Deck 3: Additional Derivative Topics

Question 1

Find f‛(x).

A)
B)
C)
D)

Answer 1

Question 2

Solve the problem.
If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10 years?

A) $8452.29
B) $8442.52
C) $7625.00
D) $7420.65

Answer 2

$8452.29

Question 3

Solve the problem.
How long will it take for the value of an account to be $890 if $350 is deposited at 11% interest compounded continuously? Round your answer to the nearest hundredth.

A) 10.41 yr
B) 0.93 yr
C) 9.33 yr
D) 8.48 yr

Answer 3

8.48 yr

Question 4

Provide an appropriate response.
Find t to four decimal places.

A) -70.1312
B) -66.4815
C) 42.7962
D) 44.321

Question 5

Provide an appropriate response.
Graph the function which calculates the present value of an amount of $5000 at an annual nominal rate of 7% compounded continuously for

A)
B)
C)
D)

Question 6

Find f‛(x).

A)
B)
C)
D)

Question 7

Solve the problem.
What will the value of an account (to the nearest cent) be after 8 years if $100 is invested at 6.0% interest compounded continuously?

A) $159.38
B) $175.32
C) $161.61
D) $849.47

Question 8

Provide an appropriate response.
Find:

A) 5000
B)
C) 0
D) 1

Question 9

Provide an appropriate response.
Find x to two decimal places.

A) 7813.95
B) 7975.01
C) 7831.95
D) 8320.50

Question 10

Provide an appropriate response.
A man with $9000 to invest puts the money into an account that earns 8% compounded continuously. Graph the corresponding present value function and calculate the number of years before the $9000 will be due in
Order for its present value to be $7000. Use the formula

A)
B)
C)
D)

Question 11

Find f‛(x).

A)
B)
C)
D)

Question 12

Find f‛(x).

A)
B)
C)
D)

Question 13

Solve the problem.
An investor buys 100 shares of a stock for $20,000. After 5 years the stock is sold for $32,000. If interest is compounded continuously, what annual nominal rate of interest did the original $20,000 investment earn?
(Represent the answer as a percent to three decimal places.)

A) 8.470%
B) 1.200%
C) 9.400%
D) 0.094%

Question 14

Solve the problem.
Suppose that $8000 is invested at an interest rate of 5.5% per year, compounded continuously. How long would it take to double the investment?

A) 12.6 yr
B) 11.6 yr
C) 2 yr
D) 13.6 yr

Question 15

Find f‛(x).

A)
B)
C)
D)

Question 16

Find f‛(x).

A)
B)
C)
D)

Question 17

Solve the problem.
How long will it take for $8400 to grow to $14.600 at an interest rate of 9.4% if the interest is compounded continuously? Round the number of years to the nearest hundredth.

A) 0.59 yr
B) 0.06 yr
C) 58.81 yr
D) 5.88 yr

Question 18

Provide an appropriate response.
Find t to four decimal places.

A) 2.9134
B) -2.8134
C) 2.6134
D) 2.8134

Question 19

Solve the problem.
Radioactive carbon-14 has a continuous compound rate of decay of r = -0.000124. Estimate the age of a skull uncovered at an archaeological site if 6% of the original amount of carbon-14 is still present. (Compute answer
To the nearest year.)

A) 124,027 yr
B) 20,032 yr
C) 22,689 yr
D) 470 yr

Question 20

Solve the problem.
How long will it take money to double if it is invested at 5.25%, compounded continuously? Round your answer to the nearest tenth.

A) 0.13 yr
B) 14 yr
C) 13.2 yr
D) 26.4 yr

Question 21

Find the equation of the line tangent to the graph of f at the indicated value of x.
f(x) = 5 ln x; x = 1

A) y = 5x - 1
B) y = 5x + 5
C) y = 5x
D) y = 5x - 5

Question 22

Find the equation of the line tangent to the graph of f at the indicated value of x.

A)
B)
C)
D)

Question 23

Find f‛(x).

A)
B)
C)
D)

Question 24

Find f‛(x).

