Deck 11: Derivatives of Exponential and Logarithmic Functions

Question 1

Use a graphing utility to graph the function $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } = 2 ^ { - x }$ .

A)
B)
C)
D)
E)

Answer 1

Question 2

Determine the continuity of the function below. $f ( x ) = \frac { e ^ { x } + e ^ { - x } } { 2 }$

A)discontinuous at x = 0
B)continuous on the entire real number line
C)discontinuous at x = 1
D)discontinuous at x = 2
E)discontinuous at x = 4

Answer 2

continuous on the entire real number line

Question 3

With an annual rate of inflation of 4% over the next 10 years, the approximate cost of goods or services during any year in the decade is given by $C ( t ) = P ( 1.04 ) ^ { t } , 0 \leq t \leq 10$ where is the time (in years) and is the present cost. The price of an oil change for a car is presently $24.95.Estimate the price 10 years from now.

A)$37.09
B)$36.93
C)$89.00
D)$63.90

Answer 3

$36.93

Question 4

How much more interest will be earned if $6000 is invested for 6 years at an annual rate of 9% compounded continuously, instead of at 9% compounded quarterly?

A)$20.72
B)$40.72
C)$61.44
D)$994.60
E)$1035.32

Question 5

Sketch the graph of the function $f ( x ) = e ^ { 3 x + 2 }$ .

A)
B)
C)
D)
E)

Question 6

Use a graphing utility to graph the function $g ( x ) = \frac { 11 } { 1 + e ^ { - x } }$ . Be sure to choose an appropriate viewing window.

A)
B)
C)
D)
E)

Question 7

What lump sum should be deposited in an account that will earn at an annual rate of 12%, compounded quarterly, to grow to $90,000 for retirement in 15 years?

A)$87,176.69
B)$15,010.50
C)$10,975.61
D)$32,142.86
E)$15,275.98

Question 8

Assume the population P (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function $P ( t ) = 255.82 ( 1.606 ) ^ { t }$ , where t is the time in years, with t = 2 corresponding to1992. Use the model to estimate the population in the year 2006. Round your answer to the nearest million.

A)4389 million
B)2733 million
C)7049 million
D)660 million
E)4388 million

Question 9

Use a graphing utility to graph the function $f ( x ) = 3 ^ { - x ^ { 2 } }$ .

A)
B)
C)
D)
E)

Question 10

Sketch the graph of the function $f ( x ) = e ^ { 3 x }$ .

A)
B)
C)
D)
E)

Question 11

Determine whether the function below has any horizontal asymptotes. $f ( x ) = \frac { e ^ { x } - e ^ { - x } } { 2 }$

A)horizontal asymptotes: y = 0 and y = 2
B)no horizontal asymptotes
C)horizontal asymptotes: y = 1
D)horizontal asymptotes: y = 1 and y = 3
E)horizontal asymptotes: y = 3

Question 12

After t years, the remaining mass y(in grams) of 16 grams of a radioactive element whose half-life is 32 years is given by $y = 16 \left( \frac { 1 } { 2 } \right) ^ { t / 32 }$ , for $t \geq 0$ . How much of the initial mass remains after 96 years? Round your answer to two decimal places.

A)4.00 grams
B)3.20 grams
C)4.30 grams
D)4.90 grams
E)2.00 grams

Question 13

Use the properties of exponents to simplify the expression $\left( e ^ { - 4 } \right) \left( e ^ { - 7 / 2 } \right)$ .

A) $e ^ { - 14 }$
B) $e ^ { 55 }$
C) $e ^ { 14 }$
D) $e ^ { - 55 }$
E) $e ^ { 16 }$

Question 14

Use the properties of exponents to simplify the expression $\left[ \left( 7 ^ { - 1 } \right) \left( 7 ^ { \frac { 3 } { 5 } } \right) \right] ^ { 5 }$ .

A) $\frac { 1 } { 343 }$
B)343
C) $\frac { 1 } { 16807 }$
D) $\frac { 1 } { 49 }$
E)49

Question 15

Sketch the graph of the function $f ( x ) = 3 ^ { x }$ .

