Deck 7: Exponential Functions

Question 1

Find the domain on which is one-to-one and a formula for the inverse of .

Answer 1

is one-to-one on

Question 2

Calculate where is the inverse of :

A)
B)

Answer 2

A) B)

Question 3

Let with domain . Find

Answer 3

Question 4

Evaluate the integrals:

A)
B)
C)

Question 5

Solve:

Question 6

Find the derivative of the function at the given point:

A)
B)
C)

Question 7

Solve

Question 8

Find and , such that is tangent to at .

Question 9

Find the slope of the function at .

Question 10

Calculate where is the inverse of :

A)
B)

Question 11

Use the definition of the derivative at to find if .

Question 12

Let be the inverse of . Use the geometric relation between the graphs of inverse functions to compute the integral .

Question 13

Let be the inverse of . Use the geometric relation between the graphs of inverse functions to compute the integral .

Question 14

Compute the integral .

Question 15

Let be the inverse of .
Use the geometric relation between the graphs of inverse functions to compute the integral .

Question 16

Solve:

Question 17

Calculate where is the inverse of :

A)
B)

Question 18

Solve:

Question 19

Calculate where is the inverse of :

A)
B)

Question 20

Let be the inverse of . Use the geometric relation between the graphs of inverse functions to compute the integral .

Question 21

Let . Use logarithmic differentiation to calculate .

Question 22

The derivative of is

A)
B)
C)
D)
E)

Question 23

The population in a certain country was 10,900,000 in 1990 and 17,100,000 in 2005. The doubling time of the population is:

A) 52.7 years
B) 53.6 years
C) 31.3 years
D) 23.1 years
E) none of the above

Question 24

Find an equation of the tangent line to the graph of the function for .

Question 25

Let . Calculate .

Question 26

Evaluate the integrals:

A)
B)

Question 27

The population of a country was in 1980, whereas in 1990 it was

A) Find the population in
B) Determine the population's doubling time.
Assume that the rate of growth is proportional to the population.

Question 28

Find the slope of the tangent line to the graph of the function at To find , use the equality or use a calculator.

Question 29

The rate of change of the following function is proportional to .

A)
B)
C)
D)
E)

Question 30

Evaluate the integrals using the given substitutions:

A)
B)

Question 31

Consider the Gompertz differential equation .
Define the function by . The differential equation for is,

A)
B)
C)
D)
E)

Question 32

Evaluate the definite integrals:

A)
B)
C)

Question 33

Let . Use logarithmic differentiation to calculate .

Question 34

A population is modeled by a function . Initially, the population counts 1000. The rate of change of the population is proportional to . Find the population at time years given that the doubling time is 8 years.

Question 35

The population of a country was in and in . The population doubling time is:

A) 25 years
B) 30.1 years
C) 39.5 years
D) 40.1 years
E) none of the above

Question 36

Find if .

Question 37

Compute the derivatives of the following functions:

A)
B)
C)

Question 38

Find all solutions to satisfying .

Question 39

Let . Calculate .

Question 40

Evaluate the integrals:

A)
B)
C)

Question 41

How long will it take for your money to double when it is invested at an annual rate of and compounded yearly? How long will it take for your money to double at an annual rate of ?

Question 42

A glass of cold water with initial temperature C is placed in a room held at C. If the temperature of the water is C after 1 h, what will the temperature of the water be after another hour?

Question 43

A drug leaves a person's bloodstream at a rate proportional to the amount present. Assume that half of the initial amount of the drug is still in the bloodstream 4 hours after the dose was given. When is one-tenth of the initial dose remaining in the bloodstream?

Question 44

A frozen turkey with an initial temperature of F is placed into a hot oven. After 1 h, the temperature of the turkey is F. After 3 h, the temperature is F. What is the temperature of the oven?

Question 45

David has in his account earning interest at an annual rate of compounded weekly. How much money was in his account 2 years ago (1 year = 52 weeks)?

Question 46

Bob deposits to a bank account with an annual interest rate of . The interest is compounded continuously. How much money will Bob earn after 5 years?

A)
B)
C)
D)
E) $1516

Question 47

How much would a 20-year-old have to invest today in order to have $1 million 30 years from now if the interest is compounded continuously at an annual rate of 6.4%?

Question 48

At 6:45 a.m., a glass of cold water with initial temperature C is placed in a room held at C. If the temperature of the water is C at 8:15 a.m., when will the water temperature reach C?

Question 49

A frozen turkey with an initial temperature of C is placed into a hot oven held at C. After 1 h, the temperature of the turkey has risen to C. Give an equation representing the temperature of the turkey as a function of , the number of hours since the turkey was placed in the oven.

