Question 1

Evaluate the given expression. $$\left( \frac { 625 } { 1,296 } \right) ^ { 5 / 4 }$$

Accepted Answer

A)  $\frac { 3,125 } { 1,296 }$ 
B)  $\frac { 3,125 } { 7,776 }$ 
C)  $\frac { 62.5 } { 1,296 }$ 
D)  $\frac { 5 } { 6 }$ 
A)  $\frac { 3,125 } { 1,296 }$ 
B)  $\frac { 3,125 } { 7,776 }$ 
C)  $\frac { 62.5 } { 1,296 }$ 
D)  $\frac { 5 } { 6 }$ B

Question 2

Find the effective interest rate if the nominal rate is 6.5% compounded monthly. Round to one decimal place, if necessary.

Accepted Answer

A)  6.7% 
B)  6.9% 
C)  7.1% 
D)  7.3% 
A)  6.7% 
B)  6.9% 
C)  7.1% 
D)  7.3% A

Question 3

Solve for x: $5 ^ { x } = e ^ { 3 }$

Accepted Answer

A)  $x = \frac { 3 } { \ln 5 }$ 
B)  x = ln 5 
C)  x = 3 ln 5 
D)  x = 3 - ln5 
A)  $x = \frac { 3 } { \ln 5 }$ 
B)  x = ln 5 
C)  x = 3 ln 5 
D)  x = 3 - ln5 A

Question 4

Differentiate the given function. $$y = 20 - 5 e ^ { - 0.03 x }$$

Accepted Answer

A)  $- 5 e ^ { - 0.03 x }$ 
B)  $0.15 e ^ { - 0.03 x }$ 
C)  $- 0.15 e ^ { - 0.03 x }$ 
D)  $5 e ^ { - 0.03 x }$ 
A)  $- 5 e ^ { - 0.03 x }$ 
B)  $0.15 e ^ { - 0.03 x }$ 
C)  $- 0.15 e ^ { - 0.03 x }$ 
D)  $5 e ^ { - 0.03 x }$

Question 5

The amount of a sample of a radioactive substance remaining after t years is given to be a function of the form $Q ( t ) = Q _ { 0 } e ^ { - k t }$ . If 100 grams remain after 1,000 years and 90 grams after 2,000 years, what is the initial amount $Q _ { 0 }$ ? Round your answer to two decimal places.

Accepted Answer

A)  110.21 grams 
B)  110.83 grams 
C)  111.11 grams 
D)  111.49 grams 
A)  110.21 grams 
B)  110.83 grams 
C)  111.11 grams 
D)  111.49 grams

Question 6

The consumer demand for a certain commodity is $D ( p ) = 1,000.00 e ^ { - 0.33 p }$ units per month when the market price is p dollars per unit. Express consumers' total monthly expenditure for the commodity as a function of p and determine the market price that will result in the greatest consumer expenditure.

Accepted Answer

A)  $45.45 
B)  $1,000.00 
C)  $3.03 
D)  $24.24 
A)  $45.45 
B)  $1,000.00 
C)  $3.03 
D)  $24.24

Question 7

Suppose $1,500 is invested at an annual interest rate of 8 percent compounded quarterly. Compute the balance after 12 years.

Accepted Answer

A)  $3,780.61 
B)  $3,820.61 
C)  $3,880.61 
D)  $3,890.61 
A)  $3,780.61 
B)  $3,820.61 
C)  $3,880.61 
D)  $3,890.61

Question 8

Solve for x: $$3 \ln x - \frac { 1 } { 4 } \ln x ^ { 3 } = 9$$

Accepted Answer

A)  $x = e ^ { 4 }$ 
B)  x = e 
C)  $x = e ^ { 9 }$ 
D)  $x = e ^ { 3 }$ 
A)  $x = e ^ { 4 }$ 
B)  x = e 
C)  $x = e ^ { 9 }$ 
D)  $x = e ^ { 3 }$

Question 9

A fossil is 15,000 years old. What percentage of the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ in the atmosphere is the ratio found in the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round your answer to one decimal place.

Accepted Answer

The answer of A fossil is 15,000 years old. What...

Question 10

Use logarithmic differentiation to find $\frac { d y } { d x }$ , where $y = ( 2 x - 1 ) ^ { 3 } ( x + 5 ) ^ { 2 } ( 1 - x ) ^ { 5 }$ .

Accepted Answer

The answer of Use logarithmic differentiation to find \(\frac {...

Question 11

Solve for x: $a ^ { 2 x - 2 } = b$

Accepted Answer

A)  $x = \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$ 
B)  $\frac { \ln b } { 2 \ln a }$ 
C)  $\left( \frac { \ln b } { \ln a } + 2 \right) ^ { 2 }$ 
D)  $x = 1 + \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$ 
A)  $x = \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$ 
B)  $\frac { \ln b } { 2 \ln a }$ 
C)  $\left( \frac { \ln b } { \ln a } + 2 \right) ^ { 2 }$ 
D)  $x = 1 + \frac { 1 } { 2 } \left( 2 + \frac { \ln b } { \ln a } \right)$

Question 12

It is projected that t years from now, the population of a certain country will be $P ( t ) = 30 e ^ { 0.08 t }$ million. What will be the population in 40 years? Round your answer to two decimal places, if necessary.