A)
B)
C)
D)
E)

Question 16

Suppose that the annual rate of inflation averages 4% over the next 10 years. With this rate of inflation, the approximate cost C of goods or services during any year in that decade will be given by C(t) = P(1.04)t, 0 $\leq t \leq$ 10 where t is time in years and P is the present cost. If the price of an oil change for your car is presently $\$$ 24.95, estimate the price 9 years from now. Round your answer to two decimal places.

A) $\$$ 36.93
B) $\$$ 37.51
C) $\$$ 38.93
D) $\$$ 40.51
E) $\$$ 35.51

Question 17

What is the resulting balance if $5700 is invested for 6 years at an annual rate of 7% compounded monthly?

A) $\$$ 5902.43
B) $\$$ 5936.86
C) $\$$ 8094.00
D) $\$$ 8664.60
E) $\$$ 11,811.92

Question 18

After t years, the value of a car that originally cost $\$$ 17,000 depreciates so that each year it is worth $\frac { 3 } { 4 }$ of its value for the previous year. Find a model for V(t), the value of the car after t years.

A)V(t)= 17,000 $\left( \frac { 3 ^ { t } } { 4 } \right)$
B)V(t)= 17,000t $\left( \frac { 3 } { 4 } \right)$
C)V(t)= 17,000 $\left( \frac { 3 } { 4 } \right) ^ { t }$
D)V(t)= 17,000 $\left( \frac { 3 } { 4 ^ { t } } \right)$
E)V(t)= 17,000t $\left( \frac { 3 } { 4 } \right) ^ { t }$

Question 19

To help their son buy a car on his 19th birthday, a boy's parents invest $1400 on his 10th birthday. If the investment pays an annual rate of 9% compounded continuously, how much is available on his 19th birthday?

A)$3118.94
B)$3147.07
C)$3040.65
D)$2534.00
E)$35,747.21

Question 20

Evaluate the expression $256 ^ { \frac { 2 } { 4 } }$ .

A)256
B)4
C)16
D)18
E)20

Question 21

What is the annual percentage yield (or effective annual rate) for a nominal rate of 9% compounded quarterly?

A)9.00%
B)9.38%
C)9.42%
D)9.31%
E)9.20%

Question 22

Use implicit differentiation to find $\frac { d y } { d x }$ . $9 e ^ { xy } + y ^ { 2 } = 15$

A) $\frac { d y } { d x } = \frac { 9 y e ^ { xy } } { 9 e ^ { x y } + 2 y }$
B) $\frac { d y } { d x } = \frac { 9 y e ^ { xy } } { 9 e ^ { x y } - 2 y }$
C) $\frac { d y } { d x } = - \frac { 9 e ^ { x y } } { 9 e ^ { xy } x - 2 y }$
D) $\frac { d y } { d x } = - \frac { 9 y e ^ { xy } } { 9 e ^ { xy } x + 2 y }$
E) $\frac { d y } { d x } = - \frac { 9 y e ^ { x y } } { 9 e ^ { xy } x + y }$

Question 23

Find the derivative of the following function. $y = 8 x ^ { 3 } - 6 e ^ { x }$

A) $y ^ { \prime } = 24 x ^ { 2 } - 6 x e ^ { x - 1 }$
B) $y ^ { \prime } = 8 x ^ { 2 } - 6 x e ^ { x - 1 }$
C) $y ^ { \prime } = 8 x ^ { 2 } - 6 e ^ { x }$
D) $y ^ { \prime } = 24 x ^ { 2 } - e ^ { x }$
E) $y ^ { \prime } = 24 x ^ { 2 } - 6 e ^ { x }$

Question 24

Find the future value if $5000 is invested for 2 years at an annual rate of 10% compounded quarterly.

A) $\$$ 4000.00
B) $\$$ 6077.53
C) $\$$ 5253.12
D) $\$$ 6050.00
E) $\$$ 6092.01

Question 25

Find $\frac { d y } { d x }$ if $y = e ^ { 5 x ^ { 5 } }$ .

A) $\frac { d y } { d x } = 5 e ^ { 5 x }$
B) $\frac { d y } { d x } = e ^ { 5 x ^ { 5 } }$
C) $\frac { d y } { d x } = 5 x ^ { 5 } e ^ { 5 x ^ { 5 } - 1 }$
D) $\frac { d y } { d x } = 25 x ^ { 4 } \ln \left( 5 ^ { 5 } \right)$
E) $\frac { d y } { d x } = 25 x ^ { 4 } e ^ { 5 x ^ { 5 } }$

Question 26

Find the equation of the tangent line to $f ( x ) = 9 x + e ^ { x }$ at the point (0,1).