A)
B)
C)
D)

Question 25

Find for the indicated function y.

A)
B)
C)
D)

Question 26

Find the equation of the line tangent to the graph of f at the indicated value of x.

A) y = 2x + 2
B) y = 2x - 2
C) y = x + 2
D) y = 2x

Question 27

Find f‛(x).

A)
B)
C)
D)

Question 28

Find for the indicated function y.

A)
B)
C)
D)

Question 29

Find for the indicated function y.

A)
B)
C)
D)

Question 30

Find for the indicated function y.

A)
B)
C)
D)

Question 31

Find for the indicated function y.

A)
B)
C)
D)

Question 32

Find the equation of the line tangent to the graph of f at the indicated value of x.

A) y = 6x + 5
B) y = 5x + 5
C) y = 6x - 5
D) y = 6x + 6

Question 33

Use appropriate properties of logarithms to rewrite f(x), and then find

A)
B)
C)
D)

Question 34

Find f‛(x).

A)
B)
C)
D)

Question 35

Find the equation of the line tangent to the graph of f at the indicated value of x.
f(x) = 8 + ln x; x = 1

A) y = x + 9
B) y = x - 7
C) y = x + 7
D) y = x - 9

Question 36

Provide an appropriate response.
Use graphical approximation methods to find the point(s) of intersection of to two decimal places.

A) (0.87, 0.42), (1.23, 3.41)
B) (0.44, 0.67), (4.18, 2.04)
C) (-0.87, 0.42)
D) (1.23, 3.41)

Question 37

Provide an appropriate response.
Use graphical approximation methods to find the point(s) of intersection of to two decimal places.

A) (-0.87, 0.42)
B) (-0.87, 0.42), (1.23, 3.41)
C) (0.87, 0.42), (1.23, 3.41)
D) (1.23, 3.41)

Question 38

Find for the indicated function y.

A) 4x - 7 ln 7
B)
C)
D)

Question 39

Find the equation of the line tangent to the graph of f at the indicated value of x.

A) y = 3ex - 6e + 1
B) y = 3ex + 1
C) y = 3ex + 6e + 1
D) y = 6ex - 1

Question 40

Use appropriate properties of logarithms to rewrite f(x), and then find
f(x) = 5x + 4 ln 2x

A)
B)
C)
D)

Question 41

Differentiate.
Find

A)
B)
C)
D)

Question 42

Provide an appropriate response.
Find . Do not simplify.

A)
B)
C)
D)

Question 43

Differentiate.
Find

A)
B)
C)
D)

Question 44

Solve.
A single bacterium divides every 0.5 hour to produce two complete bacteria. If we start with a colony of 6000 bacteria, after t hours there will be

A) 133,084 bacteria
B) 2,129,348 bacteria
C) 8,517,393 bacteria
D) 34,069,570 bacteria

Question 45

Differentiate.
Find

A)
B)
C)
D)

Question 46

Differentiate.
Find

A)
B)
C)
D)

Question 47

Differentiate.
Find

A)
B)
C)
D)

Question 48

Solve.
A single bacterium divides every 0.5 hour to produce two complete bacteria. If we start with a colony of 6000 bacteria, after t hours there will be A . Find A'(t) and A'(1).

A)
B)
C)
D)

Question 49

Differentiate.
Find f'(t) if f(t) = and simplify.

A)
B)
C)
D)

Question 50

Solve.
A mathematical model for the average of a group of people learning to type is given by N(t) = 10 + 6 ln t, t ≥ 1, where N(t) is the number of words per minute typed after t hours of instruction and practice (2 hours per day, 5
Days per week). What is the rate of learning after 50 hours of instruction and practice?