Question 50

By Newton's Law of Cooling, an object in air cools down at a rate proportional to the difference between the object's temperature and the air temperature. When the air temperature is C, it takes 10 min for the object to cool down from C to C. The temperature of the object decreases from C to C during:

A) 30.21 min
B) 22.37 min
C) 16.67 min
D) 19.10 min
E) 6.66 min

Question 51

is deposited into an account earning interest at an annual rate of . Find the value of the account after one year if

A) it is compounded twice a year.
B) it is compounded monthly.
C) it is compounded weekly (52 weeks a year)
D) it is compounded daily (365 days a year)
E) it is compounded continuously.
Does the method of compounding have much effect on the final value?

Question 52

Ben has in his account earning interest at an annual rate of compounded monthly. How much money was in his account 3 years ago?

Question 53

Bob's retirement account pays about $120,000 per year, continuously for 20 years. Find the present value of the income stream if the interest rate is 5.5%.

Question 54

Find all solutions to satisfying .

Question 55

How much money will an investment of $12,000 be worth in 25 years if it is invested in a bank earning 5.3% interest compounded quarterly?

Question 56

The population in an ant colony doubles every 3 months. Assuming exponential growth, how many years does it take the population to increase five fold?

Question 57

The decay constant of a 15-kg isotope is 0.137 (years) . Find the half-life of the isotope.

Question 58

How long will it take for your money to double when it is invested at an annual rate of and compounded yearly? How long will it take for your money to double at an annual rate of ?

Question 59

Jane wins the lottery. She has a choice about how to receive her winnings.
Plan A: receive continuous payments of $10,000 a year for the next 20 years.
Plan B: receive $120,000 now and no additional money.
Assume that Jane can earn 6% interest compounded continuously.

A) Compute the present value of the income stream for Plan A.
B) If the yearly amount for Plan A is changed to $10,500, which plan should
Jane use if she bases her decision solely on present value?

Question 60

A cold chicken with an initial temperature of C is placed in an oven held at C. If the temperature of the chicken is C after 10 min, how much longer after that must the chicken remain in the oven before it reaches C?

Question 61

At 7:00 a.m., a mug of hot chocolate with initial temperature of C is placed on a table in a room held at C. At 7:45 a.m., the temperature of the hot chocolate is C. Express the temperature of the hot chocolate as a function of , the number of hours since 7:00 a.m.

Question 62

Apply L'Hopital's Rule to evaluate the limit

A)
B)
C)

Question 63

A 3-kg object is dropped from a tall building. After 1 s, the object has a velocity of m/s. What is the object's terminal velocity?

Question 64

Compute the following limits:

A)
B)

Question 65

The values of and , such that the function is continuous for all are

A)
B)
C)
D)
E)

Question 66

A 2-kg object is dropped from a height of 75 m. After 1 s, the object has a velocity of m/s. What is its velocity after 2 s?

Question 67

A 1-kg object is dropped from a height of 80 m. After 2 s, the velocity of the object is m/s. How long after the ball was dropped did the object have a velocity of m/s?

Question 68

Evaluate the limits using L'Hopital's Rule

A)
B)
C)

Question 69

Find the limits using L'Hopital's Rule

A)
B)

Question 70

A mug of hot chocolate with initial temperature C is placed on a table in a room held at a constant temperature. After 20 min, the temperature of the hot chocolate is C. After another 20 min, the temperature of the hot chocolate is C. What is the temperature of the room?

Question 71

Find the horizontal asymptote(s) of the function .

Question 72

From the top of a tall building, a 2-kg object is thrown vertically downward. One second after being thrown, the object has a velocity of m/s. If the coefficient of air resistance is kg/s, what was the initial velocity of the object?

Question 73

A glass of cold water with initial temperature C is placed in a room held at a constant temperature. After 1 h, the water temperature is C. After another 2 h, the water temperature is C. What is the temperature of the room?

Question 74

From the top of a tall building, a 5-kg object is thrown vertically downward. One second after it is thrown, the object has a velocity of m/s. Two seconds after it is thrown, the object has a velocity of m/s. What was the initial velocity of the object?

Question 75

Find the horizontal asymptote(s) of the function .

Question 76

From the top of a tall building, a 1-kg object is thrown vertically downward. One second after being thrown, the object has a velocity of m/s. Two seconds after being thrown, the object has a velocity of m/s. What is the coefficient of air resistance ?

Question 77

A mug of hot chocolate with initial temperature is placed on a table in a room held at C. After 15 min, the temperature of the hot chocolate is C. After an additional 30 min, the temperature of the hot chocolate is C. What was the initial temperature ?

Question 78

At 7:30 a.m., a mug of hot chocolate with initial temperature of C is placed on a table in a room held at C. At 8:45 a.m., the temperature of the hot chocolate is C. At what time did the temperature of the hot chocolate reach C?

Question 79

Evaluate the limits using L'Hopital's Rule

A)
B)
C)

Question 80

A mug of hot chocolate with initial temperature C is in a room held at C. After 30 min, the temperature of the hot chocolate is C. What is the temperature of the hot chocolate after another 10 min?