A) $y = - 10 x - 10$
B) $y = 10 x - 10$
C) $y = 10 x - 1$
D) $y = 10 x + 1$
E) $y = - 10 x - 1$

Question 27

Find the derivative of $f ( x ) = x ^ { - 3 } - 10 e ^ { x }$

A) $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 e ^ { x }$
B) $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x - 1 }$
C) $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 e ^ { x }$
D) $f ^ { \prime } ( x ) = 3 x ^ { 2 } - 10 x e ^ { x - 1 }$
E) $f ^ { \prime } ( x ) = - \frac { 3 } { x ^ { 4 } } - 10 x e ^ { x }$

Question 28

If $x ^ { 5 } y = 3 e ^ { x + y } , \text { find } d y / d x$

A) $\frac { d y } { d x } = \frac { 3 e ^ { x + y } - 5 x ^ { 5 } y } { x ^ { 5 } + 3 e ^ { x + y } }$
B) $\frac { d y } { d x } = \frac { 3 e ^ { x + y } - 5 x ^ { 4 } y } { x ^ { 5 } - 3 e ^ { x + y } }$
C) $\frac { d y } { d x } = \frac { 3 e ^ { x + y } + x ^ { 4 } y } { x ^ { 5 } - 3 e ^ { x + y } }$
D) $\frac { d y } { d x } = \frac { e ^ { x + y } - 5 x ^ { 4 } y } { x ^ { 5 } - e ^ { x + y } }$
E) $\frac { d y } { d x } = \frac { 3 e ^ { x + y } + x ^ { 5 } y } { x ^ { 5 } - e ^ { x + y } }$

Question 29

If $4 x + e ^ { 3 x y } = 9 , \text { find } d y / d x$

A) $\frac { d y } { d x } = - \frac { 4 + 3 y e ^ { 3 xy } } { 3 x e ^ { 3 xy } }$
B) $\frac { d y } { d x } = - \frac { 4 + 3 x e ^ { 3 xy } } { 3 y e ^ { 3 xy } }$
C) $\frac { d y } { d x } = \frac { 4 + 3 y e ^ { 3 x y } } { 4 x e ^ { 3 xy } }$
D) $\frac { d y } { d x } = \frac { 4 + 3 x e ^ { 3 x y } } { 3 y e ^ { 3 x y } }$
E) $\frac { d y } { d x } = \frac { 4 + 3 y e ^ { 3 x y } } { x e ^ { 3 x y } }$

Question 30

Find the derivative of the following function. $y = 5 e ^ { 3 x ^ { 2 } - 3 }$

A) $y ^ { \prime } = 15 x e ^ { 3 x ^ { 2 } - 3 }$
B) $y ^ { \prime } = 30 x e ^ { 3 x ^ { 2 } - 3 }$
C) $y ^ { \prime } = 30 e ^ { 3 x ^ { 2 } - 3 }$
D) $y ^ { \prime } = 15 e ^ { 3 x ^ { 2 } - 3 }$
E) $y ^ { \prime } = 10 x e ^ { 3 x ^ { 2 } - 3 }$

Question 31

Find $\frac { d y } { d x }$ if $y = x ^ { 4 } e ^ { x ^ { 10 } }$ .

A) $\frac { d y } { d x } = e ^ { x ^ { 10 } } \left( \frac { x ^ { 5 } } { 5 } + \frac { x ^ { 11 } } { 11 } \right)$
B) $\frac { d y } { d x } = e ^ { x ^ { 10 } } \left( x ^ { 4 } + x ^ { 10 } \right)$
C) $\frac { d y } { d x } = e ^ { x ^ { 10 } } \left( 4 x ^ { 3 } + 10 x ^ { 13 } \right)$
D) $\frac { d y } { d x } = 4 x ^ { 3 } e ^ { x ^ { 10 } - 1 }$
E) $\frac { d y } { d x } = 4 x ^ { 3 } e ^ { x ^ { 10 } }$