Accepted Answer

A)  735.98 million 
B)  330.7 million 
C)  711.44 million 
D)  760.51 million 
A)  735.98 million 
B)  330.7 million 
C)  711.44 million 
D)  760.51 million

Question 13

If log4 x - 2 = 5, then x = 3.161, when rounded to three decimal places.

Accepted Answer

A) True 
 B)False

Question 14

If $\frac { 1 } { 2 } A = 2 A e ^ { - 0.06 x }$ , then x is $\frac { \ln 4 } { 0.06 }$ .

Accepted Answer

A) True 
 B)False

Question 15

A certain machine depreciates so that its value after t years is $Q ( t ) = 10,000 e ^ { - 0.5 t }$ dollars. At what rate is the value of the machine changing with respect to time after 3 years?

Accepted Answer

Decreasing

Question 16

Let $f ( x ) = 8 x ^ { 5 } - 80 \ln x$ , for x > 0. Find the minimum value of f for x > 0.

Accepted Answer

A)  $4 \left( 2 ^ { 5 } - 10 \ln 2 \right)$ 
B)  $8 \left( 2 ^ { 5 } - 10 \ln 2 \right)$ 
C)  0 
D)  16(1 - ln 2) 
A)  $4 \left( 2 ^ { 5 } - 10 \ln 2 \right)$ 
B)  $8 \left( 2 ^ { 5 } - 10 \ln 2 \right)$ 
C)  0 
D)  16(1 - ln 2)

Question 17

The effective interest rate (rounded to two decimal places) if the nominal rate is 7 percent compounded quarterly is 7.30%.

Accepted Answer

A) True 
 B)False

Question 18

Solve for x to two decimal places: $$\ln \sqrt { x } = \frac { 1 } { 8 } ( \ln 7 + 2 \ln 4 )$$

Accepted Answer

The answer of Solve for x to two decimal places:...

Question 19

Solve for x if log4 (2x) = 3.

Accepted Answer

Question 20

Simplify $\left( 16 ^ { 1 / 2 } - 4 ^ { 1 / 2 } \right) ^ { - 2 }$ .

Accepted Answer

The answer of Simplify \(\left( 16 ^ { 1 /...

Evaluate the given expression. $\left( \frac { 625 } { 1,296 } \right) ^ { 5 / 4 }$

Find the effective interest rate if the nominal rate is 6.5% compounded monthly. Round to one decimal place, if necessary.

Solve for x: $5 ^ { x } = e ^ { 3 }$

Differentiate the given function. $y = 20 - 5 e ^ { - 0.03 x }$

Suppose $1,500 is invested at an annual interest rate of 8 percent compounded quarterly. Compute the balance after 12 years.

Solve for x: $3 \ln x - \frac { 1 } { 4 } \ln x ^ { 3 } = 9$

A fossil is 15,000 years old. What percentage of the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ in the atmosphere is the ratio found in the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round your answer to one decimal place.

Use logarithmic differentiation to find $\frac { d y } { d x }$ , where $y = ( 2 x - 1 ) ^ { 3 } ( x + 5 ) ^ { 2 } ( 1 - x ) ^ { 5 }$ .

Solve for x: $a ^ { 2 x - 2 } = b$

It is projected that t years from now, the population of a certain country will be $P ( t ) = 30 e ^ { 0.08 t }$ million. What will be the population in 40 years? Round your answer to two decimal places, if necessary.

If log₄x - 2 = 5, then x = 3.161, when rounded to three decimal places.

If $\frac { 1 } { 2 } A = 2 A e ^ { - 0.06 x }$ , then x is $\frac { \ln 4 } { 0.06 }$ .

A certain machine depreciates so that its value after t years is $Q ( t ) = 10,000 e ^ { - 0.5 t }$ dollars. At what rate is the value of the machine changing with respect to time after 3 years?

Let $f ( x ) = 8 x ^ { 5 } - 80 \ln x$ , for x > 0. Find the minimum value of f for x > 0.

The effective interest rate (rounded to two decimal places) if the nominal rate is 7 percent compounded quarterly is 7.30%.

Solve for x to two decimal places: $\ln \sqrt { x } = \frac { 1 } { 8 } ( \ln 7 + 2 \ln 4 )$

Solve for x if log₄(2x) = 3.

Simplify $\left( 16 ^ { 1 / 2 } - 4 ^ { 1 / 2 } \right) ^ { - 2 }$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Exam 4: Exponential and Logarithmic Functions

Evaluate the given expression. (6251,296)5/4\left( \frac { 625 } { 1,296 } \right) ^ { 5 / 4 }(1,296625​)5/4

Find the effective interest rate if the nominal rate is 6.5% compounded monthly. Round to one decimal place, if necessary.

