)
C)
D)
E)

Question 21

Find the indefinite integral .

A)
B)
C)
D)
E)

Question 22

Find the area of the region bounded by the graphs of the equations Round your answer to four decimal places.

A)7.2383
B)9.7889
C)10.7889
D)6.5452
E)7.5452

Question 23

Find .

A)
B)
C)
D)
E)

Question 24

The rate of change in sales S is inversely proportional to time measured in weeks. Find S as a function of t if sales after 3 and 9 weeks are 250 units and 450 units respectively.

A)
B)
C)
D)
E)

Question 25

Solve the differential equation.

A)
B)
C)
D)
E)

Question 26

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 27

Find .

A)
B)
C)
D)
E)

Question 28

Find the area of the region given below bounded by the graph of .

A)
B)
C)
D)
E)

Question 29

Suppose a population of bacteria is changing at a rate of where t is the time in days. The initial population (when t = 0) is 1000. Write an equation that gives the population at any time t.

A)
B)
C)
D)
E)

Question 30

Find the indefinite integral.

A)
B)
C)
D)
E)

Question 31

Find given .

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B)
C)
D)
E)

Question 32

Solve the differential equation.

A)
B)
C)
D)
E)

Question 33

Find the time required for an object to cool from by evaluating , where t is time in minutes. Round your answer to four decimal places.

A)7.1728 minutes
B)4.5256 minutes
C)4.9328 minutes
D)6.5208 minutes
E)4.1141 minutes

Question 34

Find if ..

A)
B)
C)
D)
E)

Question 35

Find the indefinite integral of .

A)
B)
C)
D)
E)

Question 36

Find the indefinite integral .

A)
B)
C)
D)
E)

Question 37

Find .

A)
B)
C)
D)
E)

Question 38

Suppose the demand equation for a product is Find the average price p on the interval . Round your answer to the nearest dollar.

A)$67
B)$74
C)$86
D)$106
E)$732

Question 39

Solve the differential equation.

A)
B)
C)
D)
E)

Question 40

Evaluate the definite integral .

A)
B)
C)
D)
E)

Question 41

Solve the differential equation .

A)
B)
C)
D)
E)

Question 42

Suppose that the population (in millions) of Egypt in 2007 was 80.3 and that the expected continuous annual rate of change of the population is 0.017. Find the exponential growth model for the population by letting correspond to 2000. Round your answer to four decimal places.

A)
B)
C)
D)
E)

Question 43

Write and solve the differential equation that models the following verbal statement:The rate of change of y with respect to r is proportional to .

A) ,
B) ,
C) ,
D) ,
E) ,

Question 44

Find the function passing through the point with the first derivative .

A)
B)
C)
D)
E)

Question 45

The rate of change of N is proportional to N. When and when . What is the value of N when ? Round your answer to three decimal places.

A)6,174.815
B)6,224.815
C)382.187
D)6,254.815
E)168,070.000

Question 46

Find the function passing through the point with the first derivative .

A)
B)
C)
D)
E)

Question 47

Solve the differential equation.

A)
B)
C)
D)
E)

Question 48

The number of bacteria in a culture is increasing according to the law of exponential growth. After 2 hours there are 140 bacteria in the culture and after 4 hours there are 370 bacteria in the culture. Answer the following questions, rounding numerical answers to four decimal places.(i) Find the initial population.(ii) Write an exponential growth model for the bacteria population. Let t represent time in hours.(iii) Use the model to determine the number of bacteria after 8 hours.(iv) After how many hours will the bacteria count be 25,000?

A)(i) 52.9730 ; (ii) ; (iii) 2,584.3367 ; (iv) 12.6702 hr
B)(i) 55.1230 ; (ii) ; (iii) 5,271.7880 ; (iv) 17.2327 hr
C)(i) 58.7530 ; (ii) ; (iii) 8,023.2414 ; (iv) 19.3439 hr
D)(i) 60.3130 ; (ii) ; (iii) 10,675.7049 ; (iv) 21.1233 hr
E)(i) 52.9730 ; (ii) ; (iii) 3,897.0708 ; (iv) 15.0047 hr

Question 49

Find the exponential function that passes through the two given points. Round your values of C and k to four decimal places.

A)
B)
C)
D)
E)

Question 50

Write and solve the differential equation that models the following verbal statement. Evaluate the solution at the specified value of the independent variable, rounding your answer to four decimal places:The rate of change of y is proportional to y. When x = 0, y = 4 and when x = 1, y = 16. What is the value of y when x = 7?

A)
B)
C)
D)
E)

Question 51

A container of hot liquid is placed in a freezer that is kept at a constant temperature of 15 F. The initial temperature of the liquid is 170 F. After 4 minutes, the liquid´s temperature is 64 F. How much longer will it take for its temperature to decrease to 32 F? Round your answer to two decimal places.

A)2.21 minutes
B)3.68 minutes
C)3.31 minutes
D)4.04 minutes
E)1.47 minutes

Question 52

The isotope has a half-life of 24,100 years. Given an initial amount of 9 grams of the isotope, how many grams will remain after 2,000 years? After 20,000 years? Round your answers to four decimal places.

A)5.9478 gm, 3.5442 gm
B)3.3988 gm, 2.0253 gm
C)8.4969 gm, 5.0632 gm
D)10.1963 gm, 6.0758 gm
E)5.0981 gm, 3.0379 gm

Question 53

The isotope has a half-life of 5,715 years. After 20,000 years, a sample of the isotope is reduced 0.1 grams. What was the initial size of the sample (in grams)? How large was the sample after the first 2,000 years? Round your answers to four decimal places.

A)0.6786, 0.5324
B)0.5655, 0.4437
C)0.9048, 0.7099
D)1.4703, 1.1536
E)1.1310, 0.8874

Question 54

Suppose that the population (in millions) of a Paraguay in 2007 is 6.7 and that expected continuous annual rate of change of the population is 0.024. The exponential growth model for the population by letting corresponds to 2000 is . Use the model to predict the population of the country in 2014. Round your answer to two decimal places.

A)7.93 million
B)7.74 million
C)8.12 million
D)6.86 million
E)7.03 million