A) 0.1 words per minute typed per hour of instruction and practice
B) 0.15 words per minute typed per hour of instruction and practice
C) 0.5 words per minute typed per hour of instruction and practice
D) 0.12 words per minute typed per hour of instruction and practice

Question 51

Differentiate.
Find f'(t) for

A)
B)
C)
D)

Question 52

Provide an appropriate response.
Find the derivative of the function

A)
B)
C)
D)

Question 53

Differentiate.
Find

A)
B)
C)
D)

Question 54

Solve.
An experiment was set up to find a relationship between weight and systolic blood pressure in normal children. Using hospital records for 5000 normal children, the experimenters found that the systolic blood pressure was
Given approximately by where P(x) is measured in millimeters of mercury and
X is measured in pounds. What is the rate of change of blood pressure with respect to weight at the 70-pound
Weight level?

A) 0.25 mm of mercury per pound of weight gain
B) 0.22 mm of mercury per pound of weight gain
C) 0.29 mm of mercury per pound of weight gain
D) 0.01 mm of mercury per pound of weight gain

Question 55

Differentiate.
Let f and g be functions that satisfy: h(x) = f(x)g(x) - 2f(x) + 7.

A) 5
B) -5
C) 6
D) -6

Question 56

Provide an appropriate response.
Find Do not simplify.

A)
B)
C)
D)

Question 57

Differentiate.
Find

A)
B)
C)
D)

Question 58

Solve.
The resale value R (in dollars) of a company car after t years is estimated to be given by What is the rate of depreciation (in dollars per year) after 5 years?

A) -$15,144/yr
B) -$1641/yr
C) -$1157/yr
D) -$1378/yr

Question 59

Differentiate.
Find

A)
B)
C)
D)

Question 60

Solve.
The salvage value S (in dollars) of a company airplane after t years is estimated to be given by What is the rate of depreciation (in dollars per year) after 5 years?

A) -$14,987/yr
B) -$67,625/yr
C) -$21,409/yr
D) -$42,018/yr

Question 61

Find the derivative.
Find

A)
B)
C)
D)

Question 62

Provide an appropriate response.
Find the values of x where the tangent line is horizontal for the graph of

A) x = 0, x = -2
B) x = -2
C) x = -2, x = 0, x = -4
D) x = 0, x = -4

Question 63

Provide an appropriate response.
Write composite function in the form y = f(u) and u = g(x).

Question 64

Provide an appropriate response.
Find the composition f[g(x)] if f(u) =

A)
B)
C)
D)

Question 65

Provide an appropriate response.
Find

A)
B)
C)
D)

Question 66

Find the derivative.
Find

A)
B)
C)
D)

Question 67

Provide an appropriate response.
Find

A)
B)
C)
D)

Question 68

Find the derivative.
y =

A)
B)
C)
D)

Question 69

Provide an appropriate response.
Find the composite g

A)
B)
C)
D)

Question 70

Solve the problem.
One hour after x milligrams of a particular drug are given to a person, the change in body temperature T(x), in degrees Celsius, is given approximately by: Find the sensitivity, T'(x), of the body to a dosage of three milligrams.

A)
B)
C)
D)

Question 71

Find the derivative.
Find:

A)
B)
C)
D)

Question 72

Provide an appropriate response.
Consider the function: . Choose the answer choice that includes all of the pair(s) of functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).

A)
B)
C)
D) Both A and B

Question 73

Provide an appropriate response.
Find

A)
B)
C)
D)

Question 74

Provide an appropriate response.
Find

A)
B)
C)
D)

Question 75

Provide an appropriate response.
Find

A)
B) 2(10t - 4)
C)
D)

Question 76

Provide an appropriate response.
Find

A)
B)
C)
D)

Question 77

Solve the problem.
A publishing company has published a new magazine for young adults. The monthly sales S (in thousands) is given by where t is the number of months since the first issue was published. Find S(3) and S'(3) and
Interpret the results.

A) At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per month.
B) At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per month.
C) At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per month.
D) At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per month.

Question 78

Find the derivative.
Find

A)
B)
C)
D)

Question 79

Provide an appropriate response.
Consider the function: . Choose the answer choice that includes all of the pair(s) of functions from the list so that f(x) can be written as a composition: f(x) = g(h(x)).

A)
B)
C)
D) Both A and C

Question 80

Provide an appropriate response.
Write composite function in the form y = f(u) and u = g(x).

A)
B)
C)
D)