Deck 10: Limits and the Derivative

Question 1

Find f ' (x).

A)
B)
C)
D)

Answer 1

Question 2

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)$161.61
C)$849.47
D)$175.32

Answer 2

$161.61

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

Answer 3

8.48 yr

Question 4

Find f ' (x).

A) -6xe^x-1 + 7
B) -6e^x + 3
C) -6e^x + 7x
D) -6e^x + 7

Question 5

Find f ' (x).

A) - 4x³
B) - 4x³
C) - 4x
D) - 4x³

Question 6

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

Question 7

Find f ' (x).

A)
B)
C)
D)

Question 8

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

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

Question 9

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

Question 10

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)26.4 yr
C)13.2 yr
D)14 yr

Question 11

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

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

Question 12

Provide an appropriate response.
Find

A) 1
B) 5000
C) ∞
D) 0

Question 13

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 P = Ae-rt.

A) 3.14 years
B) 6.28 years
C) 7.45 years
D) 0.84 years

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

Question 15

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

Question 16

Find f ' (x).

A)
B)
C)
D)

Question 17

Find f ' (x).

A)
B)
C)
D)

Question 18

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 0 c t c 10.Use the formula P = Ae-rt.

A)
B)
C)
D)

Question 19

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

A) 2.6134
B) 2.8134
C) -2.8134
D) 29134

Question 20

Provide an appropriate response.
Find t to four decimal places.e -0.07t = 0.05

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

Question 21

Differentiate.
Find f'(x) for f(x) = (2x - 4) (2x³ - x² + 1).

A) f'(x) = 16x³ - 10x² + 30x + 2
B) f'(x) = 12x³ + 30x² - 10x + 2
C) f'(x) = 16x³ - 30x² + 8x + 2
D) f'(x) = 4x³ - 10x² - 30x + 2

Question 22

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

A) y = 2ex - 4e + 1
B) y = 2ex + 4e + 1
C) y = 2ex + 1
D) y = 4ex - 1

Question 23

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 3 years?

A) -$43,693/yr
B) -$30,585/yr
C) -$85,750/yr
D) -$94,206/yr

Question 24

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

A) y = 5x - 4
B) y = 4x + 4
C) y = 5x + 4
D) y = 5x + 5

Question 25

Solve.
A mathematical model for the average of a group of people learning to type is given by 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) 0.5 words per minute typed per hour of instruction and practice
B) 0.12 words per minute typed per hour of instruction and practice
C) 0.1 words per minute typed per hour of instruction and practice
D) 0.15 words per minute typed per hour of instruction and practice

Question 26

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 P(x)= 17.5(1 + ln x),10 ≤ x ≤ 100,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 80-pound weight level?

A)0.19 mm of mercury per pound of weight gain
B)0.01 mm of mercury per pound of weight gain
C)0.22 mm of mercury per pound of weight gain
D)0.25 mm of mercury per pound of weight gain

Question 27

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(t) = 6000 · 2^2t = 6000 · 4^t bacteria. Find A'(t) and A'(1).

A) A'(t) = 6000(ln 4)4^t; A'(1) = 33,271 bacteria
B) A'(t) = 6000(ln 2)2^t; A'(1) = 8317 bacteria
C) A'(t) = 6000(ln 4)2^t; A'(1) = 16,635 bacteria
D) A'(t) = 6000(ln 2)4^t; A'(1) = 16,635 bacteria

Question 28

Find f ' (x).
f(x) = 4 ln x + ln x⁷ + 2e^x

A)
B)
C)
D)

Question 29

Find the equation of the line tangent to the graph of f at the indicated value of x.
f(x) = 9 ln x; x = 1

A) y = 9x + 9
B) y = 9x
C) y = 9x - 9
D) y = 9x - 1

Question 30

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 4 years?

A) -$1641/yr
B) -$1953/yr
C) -$15,529/yr
D) -$1378/yr

Question 31

Find the equation of the line tangent to the graph of f at the indicated value of x.
f(x) = 2 + ln x⁶; x = e

A)
B)
C)
D)

Question 32

Differentiate.
Find f'(t) for f(x) = (3x - 3) (3x³ - x² + 1)

A) f'(x) = 27x³ + 36x² - 12x + 3
B) f'(x) = 9x³ + 12x² - 36x + 3
C) f'(x) = 36x³ - 36x² + 6x + 3
D) f'(x) = 36x³ - 12x² + 36x + 3

Question 33

Find the equation of the line tangent to the graph of f at the indicated value of x.
f(x) = 6 + ln x; x = 1

A) y = x - 5
B) y = x - 7
C) y = x + 7
D) y = x + 5

Question 34

Provide an appropriate response.
Use graphical approximation methods to find the point(s) of intersection of f(x) = (ln x)² and to two decimal places.

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

Question 35

Provide an appropriate response.
Use graphical approximation methods to find the point(s) of intersection of f(x) = ex and g(x) = x6 to two
decimal places.

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

Question 36

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

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

Question 37

Differentiate.
Find f'(x) for f(x) = (5x³ + 4)(3x⁷ - 5).

A) f'(x) = 150x⁹ + 84x⁶ - 75x²
B) f'(x) = 20x⁹ + 84x⁶ - 75x
C) f'(x) = 150x⁹ + 84x⁶ - 75x
D) f'(x) = 20x⁹ + 84x⁶ - 75x²