Question 1

Solve for x: $\log _ { 5 } ( x - 1 ) = 3$ .

Accepted Answer

A) 125 
B) 244 
C) 126 
D) 124 
A) 125 
B) 244 
C) 126 
D) 124 C

Question 2

A traffic accident was witnessed by $\frac { 1 } { 14 }$ of the residents of a small town.The number of residents who had heard about the accident t hours later is given by a function of the form $\frac { B } { 1 + C e ^ { - k t } }$ ,where B is the population of the town.If $\frac { 1 } { 4 }$ of the residents had heard about the accident after 1 hours,how long did it take for $\frac { 1 } { 2 }$ of the residents to hear the news? Round your answer to two decimal places.

Accepted Answer

A) 0.87 hours 
B) 0.44 hour 
C) 1.75 hours 
D) 3.50 hours 
A) 0.87 hours 
B) 0.44 hour 
C) 1.75 hours 
D) 3.50 hours C

Question 3

An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: Experience (months)
$$\begin{array} { l l l } 
\text { Experience (months) } & 0 & 2 \
\text { Output (units per hour) } & 300 & 510
\end{array}$$ 
The expert believes that the output Q is related to experience t by a function of the form $Q ( t ) = 600 - A e ^ { - k t }$ .Find the function of this form that fits the data.Round numbers to four decimal places,if necessary.

Accepted Answer

A)  $Q ( t ) = 600 - 510 e ^ { 0.6020 t }$ 
B)  $Q ( t ) = 600 - 510 e ^ { - 0.6020 t }$ 
C)  $Q ( t ) = 600 - 300 e ^ { - 0.6020 t }$ 
D)  $Q ( t ) = 600 - 300 e ^ { 0.6020 t }$ 
A)  $Q ( t ) = 600 - 510 e ^ { 0.6020 t }$ 
B)  $Q ( t ) = 600 - 510 e ^ { - 0.6020 t }$ 
C)  $Q ( t ) = 600 - 300 e ^ { - 0.6020 t }$ 
D)  $Q ( t ) = 600 - 300 e ^ { 0.6020 t }$ C

Question 4

It is projected that t years from now,the population of a certain country will be $P ( t ) = 70 e ^ { 0.01 t }$ million.What will be the population in 20 years? Round your answer to two decimal places,if necessary.

Accepted Answer

A) 140.96 million 
B) 86.72 million 
C) 85.5 million 
D) 84.28 million 
A) 140.96 million 
B) 86.72 million 
C) 85.5 million 
D) 84.28 million

Question 5

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

Accepted Answer

A)  $\frac { 42 } { x \ln x }$ 
B)  $\frac { 42 } { \ln ( \ln x ) }$ 
C)  $\frac { 7 } { x \ln x }$ 
D)  $\frac { 42 } { x + \ln x }$ 
A)  $\frac { 42 } { x \ln x }$ 
B)  $\frac { 42 } { \ln ( \ln x ) }$ 
C)  $\frac { 7 } { x \ln x }$ 
D)  $\frac { 42 } { x + \ln x }$

Question 6

Differentiate the given function. $f ( x ) = 35 - 3 e ^ { - 0.06 x }$

Accepted Answer

A)  $3 x e ^ { - 0.06 x }$ 
B)  $- 0.18 e ^ { - 0.06 x }$ 
C)  $0.18 e ^ { - 006 x }$ 
D)  $- 3 x e ^ { - 0.06 x }$ 
A)  $3 x e ^ { - 0.06 x }$ 
B)  $- 0.18 e ^ { - 0.06 x }$ 
C)  $0.18 e ^ { - 006 x }$ 
D)  $- 3 x e ^ { - 0.06 x }$

Question 7

Let $f ( x ) = 6 x ^ { 5 } - 90 \ln x$ ,for x > 0.Find the minimum value of f for x > 0.

