Question 1

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 2

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and this interest is compounded continuously. After 4 years, the money has doubled. The balance at the end of 8 years is $4 A _ { 0 }$ .

Accepted Answer

A) True 
 B)False

Question 3

How quickly will money triple if it is invested at 7% interest compounded continuously? Round your answer to two decimal places.

Accepted Answer

A)  15.69 years 
B)  15.71 years 
C)  15.73 years 
D)  15.75 years 
A)  15.69 years 
B)  15.71 years 
C)  15.73 years 
D)  15.75 years

Question 4

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 5

If $3,000 is invested at 2 percent compounded continuously, what is the balance after 11 years?

Accepted Answer

The answer of If $3,000 is invested at 2 percent...

Question 6

$\frac { 3 ^ { 4 } } { 3 ^ { 6 } } = \frac { 1 } { 9 }$

Accepted Answer

A) True 
 B)False

Question 7

Differentiate the given function. $f ( x ) = 35 - 4 e ^ { - 0.06 x }$

Accepted Answer

The answer of Differentiate the given function. \(f ( x...

Question 8

Solve for x if log3 x = 5.

Accepted Answer

Question 9

A traffic accident was witnessed by $\frac { 1 } { 12 }$ 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 $\frac { 1 } { 5 }$ of the residents had heard about the accident after 1 hours, how long did it take for $\frac { 1 } { 4 }$ of the residents to hear the news? Round your answer to two decimal places.

Accepted Answer

The answer of A traffic accident was witnessed by \(\frac...

Question 10

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ $$f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }$$

Accepted Answer

A)  $f ^ { \prime } ( x ) = \frac { 1 } { 8 } \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$ 
B)  $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$ 
C)  $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$ 
D)  $f ^ { \prime } ( x ) = \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$ 
A)  $f ^ { \prime } ( x ) = \frac { 1 } { 8 } \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$ 
B)  $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$ 
C)  $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$ 
D)  $f ^ { \prime } ( x ) = \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$

Question 11

How many relative extrema does $x ^ { 10 } e ^ { x }$ have on the interval (-7, 7)?

Accepted Answer

The answer of How many relative extrema does \(x ^...

Question 12

Solve for x: $3 \ln x - \frac { 1 } { 5 } \ln x ^ { 3 } = 12$

Accepted Answer

The answer of Solve for x: \(3 \ln x -...

Question 13

How much money should be invested now at a yearly rate of 8 percent compounded quarterly so that 10 years from now the account will be worth $11,000?

Accepted Answer

A)  $506.34 
B)  $5,095.13 
C)  $9,023.83 
D)  $4,981.79 
A)  $506.34 
B)  $5,095.13 
C)  $9,023.83 
D)  $4,981.79

Question 14

Find all real numbers x that satisfy the given equation. $$\left( \frac { 1 } { 81 } \right) ^ { x - 1 } = 3 ^ { 4 - 3 x ^ { 2 } }$$

Accepted Answer

A)  0, $- \frac { 4 } { 3 }$ 
B)  0, $\frac { 4 } { 3 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  0 
A)  0, $- \frac { 4 } { 3 }$ 
B)  0, $\frac { 4 } { 3 }$ 
C)  $- \frac { 4 } { 3 }$ 
D)  0

Question 15

Solve for x. Answer exactly and then round to three decimal places. ln(-3x + 38) = -1

Accepted Answer

The answer of Solve for x. Answer exactly and then...

Question 16

If $f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }$ , then $f ^ { \prime } ( x ) = 1 - 2 e ^ { - x }$ .

Accepted Answer

A) True 
 B)False

Question 17

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)  $x = - 6 e ^ { - 6 x }$

Question 18

Find the derivative of $\ln \left[ \left( \ln x ^ { 2 } \right) ^ { 7 } \right]$ .

Accepted Answer

The answer of Find the derivative of \(\ln \left[ \left(...

Question 19

If the amount of a sample of radioactive substance that remains after t years is given by $Q ( t ) = Q _ { 0 } e ^ { - 0.0003 t }$ and 500 grams of the substance remain at the end of 7,000 years, then there were about 4,083 grams present initially.

Accepted Answer

A) True 
 B)False

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

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and this interest is compounded continuously. After 4 years, the money has doubled. The balance at the end of 8 years is $4 A _ { 0 }$ .

How quickly will money triple if it is invested at 7% interest compounded continuously? Round your answer to two decimal places.

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

If $3,000 is invested at 2 percent compounded continuously, what is the balance after 11 years?

$\frac { 3 ^ { 4 } } { 3 ^ { 6 } } = \frac { 1 } { 9 }$

Differentiate the given function. $f ( x ) = 35 - 4 e ^ { - 0.06 x }$

Solve for x if log₃x = 5.

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ $f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }$

How many relative extrema does $x ^ { 10 } e ^ { x }$ have on the interval (-7, 7)?