Question 32

Find the derivative of the following function. $y = 3 e ^ { 6 \sqrt { x } } + 6$

A) $y ^ { \prime } = \frac { 9 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$
B) $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$
C) $y ^ { \prime } = \frac { 18 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$
D) $y ^ { \prime } = \frac { 3 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$ .
E) $y ^ { \prime } = \frac { 6 e ^ { 6 \sqrt { x } } } { \sqrt { x } }$

Question 33

If $x - 9 x e ^ { 4 y } = 5 , \text { find } d y / d x$

A) $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 72 e ^ { 4 y } }$
B) $\frac { d y } { d x } = \frac { 1 + 4 e ^ { 4 y } } { 36 x e ^ { 4 y } }$
C) $\frac { d y } { d x } = \frac { 1 - 9 e ^ { 4 y } } { 36 x e ^ { 4 y } }$
D) $\frac { d y } { d x } = \frac { 1 + 9 e ^ { 4 y } } { 72 x e ^ { 4 y } }$
E) $\frac { d y } { d x } = \frac { 1 - 4 e ^ { 4 y } } { 36 e ^ { 4 y } }$

Question 34

Find an equation of the tangent line to the graph of $y = e ^ { 5 x }$ at the point (0,1) .

A) $y = x + 1$
B) $y = \ln ( 5 ) x + 1$
C) $y = 6 x + 1$
D) $y = 5 x + 1$
E) $y = 5 x - 1$

Question 35

Find the derivative of the following function. $y = 1 - 6 e ^ { - x ^ { 6 } }$

A) $y ^ { \prime } = 36 x ^ { 5 } e ^ { - x ^ { 6 } }$
B) $y ^ { \prime } = - 36 x ^ { 5 } e ^ { - x ^ { 6 } }$
C) $y ^ { \prime } = 6 e ^ { - x ^ { 6 } }$
D) $y ^ { \prime } = 6 x ^ { 6 } e ^ { - x ^ { 6 } }$
E) $y ^ { \prime } = - 6 x ^ { 6 } e ^ { - x ^ { 6 } }$

Question 36

Find the derivative of the following function. $p = 5 q e ^ { q ^ { 4 } }$

A) $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( 4 q ^ { 4 } + 1 \right)$
B) $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( 4 q ^ { 3 } + 1 \right)$
C) $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( q ^ { 4 } + 5 \right)$
D) $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( 4 q ^ { 4 } + 5 \right)$
E) $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( 4 q ^ { 3 } + 5 \right)$

Question 37

Write the equation of the line tangent to the graph of $y = 2 x e ^ { - x } + 6$ at $x = 1$

A) $y = 2 e ^ { - 1 } x + 6$
B) $y = 2 e ^ { - 1 } x - 6$
C) $y = 2 e ^ { - 1 } x$
D) $y = 2 e ^ { - 1 } + 6$
E) $y = 2 e ^ { - 1 } - 6$

Question 38

The average time between incoming calls at a switchboard is 3 minutes. If a call has just come in, the probability that the next call will come within the next t minutes is $P ( t ) = 1 - e ^ { - t / 3 }$ . Find the probability that the next call will come within the next $\frac { 3 } { 6 }$ minute. Round your answer to two decimal places.

A)15.35%
B)1.54%
C)184.65%
D)17.58%
E)5.08%

Question 39

The demand function for a product is modeled by $p = 3000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0002 x } } \right)$ . Find the price of the product if the quantity demanded is x = 100. Round your answer to two decimal places where applicable.

A) $\$$ 422.12
B) $\$$ 2518.48
C) $\$$ 491.72
D) $\$$ 502.02
E) $\$$ 2508.28

Question 40

The demand function for a product is modeled by $p = 4000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0003 x } } \right)$ . What is the limit of the price as x increases without bound? Round your answer to two decimal places where applicable.

A)The limit of the price as x increases without bound is -1.
B)The limit of the price as x increases without bound is 1.
C)The limit of the price as x increases without bound is 0.
D)The limit of the price as x increases without bound is $4000$ .
E)The limit of the price as x increases without bound is $5$ .

