Question 1

Find all real numbers x that satisfy the given equation.3x23x = 24

Accepted Answer

Question 2

If $\log _ { 5 } ( 7 x ) = 4$ , then $x = \frac { 62.5 } { 7 }$ .

Accepted Answer

A) True 
 B)False

Question 3

Simplify $\left( 49 ^ { 1 / 2 } - 9 ^ { 1 / 2 } \right) ^ { - 2 }$ .

Accepted Answer

The answer of Simplify \(\left( 49 ^ { 1 /...

Question 4

If $f ( x ) = \sqrt [ 3 ] { 1 + e ^ { 5 x } }$ , then $f ^ { \prime } ( x ) = \frac { 5 e ^ { 5 x } } { 3 \left( 1 + e ^ { 5 x } \right) ^ { 2 / 3 } }$ .

Accepted Answer

A) True 
 B)False

Question 5

Biologists estimate that the population of a biological colony at time t (in minutes) is $P ( t ) = P _ { 0 } e ^ { k t }$ thousand, where $P _ { 0 }$ and k are positive constants. If the population is 1,000 after 20 minutes and is known to double every hour, what is the population after 40 minutes? Round to the nearest whole number.

Accepted Answer

A)  1,258 
B)  1,259 
C)  1,260 
D)  1,261 
A)  1,258 
B)  1,259 
C)  1,260 
D)  1,261

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

Question 7

Differentiate the given function. $f ( x ) = \frac { x ^ { 2 } } { e ^ { 3 x } }$

Accepted Answer

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

Question 8

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $$f ( x ) = \sqrt [ 3 ] { \frac { 6 x + 3 } { 4 + 4 x } }$$

Accepted Answer

A)  $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$ 
B)  $f ^ { \prime } ( x ) = \frac { 1 } { 3 } \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$ 
C)  $f ^ { \prime } ( x ) = \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$ 
D)  $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$ 
A)  $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$ 
B)  $f ^ { \prime } ( x ) = \frac { 1 } { 3 } \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$ 
C)  $f ^ { \prime } ( x ) = \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$ 
D)  $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$

Question 9

The function y = ln 3x is concave downward everywhere.

Accepted Answer

A) True 
 B)False

Question 10

The equation of the tangent line to $f ( x ) = e ^ { x ^ { 5 } }$ at x = 2 is

Accepted Answer

A)  $y = 80 e ^ { 32 } x - 159 e ^ { 32 }$ 
B)  $y = 32 e ^ { 32 x }$ 
C)  $y = 32 e ^ { 32 }$ 
D)  $y = 80 e ^ { 32 } x + 161 e ^ { 32 }$ 
A)  $y = 80 e ^ { 32 } x - 159 e ^ { 32 }$ 
B)  $y = 32 e ^ { 32 x }$ 
C)  $y = 32 e ^ { 32 }$ 
D)  $y = 80 e ^ { 32 } x + 161 e ^ { 32 }$

Question 11

The graph of $x ^ { 3 } e ^ { 3 }$ has

Accepted Answer

A)  a relative maximum at x = 0 
B)  a relative minimum at x = 0 
C)  a point of inflection at x = 0 
D)  nothing significant at x = 0 
A)  a relative maximum at x = 0 
B)  a relative minimum at x = 0 
C)  a point of inflection at x = 0 
D)  nothing significant at x = 0

Question 12

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 13

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 14

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

Accepted Answer

A) True 
 B)False

Question 15

At what interest rate, compounded continuously, should $2,500 be invested today so that 10 years from now the account will be worth $5,000? Round your answer to two decimal places.

Accepted Answer

A)  13.86% 
B)  0.07% 
C)  3.47% 
D)  6.93% 
A)  13.86% 
B)  0.07% 
C)  3.47% 
D)  6.93%

Question 16

Differentiate the given function. $f ( x ) = \ln x ^ { 4 }$

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

The equation of the tangent line to $f ( x ) = 7 \ln x ^ { 4 }$ at x = e is

Accepted Answer

A)  $y = \frac { 28 } { e } x$ 
B)  $y = \frac { 28 } { e } x + 56$ 
C)  $y = \frac { 28 } { e } x - 56$ 
D)  y = 28x 
A)  $y = \frac { 28 } { e } x$ 
B)  $y = \frac { 28 } { e } x + 56$ 
C)  $y = \frac { 28 } { e } x - 56$ 
D)  y = 28x

Question 19

The consumer demand for a certain commodity is $D ( p ) = 1,000.00 e ^ { - 0.19 p }$ 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

The answer of The consumer demand for a certain commodity...

Question 20

An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: $$\begin{array} { l l l } 
\text { Experience (months) } & 0 & 2 \
\text { Output (units per hour) } & 300 & 510
\end{array}$$
 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)  $Q ( t ) = 600 - 300 e ^ { 0.6020 t }$

Find all real numbers x that satisfy the given equation.3^x2³^x = 24

If $\log _ { 5 } ( 7 x ) = 4$ , then $x = \frac { 62.5 } { 7 }$ .