Accepted Answer

A) 0 
B)  $3 \left( 3 ^ { 5 } - 15 \ln 3 \right)$ 
C) 18(1 - ln 3) 
D)  $6 \left( 3 ^ { 5 } - 15 \ln 3 \right)$ 
A) 0 
B)  $3 \left( 3 ^ { 5 } - 15 \ln 3 \right)$ 
C) 18(1 - ln 3) 
D)  $6 \left( 3 ^ { 5 } - 15 \ln 3 \right)$

Question 8

Let $f ( t ) = \frac { 19 } { 3 + 4 e ^ { - t / 10 } }$ .Which value of t corresponds to a possible inflection point for f (t)?

Accepted Answer

A) There is no inflection point. 
B)  $\frac { 3 } { 4 } \ln ( 110 )$ 
C)  $\ln \left( \frac { 3 } { 4 } \right)$ 
D)  $110 \ln \left( \frac { 4 } { 3 } \right)$ 
A) There is no inflection point. 
B)  $\frac { 3 } { 4 } \ln ( 110 )$ 
C)  $\ln \left( \frac { 3 } { 4 } \right)$ 
D)  $110 \ln \left( \frac { 4 } { 3 } \right)$

Question 9

Suppose your family owns a rare book whose value t years from now will be $V ( t ) = 9 e ^ { \sqrt { 0.8 t } }$ dollars.If the prevailing interest rate remains constant at 6% per year compounded continuously,when will it be most advantageous for your family to sell the book and invest the proceeds? Round your answer to two decimal places.

Accepted Answer

A) 125.00 years 
B) 48.61 years 
C) 194.44 years 
D) 55.56 years 
A) 125.00 years 
B) 48.61 years 
C) 194.44 years 
D) 55.56 years

Question 10

If $1,500 is invested at 9 percent compounded continuously,what is the balance after 13 years?

Accepted Answer

A) $4,598.71 
B) $465.55 
C) $1,635.00 
D) $4,832.99 
A) $4,598.71 
B) $465.55 
C) $1,635.00 
D) $4,832.99

Question 11

Solve for x.Round to three decimal places,if necessary. $\log _ { 4 } x = 5$

Accepted Answer

A) 1)495 
B) 1)32 
C) 1,024 
D) 625 
A) 1)495 
B) 1)32 
C) 1,024 
D) 625

Question 12

Solve the given equation for x. $- 6 = - 7 + 5 e ^ { - 7 x }$

Accepted Answer

A)  $- \frac { e ^ { 5 } } { 7 }$ 
B)  $\frac { \ln 5 } { 7 }$ 
C)  $- 7 e ^ { 5 }$ 
D)  $- \frac { \ln 5 } { 7 }$ 
A)  $- \frac { e ^ { 5 } } { 7 }$ 
B)  $\frac { \ln 5 } { 7 }$ 
C)  $- 7 e ^ { 5 }$ 
D)  $- \frac { \ln 5 } { 7 }$

Question 13

Evaluate the given expression. $\left( \frac { 256 } { 625 } \right) ^ { 5 / 4 }$

Accepted Answer

A)  $\frac { 1,024 } { 3,125 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 1,024 } { 625 }$ 
D)  $\frac { 256 } { 625 }$ 
A)  $\frac { 1,024 } { 3,125 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 1,024 } { 625 }$ 
D)  $\frac { 256 } { 625 }$

Question 14

Differentiate the given function. $f ( x ) = \ln x ^ { 3 }$

Accepted Answer

A)  $\frac { x } { 3 }$ 
B)  $\frac { 1 } { 3 x }$ 
C) 3x 
D)  $\frac { 3 } { x }$ 
A)  $\frac { x } { 3 }$ 
B)  $\frac { 1 } { 3 x }$ 
C) 3x 
D)  $\frac { 3 } { x }$

Question 15

A radioactive substance decays exponentially.If 700 grams were present initially and 300 grams are present 100 years later,how many grams will be present after 400 years? Round your answer to two decimal places,if necessary.