Question 41

Use the properties of logarithms to approximate $\ln \frac { 5 } { 31 },$ given that $\ln 5 \approx 1.6094$ and $\ln 31 \approx 3.4340$

A)-5.0434
B)-1.8246
C)0.4687
D)5.5267
E)5.0434

Question 42

Find $f ^ { \prime \prime } ( x )$ if $f ( x ) = ( 9 + 5 x ) e ^ { - 8 x }$ .

A) $f ^ { \prime \prime } ( x ) = ( 496 - 320 x ) e ^ { - 8 x }$
B) $f ^ { \prime \prime } ( x ) = ( - 496 - 320 x ) e ^ { - 8 x }$
C) $f ^ { \prime \prime } ( x ) = - 496 ( 9 + 5 x ) e ^ { - 8 x }$
D) $f ^ { \prime \prime } ( x ) = - ( 67 + 40 x ) e ^ { - 8 x }$
E) $f ^ { \prime \prime } ( x ) = ( 496 + 320 x ) e ^ { - 8 x }$

Question 43

Sketch the graph of the function $f ( x ) = \ln | x |$ .

A)
B)
C)
D)
E)

Question 44

Use the properties of logarithms to expand $\ln \frac { 3 } { 10 }$

A) $\ln 3 + \ln 10$
B) $\ln 3 - \ln 10$
C) $\frac { \ln 3 } { \ln 10 }$
D) $( \ln 3 ) ( \ln 10 )$
E)none of the above

Question 45

Find the extrema of the function $f ( x ) = \frac { 1 } { 3 - e ^ { - x } }$ by analyzing its graph below.

A)(0, 1)
B)no relative extrema
C) $\left( 3 , e ^ { 3 } \right)$ , (0, 0)
D) $( 1,3 ) , \left( 3 , e ^ { - 3 } \right)$
E) $\left( 3 , e ^ { - 3 } \right)$

Question 46

Simplify $e ^ { \ln 8 x ^ { 5 } }$ .

A) $- x ^ { 5 }$
B) $- 8 x ^ { 5 }$
C) $8 x$
D) $8 x ^ { 5 }$
E) $x ^ { 5 }$

Question 47

Solve for the equation $e ^ { \sqrt { x } } = e ^ { 16 }$ for $x$ .

A) $x = 16$
B) $x = 4$
C) $x = 4096$
D) $x = 256$
E) $x = 4097$

Question 48

Sketch the graph of the function $y = 5 + \ln x$ .

A)
B)
C)
D)
E)

Question 49

Use the properties of logarithms to expand $\ln \left( \frac { x ^ { 2 } - 25 } { x ^ { 10 } } \right) ^ { 9 }$ .

A) $9 [ \ln ( x + 5 ) - \ln ( x - 5 ) - 10 \ln x ]$
B) $9 [ \ln ( x + 5 ) + \ln ( x - 5 ) + 10 \ln x ]$
C) $9 [ \ln ( x + 5 ) - \ln ( x - 5 ) - \ln x ]$
D) $9 [ \ln ( x + 5 ) + \ln ( x - 5 ) + \ln x ]$
E) $9 [ \ln ( x + 5 ) + \ln ( x - 5 ) - 10 \ln x ]$

Question 50

A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 650 with a standard deviation of 15.5. Assuming the data can be modeled by a normal probability density function, find a model for these data.

A) $f ( x ) = \frac { 1 } { 15.5 \sqrt { \pi } } e ^ { - ( x - 650 ) / 240.25 }$
B) $f ( x ) = \frac { 1 } { 15.5 \sqrt { 2 \pi } } e ^ { - ( x - 650 ) / 480.5 }$
C) $f ( x ) = \frac { 1 } { 15.5 \sqrt { \pi } } e ^ { - ( x - 650 ) ^ { 2 } / 480.5 }$
D) $f ( x ) = \frac { 1 } { 15.5 \sqrt { 2 \pi } } e ^ { - ( x - 650 ) ^ { 2 } / 480.5 }$
E) $f ( x ) = \frac { 1 } { 15.5 \sqrt { 2 \pi } } e ^ { - ( x - 650 ) ^ { 2 } / 240.25 }$

Question 51

Simplify $\ln e ^ { - 6 x ^ { 4 } }$

A) $- 6 x ^ { 4 }$
B) $- 24 x$
C) $- 24 x ^ { 4 }$
D) $- 6 x$
E) $x ^ { 4 }$

Question 52

The average typing speed N (in words per minute) after t weeks of lessons is modeled by $N = \frac { 91 } { 1 + 7.5 e ^ { - 0.15 t } }$ . Find the rate at which the typing speed is changing when t = 20 weeks. Round your answer to two decimal places.