Simplify $\left( 49 ^ { 1 / 2 } - 9 ^ { 1 / 2 } \right) ^ { - 2 }$ .

If $f ( x ) = \sqrt [ 3 ] { 1 + e ^ { 5 x } }$ , then $f ^ { \prime } ( x ) = \frac { 5 e ^ { 5 x } } { 3 \left( 1 + e ^ { 5 x } \right) ^ { 2 / 3 } }$ .

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

Differentiate the given function. $f ( x ) = \frac { x ^ { 2 } } { e ^ { 3 x } }$

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 3 ] { \frac { 6 x + 3 } { 4 + 4 x } }$

The function y = ln 3x is concave downward everywhere.

The equation of the tangent line to $f ( x ) = e ^ { x ^ { 5 } }$ at x = 2 is

The graph of $x ^ { 3 } e ^ { 3 }$ has

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

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

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

At what interest rate, compounded continuously, should $2,500 be invested today so that 10 years from now the account will be worth $5,000? Round your answer to two decimal places.

Differentiate the given function. $f ( x ) = \ln x ^ { 4 }$

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

The equation of the tangent line to $f ( x ) = 7 \ln x ^ { 4 }$ at x = e is

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

Find all real numbers x that satisfy the given equation.3x23x = 24

If log⁡5(7x)=4\log _ { 5 } ( 7 x ) = 4log5​(7x)=4 , then x=62.57x = \frac { 62.5 } { 7 }x=762.5​ .

Simplify (491/2−91/2)−2\left( 49 ^ { 1 / 2 } - 9 ^ { 1 / 2 } \right) ^ { - 2 }(491/2−91/2)−2 .

If f(x)=1+e5x3f ( x ) = \sqrt [ 3 ] { 1 + e ^ { 5 x } }f(x)=31+e5x​ , then f′(x)=5e5x3(1+e5x)2/3f ^ { \prime } ( x ) = \frac { 5 e ^ { 5 x } } { 3 \left( 1 + e ^ { 5 x } \right) ^ { 2 / 3 } }f′(x)=3(1+e5x)2/35e5x​ .

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

Differentiate the given function. f(x)=x2e3xf ( x ) = \frac { x ^ { 2 } } { e ^ { 3 x } }f(x)=e3xx2​

Use logarithmic differentiation to find f′(x)f ^ { \prime } ( x )f′(x) . f(x)=6x+34+4x3f ( x ) = \sqrt [ 3 ] { \frac { 6 x + 3 } { 4 + 4 x } }f(x)=34+4x6x+3​​

The function y = ln 3x is concave downward everywhere.

The equation of the tangent line to f(x)=ex5f ( x ) = e ^ { x ^ { 5 } }f(x)=ex5 at x = 2 is

The graph of x3e3x ^ { 3 } e ^ { 3 }x3e3 has

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

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

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

At what interest rate, compounded continuously, should $2,500 be invested today so that 10 years from now the account will be worth $5,000? Round your answer to two decimal places.

Differentiate the given function. f(x)=ln⁡x4f ( x ) = \ln x ^ { 4 }f(x)=lnx4

Differentiate the given function. f(x)=x3e−3xf ( x ) = x ^ { 3 } e ^ { - 3 x }f(x)=x3e−3x

The equation of the tangent line to f(x)=7ln⁡x4f ( x ) = 7 \ln x ^ { 4 }f(x)=7lnx4 at x = e is

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

Find all real numbers x that satisfy the given equation.3^x2³^x = 24

If $\log _ { 5 } ( 7 x ) = 4$ , then $x = \frac { 62.5 } { 7 }$ .

Simplify $\left( 49 ^ { 1 / 2 } - 9 ^ { 1 / 2 } \right) ^ { - 2 }$ .

If $f ( x ) = \sqrt [ 3 ] { 1 + e ^ { 5 x } }$ , then $f ^ { \prime } ( x ) = \frac { 5 e ^ { 5 x } } { 3 \left( 1 + e ^ { 5 x } \right) ^ { 2 / 3 } }$ .

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

Differentiate the given function. $f ( x ) = \frac { x ^ { 2 } } { e ^ { 3 x } }$

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 3 ] { \frac { 6 x + 3 } { 4 + 4 x } }$

The equation of the tangent line to $f ( x ) = e ^ { x ^ { 5 } }$ at x = 2 is

The graph of $x ^ { 3 } e ^ { 3 }$ has

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

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

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

Differentiate the given function. $f ( x ) = \ln x ^ { 4 }$

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

The equation of the tangent line to $f ( x ) = 7 \ln x ^ { 4 }$ at x = e is