Solve for x: $3 \ln x - \frac { 1 } { 5 } \ln x ^ { 3 } = 12$

How much money should be invested now at a yearly rate of 8 percent compounded quarterly so that 10 years from now the account will be worth $11,000?

Find all real numbers x that satisfy the given equation. $\left( \frac { 1 } { 81 } \right) ^ { x - 1 } = 3 ^ { 4 - 3 x ^ { 2 } }$

Solve for x. Answer exactly and then round to three decimal places. ln(-3x + 38) = -1

If $f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }$ , then $f ^ { \prime } ( x ) = 1 - 2 e ^ { - x }$ .

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

Find the derivative of $\ln \left[ \left( \ln x ^ { 2 } \right) ^ { 7 } \right]$ .

If the amount of a sample of radioactive substance that remains after t years is given by $Q ( t ) = Q _ { 0 } e ^ { - 0.0003 t }$ and 500 grams of the substance remain at the end of 7,000 years, then there were about 4,083 grams present initially.

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

A sum of money, A0A _ { 0 }A0​ , is invested at a certain fixed interest rate, and this interest is compounded continuously. After 4 years, the money has doubled. The balance at the end of 8 years is 4A04 A _ { 0 }4A0​ .

How quickly will money triple if it is invested at 7% interest compounded continuously? Round your answer to two decimal places.

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.

If $3,000 is invested at 2 percent compounded continuously, what is the balance after 11 years?

3436=19\frac { 3 ^ { 4 } } { 3 ^ { 6 } } = \frac { 1 } { 9 }3634​=91​

Differentiate the given function. f(x)=35−4e−0.06xf ( x ) = 35 - 4 e ^ { - 0.06 x }f(x)=35−4e−0.06x

Solve for x if log3 x = 5.

Use logarithmic differentiation to find f′(x)f ^ { \prime } ( x )f′(x) f(x)=6x+58+6x8f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }f(x)=88+6x6x+5​​

How many relative extrema does x10exx ^ { 10 } e ^ { x }x10ex have on the interval (-7, 7)?

Solve for x: 3ln⁡x−15ln⁡x3=123 \ln x - \frac { 1 } { 5 } \ln x ^ { 3 } = 123lnx−51​lnx3=12

How much money should be invested now at a yearly rate of 8 percent compounded quarterly so that 10 years from now the account will be worth $11,000?

Find all real numbers x that satisfy the given equation. (181)x−1=34−3x2\left( \frac { 1 } { 81 } \right) ^ { x - 1 } = 3 ^ { 4 - 3 x ^ { 2 } }(811​)x−1=34−3x2

Solve for x. Answer exactly and then round to three decimal places. ln(-3x + 38) = -1

If f(x)=(x+2e−x)3f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }f(x)=(x+2e−x)3 , then f′(x)=1−2e−xf ^ { \prime } ( x ) = 1 - 2 e ^ { - x }f′(x)=1−2e−x .

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

Find the derivative of ln⁡[(ln⁡x2)7]\ln \left[ \left( \ln x ^ { 2 } \right) ^ { 7 } \right]ln[(lnx2)7] .

If the amount of a sample of radioactive substance that remains after t years is given by Q(t)=Q0e−0.0003tQ ( t ) = Q _ { 0 } e ^ { - 0.0003 t }Q(t)=Q0​e−0.0003t and 500 grams of the substance remain at the end of 7,000 years, then there were about 4,083 grams present initially.

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

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and this interest is compounded continuously. After 4 years, the money has doubled. The balance at the end of 8 years is $4 A _ { 0 }$ .

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

$\frac { 3 ^ { 4 } } { 3 ^ { 6 } } = \frac { 1 } { 9 }$

Differentiate the given function. $f ( x ) = 35 - 4 e ^ { - 0.06 x }$

Solve for x if log₃x = 5.

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ $f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }$

How many relative extrema does $x ^ { 10 } e ^ { x }$ have on the interval (-7, 7)?

Solve for x: $3 \ln x - \frac { 1 } { 5 } \ln x ^ { 3 } = 12$

Find all real numbers x that satisfy the given equation. $\left( \frac { 1 } { 81 } \right) ^ { x - 1 } = 3 ^ { 4 - 3 x ^ { 2 } }$

If $f ( x ) = \left( x + 2 e ^ { - x } \right) ^ { 3 }$ , then $f ^ { \prime } ( x ) = 1 - 2 e ^ { - x }$ .

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

Find the derivative of $\ln \left[ \left( \ln x ^ { 2 } \right) ^ { 7 } \right]$ .

If the amount of a sample of radioactive substance that remains after t years is given by $Q ( t ) = Q _ { 0 } e ^ { - 0.0003 t }$ and 500 grams of the substance remain at the end of 7,000 years, then there were about 4,083 grams present initially.