A)2.40 words/min/week
B)2.70 words/min/week
C)3.71 words/min/week
D)4.93 words/min/week
E)6.24 words/min/week

Question 53

Sketch the graph of the function $f ( x ) = 2 + \ln ( x )$ .

A)
B)
C)
D)
E)

Question 54

Write the logarithmic equation $\ln$ $1.3 = 0.2624 \mathrm {~K}$ as an exponential equation.

A) $e ^ { - 0.2624 \mathrm {~K} } = 1.3$
B) $e ^ { - 1.3 } = 0.2624 \mathrm {~K}$
C) $e ^ { 0.2624 \mathrm {~K} } = 1.3$
D) $e ^ { 1.3 } = 0.2624 \mathrm {~K}$
E) $e ^ { 0.2624 \mathrm {~K} } = - 1.3$

Question 55

Use the properties of logarithms to write the expression as a single logarithm. $\ln ( 2 x ) - \ln ( 4 y )$

A) $\ln \left( \frac { 2 x } { 4 y } \right)$
B) $\ln ( 4 y - 2 x )$
C) $\ln ( 2 x - 4 y )$
D) $\frac { \ln ( 2 x ) } { \ln ( 4 y ) }$
E) $\ln \left( \frac { 2 y } { 4 x } \right)$

Question 56

A survey of high school seniors from a certain school district who took the SAT has determined that the mean score on the mathematics portion was 700 with a standard deviation of 13.5. By a normal probability density function the data can be modeled as $f ( x ) = \frac { 1 } { 13.5 \sqrt { 2 \pi } } e ^ { - ( x - 700 ) ^ { 2 } /3 64.5 }$ . Find the derivative of the model.

A) $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$
B) $f ^ { \prime } ( x ) = \frac { - 2 \sqrt { 2 } ( x - 700 ) e ^ { - ( x -700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$
C) $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 4,921 \sqrt { \pi } }$
D) $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 700 ) ^ { 2 } / 364.5 } } { 9,842 \sqrt { \pi } }$
E) $f ^ { \prime } ( x ) = \frac { \sqrt { 2 } ( x - 700 ) e ^ { - ( x - 70 ) ^ { 2 } / 182.25 } } { 4,921 \sqrt { \pi } }$

Question 57

Use the properties of logarithms to write the expression $\ln \left( x \sqrt [ 5 ] { x ^ { 2 } + 8 } \right)$ as a sum, difference, or multiple of logarithms.

A) $\ln x + \frac { 1 } { 8 } \ln \left( x ^ { 2 } + 5 \right)$
B) $\ln \left( x ^ { 2 } + 8 \right) + \frac { 1 } { 5 } \ln x$
C) $\ln x + \frac { 1 } { 5 } \ln \left( x ^ { 2 } + 8 \right)$
D) $\ln \left( x ^ { 2 } + 5 \right) + \frac { 1 } { 8 } \ln x$
E) $\ln x + \ln \left( x ^ { 2 } + 8 \right)$

Question 58

Future value. The future value that accrues when $900 is invested at 5%, compounded continuously, is $s ( t ) = 900 e ^ { 005 t }$ , where t is the number of years. At what rate is the money in this account growing when $t = 9 ?$

A)$14.11 per year
B)$49.24 per year
C)$1411.48 per year
D)$941.43 per year
E)$70.57 per year

Question 59

Find the extrema of the function $f ( x ) = \frac { 1 } { 11 - e ^ { - x } }$ .

A)(0, 1)
B) $\left( 11 , e ^ { 11 } \right)$ , (0, 0)
C) $( 1,11 ) , \left( 11 , e ^ { - 11 } \right)$
D)no relative extrema
E) $\left( 11 , e ^ { - 11 } \right)$

Question 60

Write the exponential equation $e ^ { 10 } = 22026.4658 \mathrm {~K}$ as a logarithmic equation.