Accepted Answer

A) 22.37 grams 
B) 21.12 grams 
C) 0)00 grams 
D) 23.62 grams 
A) 22.37 grams 
B) 21.12 grams 
C) 0)00 grams 
D) 23.62 grams

Question 16

Find all real numbers x that satisfy the given equation. 3x23x = 13,824

Accepted Answer

A) 15 
B) 12 
C) 9 
D) 3 
A) 15 
B) 12 
C) 9 
D) 3

Question 17

Solve for x: $4 \ln x - \frac { 1 } { 5 } \ln x ^ { 4 } = 16$

Accepted Answer

A) x = e 
B)  $x = e ^ { 16 }$ 
C)  $x = e ^ { 4 }$ 
D)  $x = e ^ { 5 }$ 
A) x = e 
B)  $x = e ^ { 16 }$ 
C)  $x = e ^ { 4 }$ 
D)  $x = e ^ { 5 }$

Question 18

Solve for x: $a ^ { 7 x - 1 } = b$

Accepted Answer

A)  $\left( \frac { \ln b } { \ln a } + 1 \right) ^ { 7 }$ 
B)  $x = \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$ 
C)  $x = 1 + \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$ 
D)  $\frac { \ln b } { 7 \ln a }$ 
A)  $\left( \frac { \ln b } { \ln a } + 1 \right) ^ { 7 }$ 
B)  $x = \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$ 
C)  $x = 1 + \frac { 1 } { 7 } \left( 1 + \frac { \ln b } { \ln a } \right)$ 
D)  $\frac { \ln b } { 7 \ln a }$

Question 19

The consumer demand for a certain commodity is $D ( p ) = 8,000.00 e ^ { - 057 p }$ 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) $8,000.00 
B) $14.04 
C) $26.32 
D) $1.75 
A) $8,000.00 
B) $14.04 
C) $26.32 
D) $1.75

Question 20

Differentiate the given function. $f ( x ) = e ^ { - 6 x }$

Accepted Answer

A)  $x = - 6 x e ^ { - 6 x }$ 
B)  $x = e ^ { - 6 x }$ 
C)  $x = - 6 e ^ { - 6 x - 1 }$ 
D)  $x = - 6 e ^ { - 6 x }$ 
A)  $x = - 6 x e ^ { - 6 x }$ 
B)  $x = e ^ { - 6 x }$ 
C)  $x = - 6 e ^ { - 6 x - 1 }$ 
D)  $x = - 6 e ^ { - 6 x }$

Solve for x: $\log _ { 5 } ( x - 1 ) = 3$ .

It is projected that t years from now,the population of a certain country will be $P ( t ) = 70 e ^ { 0.01 t }$ million.What will be the population in 20 years? Round your answer to two decimal places,if necessary.

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

Differentiate the given function. $f ( x ) = 35 - 3 e ^ { - 0.06 x }$

Let $f ( x ) = 6 x ^ { 5 } - 90 \ln x$ ,for x > 0.Find the minimum value of f for x > 0.

Let $f ( t ) = \frac { 19 } { 3 + 4 e ^ { - t / 10 } }$ .Which value of t corresponds to a possible inflection point for f (t)?

If $1,500 is invested at 9 percent compounded continuously,what is the balance after 13 years?

Solve for x.Round to three decimal places,if necessary. $\log _ { 4 } x = 5$

Solve the given equation for x. $- 6 = - 7 + 5 e ^ { - 7 x }$

Evaluate the given expression. $\left( \frac { 256 } { 625 } \right) ^ { 5 / 4 }$

Differentiate the given function. $f ( x ) = \ln x ^ { 3 }$

A radioactive substance decays exponentially.If 700 grams were present initially and 300 grams are present 100 years later,how many grams will be present after 400 years? Round your answer to two decimal places,if necessary.