Solve for x: 5x=e35 ^ { x } = e ^ { 3 }5x=e3

Differentiate the given function. y=20−5e−0.03xy = 20 - 5 e ^ { - 0.03 x }y=20−5e−0.03x

Suppose $1,500 is invested at an annual interest rate of 8 percent compounded quarterly. Compute the balance after 12 years.

Solve for x: 3ln⁡x−14ln⁡x3=93 \ln x - \frac { 1 } { 4 } \ln x ^ { 3 } = 93lnx−41​lnx3=9

A fossil is 15,000 years old. What percentage of the ratio of 14C{ } ^ { 14 } \mathrm { C }14C to 12C{ } ^ { 12 } \mathrm { C }12C in the atmosphere is the ratio found in the fossil? The half-life of 14C{ } ^ { 14 } \mathrm { C }14C is 5,730 years. Round your answer to one decimal place.

Use logarithmic differentiation to find dydx\frac { d y } { d x }dxdy​ , where y=(2x−1)3(x+5)2(1−x)5y = ( 2 x - 1 ) ^ { 3 } ( x + 5 ) ^ { 2 } ( 1 - x ) ^ { 5 }y=(2x−1)3(x+5)2(1−x)5 .

Solve for x: a2x−2=ba ^ { 2 x - 2 } = ba2x−2=b

It is projected that t years from now, the population of a certain country will be P(t)=30e0.08tP ( t ) = 30 e ^ { 0.08 t }P(t)=30e0.08t million. What will be the population in 40 years? Round your answer to two decimal places, if necessary.

If log4 x - 2 = 5, then x = 3.161, when rounded to three decimal places.

If 12A=2Ae−0.06x\frac { 1 } { 2 } A = 2 A e ^ { - 0.06 x }21​A=2Ae−0.06x , then x is ln⁡40.06\frac { \ln 4 } { 0.06 }0.06ln4​ .

A certain machine depreciates so that its value after t years is Q(t)=10,000e−0.5tQ ( t ) = 10,000 e ^ { - 0.5 t }Q(t)=10,000e−0.5t dollars. At what rate is the value of the machine changing with respect to time after 3 years?

Let f(x)=8x5−80ln⁡xf ( x ) = 8 x ^ { 5 } - 80 \ln xf(x)=8x5−80lnx , for x > 0. Find the minimum value of f for x > 0.

The effective interest rate (rounded to two decimal places) if the nominal rate is 7 percent compounded quarterly is 7.30%.

Solve for x to two decimal places: ln⁡x=18(ln⁡7+2ln⁡4)\ln \sqrt { x } = \frac { 1 } { 8 } ( \ln 7 + 2 \ln 4 )lnx​=81​(ln7+2ln4)

Solve for x if log4 (2x) = 3.

Simplify (161/2−41/2)−2\left( 16 ^ { 1 / 2 } - 4 ^ { 1 / 2 } \right) ^ { - 2 }(161/2−41/2)−2 .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

Evaluate the given expression. $\left( \frac { 625 } { 1,296 } \right) ^ { 5 / 4 }$

Solve for x: $5 ^ { x } = e ^ { 3 }$

Differentiate the given function. $y = 20 - 5 e ^ { - 0.03 x }$

Solve for x: $3 \ln x - \frac { 1 } { 4 } \ln x ^ { 3 } = 9$

A fossil is 15,000 years old. What percentage of the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ in the atmosphere is the ratio found in the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round your answer to one decimal place.

Use logarithmic differentiation to find $\frac { d y } { d x }$ , where $y = ( 2 x - 1 ) ^ { 3 } ( x + 5 ) ^ { 2 } ( 1 - x ) ^ { 5 }$ .

Solve for x: $a ^ { 2 x - 2 } = b$

It is projected that t years from now, the population of a certain country will be $P ( t ) = 30 e ^ { 0.08 t }$ million. What will be the population in 40 years? Round your answer to two decimal places, if necessary.

If log₄x - 2 = 5, then x = 3.161, when rounded to three decimal places.

If $\frac { 1 } { 2 } A = 2 A e ^ { - 0.06 x }$ , then x is $\frac { \ln 4 } { 0.06 }$ .

A certain machine depreciates so that its value after t years is $Q ( t ) = 10,000 e ^ { - 0.5 t }$ dollars. At what rate is the value of the machine changing with respect to time after 3 years?

Let $f ( x ) = 8 x ^ { 5 } - 80 \ln x$ , for x > 0. Find the minimum value of f for x > 0.

Solve for x to two decimal places: $\ln \sqrt { x } = \frac { 1 } { 8 } ( \ln 7 + 2 \ln 4 )$

Solve for x if log₄(2x) = 3.

Simplify $\left( 16 ^ { 1 / 2 } - 4 ^ { 1 / 2 } \right) ^ { - 2 }$ .