A) $\ln$ $22026.4658 \mathrm {~K} = 10$
B) $\ln$ $22026.4658 \mathrm {~K} =$ $20$
C) $\ln$ $10 = 22026.4658 \mathrm {~K}$
D) $\ln$ $10 \mathrm {~K} =$ $44052.9316 \mathrm {~K}$
E) $\ln$ $44052.9316 \mathrm {~K} = 20$

Question 61

Write the following expression as a logarithm of a single quantity. $5 \ln x - 3 \ln \left( x ^ { 2 } + 3 \right)$

A) $\ln \left( \frac { x ^ { 5 } } { 3 \left( x ^ { 2 } + 3 \right) } \right)$
B) $\ln \left( \frac { x ^ { 5 } } { \left( x ^ { 2 } + 3 \right) ^ { 3 } } \right)$
C) $\ln \left( x ^ { 5 } - \left( x ^ { 2 } + 3 \right) ^ { 3 } \right)$
D) $\ln \left( 5 x - 3 \left( x ^ { 2 } + 3 \right) \right)$
E)none of the above

Question 62

Solve the following equation for $x$ accurate to three decimal places. $\ln x ^ { - 2 } = 3$

A) $x = - 1.500$
B) $x = 0.223$
C) $x = 1.405$
D) $x = 0.513$
E) $x = 4.621$

Question 63

Solve the following equation for $x$ accurate to three decimal places. $e ^ { \ln ( 4 x ) } = 5$

A) $x = 20.000$
B) $x = 0.800$
C) $x = 1.250$
D) $x = 0.402$
E) $x = - 0.402$

Question 64

Solve $\left( 14 - \frac { 0.978 } { 22 } \right) ^ { 2 t } = 50$ for t. Round your answer to four decimal places.

A)1.4841
B)0.7412
C)3.0151
D)0.7421
E)2.7571

Question 65

Solve the exponential equation. Give answers correct to 3 decimal places. $5 ^ { 7 x } = 3125$

A)625
B)0.714
C)1.000
D)0.453
E)313

Question 66

Find the derivative of the following function. $y = \ln 5 x$

A) $\frac { 1 } { x }$
B) $\frac { 5 } { x }$
C) $\frac { 1 } { 5 x }$
D) $\frac { 1 } { x ^ { 2 } }$
E) $\frac { 1 } { 5 x ^ { 2 } }$

Question 67

Solve the exponential equation. Give the answer correct to 3 decimal places. $10,000 = 3500 e ^ { 0.3 x }$

A)0.886
B)2.953
C)3.762
D)0.315
E)3.499

Question 68

Solve the following equation for $x$ accurate to three decimal places. $2 \left( 2 ^ { 2 x - 3 } \right) = 290$

A) $x = - 34.750$
B) $x = 37.750$
C) $x = - 2.090$
D) $x = 5.090$
E) $x = - 32.707$

Question 69

Solve the exponential equation. Give the answer correct to 3 decimal places. $72 = 300 - 300 e ^ { - 0.07 x }$

A)-0.274
B)20.387
C)3.921
D)-20.387
E)-3.073

Question 70

Find the derivative of $y = 2 - 15 \ln x$

A) $\frac { d y } { d x } = - \frac { 1 } { 15 x }$
B) $\frac { d y } { d x } = - \frac { 15 } { x }$
C) $\frac { d y } { d x } = - \frac { 13 } { x }$
D) $\frac { d y } { d x } = \frac { 15 } { x }$
E) $\frac { d y } { d x } = \frac { 13 } { x }$

Question 71

Write the expression $5 \ln ( x ) + 2 \ln ( x + 3 ) - 5 \ln ( x - 3 )$ as the logarithm of a single quantity.