Find all real numbers x that satisfy the given equation. 3^x2³^x = 13,824

Solve for x: $4 \ln x - \frac { 1 } { 5 } \ln x ^ { 4 } = 16$

Solve for x: $a ^ { 7 x - 1 } = b$

Differentiate the given function. $f ( x ) = e ^ { - 6 x }$

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

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Exam 4: Exponential and Logarithmic Functions

Solve for x: log⁡5(x−1)=3\log _ { 5 } ( x - 1 ) = 3log5​(x−1)=3 .

It is projected that t years from now,the population of a certain country will be P(t)=70e0.01tP ( t ) = 70 e ^ { 0.01 t }P(t)=70e0.01t million.What will be the population in 20 years? Round your answer to two decimal places,if necessary.

Find the derivative of ln⁡[(ln⁡x6)7]\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]ln[(lnx6)7] .

Differentiate the given function. f(x)=35−3e−0.06xf ( x ) = 35 - 3 e ^ { - 0.06 x }f(x)=35−3e−0.06x

Let f(x)=6x5−90ln⁡xf ( x ) = 6 x ^ { 5 } - 90 \ln xf(x)=6x5−90lnx ,for x > 0.Find the minimum value of f for x > 0.

Let f(t)=193+4e−t/10f ( t ) = \frac { 19 } { 3 + 4 e ^ { - t / 10 } }f(t)=3+4e−t/1019​ .Which value of t corresponds to a possible inflection point for f (t)?

If $1,500 is invested at 9 percent compounded continuously,what is the balance after 13 years?

Solve for x.Round to three decimal places,if necessary. log⁡4x=5\log _ { 4 } x = 5log4​x=5

Solve the given equation for x. −6=−7+5e−7x- 6 = - 7 + 5 e ^ { - 7 x }−6=−7+5e−7x

Evaluate the given expression. (256625)5/4\left( \frac { 256 } { 625 } \right) ^ { 5 / 4 }(625256​)5/4

Differentiate the given function. f(x)=ln⁡x3f ( x ) = \ln x ^ { 3 }f(x)=lnx3

A radioactive substance decays exponentially.If 700 grams were present initially and 300 grams are present 100 years later,how many grams will be present after 400 years? Round your answer to two decimal places,if necessary.

Find all real numbers x that satisfy the given equation. 3x23x = 13,824

Solve for x: 4ln⁡x−15ln⁡x4=164 \ln x - \frac { 1 } { 5 } \ln x ^ { 4 } = 164lnx−51​lnx4=16

Solve for x: a7x−1=ba ^ { 7 x - 1 } = ba7x−1=b

Differentiate the given function. f(x)=e−6xf ( x ) = e ^ { - 6 x }f(x)=e−6x

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Solve for x: $\log _ { 5 } ( x - 1 ) = 3$ .

It is projected that t years from now,the population of a certain country will be $P ( t ) = 70 e ^ { 0.01 t }$ million.What will be the population in 20 years? Round your answer to two decimal places,if necessary.

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

Differentiate the given function. $f ( x ) = 35 - 3 e ^ { - 0.06 x }$

Let $f ( x ) = 6 x ^ { 5 } - 90 \ln x$ ,for x > 0.Find the minimum value of f for x > 0.

Let $f ( t ) = \frac { 19 } { 3 + 4 e ^ { - t / 10 } }$ .Which value of t corresponds to a possible inflection point for f (t)?

Solve for x.Round to three decimal places,if necessary. $\log _ { 4 } x = 5$

Solve the given equation for x. $- 6 = - 7 + 5 e ^ { - 7 x }$

Evaluate the given expression. $\left( \frac { 256 } { 625 } \right) ^ { 5 / 4 }$

Differentiate the given function. $f ( x ) = \ln x ^ { 3 }$

Find all real numbers x that satisfy the given equation. 3^x2³^x = 13,824

Solve for x: $4 \ln x - \frac { 1 } { 5 } \ln x ^ { 4 } = 16$

Solve for x: $a ^ { 7 x - 1 } = b$

Differentiate the given function. $f ( x ) = e ^ { - 6 x }$