A) $\ln ( 2 x + 21 )$
B) $\ln \left[ \frac { 10 \left( x ^ { 2 } + x \right) } { 5 ( x - 3 ) } \right]$
C) $\ln \left( x ^ { 5 } ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } \right)$
D) $\ln \left[ \frac { x ^ { 5 } ( x + 3 ) ^ { 2 } } { ( x - 3 ) ^ { 5 } } \right]$
E) $\ln \left[ \frac { x ^ { 5 } } { ( x + 3 ) ^ { 2 } ( x - 3 ) ^ { 5 } } \right]$

Question 72

Find the derivative of the following function. $y = 9 \ln \left( x ^ { 8 } - 4 \right)$

A) $\frac { 9 } { x ^ { 8 } - 4 }$
B) $\frac { 8 x ^ { 7 } } { x ^ { 8 } - 4 }$
C) $\frac { 8 } { x ^ { 8 } - 4 }$
D) $\frac { 72 x ^ { 7 } } { x ^ { 8 } - 4 }$
E) $\frac { 9 x ^ { 7 } } { x ^ {8 } - 4 }$

Question 73

Write the following expression as a logarithm of a single quantity. $\ln x - 16 \ln \left( x ^ { 2 } + 1 \right)$

A) $\ln \left( \frac { - 16 x } { \left( x ^ { 2 } + 1 \right) } \right)$
B) $\ln \left( \frac { x } { 16 \left( x ^ { 2 } + 1 \right) } \right)$
C) $\ln \left( x - 16 \left( x ^ { 2 } + 1 \right) \right)$
D) $\ln \left( \frac { x } { \left( x ^ { 2 } + 1 \right) ^ { 16 } } \right)$
E) $\ln \left( \frac { - 16 x } { x ^ { 2 } + 1 } \right)$

Question 74

Find the derivative of the following function. $y = \ln ( 4 x + 7 )$

A) $\frac { 7 x } { 4 x + 7 }$
B) $\frac { 4 x } { 4 x + 7 }$
C) $\frac { 1 } { 4 x + 7 }$
D) $\frac { 4 } { 4 x + 7 }$
E) $\frac { 7 } { 4 x + 7 }$

Question 75

Write the expression $3 \ln ( 2 ) - \frac { 1 } { 5 } \ln \left( x ^ { 2 } + 4 \right)$ as the logarithm of a single quantity.

A) $\ln \left[ \frac { 8 } { \left( x ^ { 2 } + 4 \right) ^ { 5 } } \right]$
B) $\ln \left( 8 \sqrt [ 5 ] { x ^ { 2 } + 4 } \right)$
C) $\ln \left[ \frac { 9 } { \sqrt [ 5 ] { x ^ { 2 } + 4 } } \right]$
D) $\ln \left( 9 \sqrt [ 5 ] { x ^ { 2 } + 4 } \right)$
E) $\ln \left[ \frac { 8 } { \sqrt [ 5 ] { x ^ { 2 } + 4 } } \right]$

Question 76

Find the derivative of the following function. $y = \ln \left( 4 x ^ { 3 } - 9 x \right) - 3 x$

A) $\frac { 12 x ^ { 2 } - 9 x } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$
B) $\frac { 1 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$
C) $\frac { 12 x ^ { 2 } - 9 } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$
D) $\frac { 12 x ^ { 2 } } { x \left( 4 x ^ { 2 } - 9 \right) } - 3$
E) $\frac { 1 } { x \left( 4 x ^ { 2 } - 1 \right) } - 3$

Question 77

Solve the exponential equation. Give the answer correct to 3 decimal places. $57 = \frac { 77 } { 1 + 4 e ^ { - 0.1 x } }$

A)-2.434
B)-0.877
C)5.315
D)-0.532
E)24.336

Question 78

Find the derivative of the following function. $y = 8 + \ln 5 x$

A) $\frac { 1 } { x }$
B) $\frac { 5 } { x }$
C) $\frac { 1 } { 5 x }$
D) $\frac { 1 } { x ^ { 2 } }$
E) $\frac { 1 } { 5 x ^ { 2 } }$

Question 79

How long (in years) would $450 have to be invested at an annual rate of 12%, compounded continuously, to amount to $790?

A)6.30 years
B)4.97 years
C)1.13 years
D)5.68 years
E)4.69 years

Question 80

Find the derivative of the following function. $y = \ln x ^ { 5 }$

A) $\frac { 1 } { x }$
B) $\frac { 5 } { x }$
C) $\frac { 1 } { 5 x }$
D) $\frac { 1 } { x ^ { 2 } }$
E) $\frac { 1 } { 5 x ^ { 2 } }$