Question 1

Find $y ^ { \prime }$ . $y = \frac { 8 + \ln x } { x ^ { 6 } }$

Accepted Answer

A)  $\frac { 6 \ln x + 47 } { x ^ { 5 } }$ 
B)  $- \frac { 48 \ln x + 5 } { x ^ { 5 } }$ 
C)  $- \frac { 6 \ln x + 47 } { x ^ { 7 } }$ 
D)  $- \frac { 8 \ln x + 5 } { x ^ { 7 } }$ 
E)  $\frac { 48 \ln x - 47 } { x ^ { 7 } }$ 
A)  $\frac { 6 \ln x + 47 } { x ^ { 5 } }$ 
B)  $- \frac { 48 \ln x + 5 } { x ^ { 5 } }$ 
C)  $- \frac { 6 \ln x + 47 } { x ^ { 7 } }$ 
D)  $- \frac { 8 \ln x + 5 } { x ^ { 7 } }$ 
E)  $\frac { 48 \ln x - 47 } { x ^ { 7 } }$

Question 2

Find $y ^ { \prime }$ . $y = 9 \log _ { 5 } x$

Accepted Answer

A)  $\frac { 9 } { x \ln 5 }$ 
B)  $\frac { 5 } { x \ln 9 }$ 
C)  $\frac { 9 } { x }$ 
D)  $\frac { 9 \ln 5 } { x }$ 
E)  $\frac { x } { 9 \ln 5 }$ 
A)  $\frac { 9 } { x \ln 5 }$ 
B)  $\frac { 5 } { x \ln 9 }$ 
C)  $\frac { 9 } { x }$ 
D)  $\frac { 9 \ln 5 } { x }$ 
E)  $\frac { x } { 9 \ln 5 }$

Question 3

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 16 }$ .

Accepted Answer

A)  $\frac { d y } { d x } = \frac { 1 } { 2 x - 16 }$ 
B)  $\frac { d y } { d x } = \frac { x } { 2 x - 16 }$ 
C)  $\frac { d y } { d x } = \frac { 1 } { x ^ { 2 } - 16 }$ 
D)  $\frac { d y } { d x } = \frac { x } { x ^ { 2 } - 16 }$ 
E)  $\frac { d y } { d x } = \frac { 2 x } { \sqrt { x ^ { 2 } - 16 } }$ 
A)  $\frac { d y } { d x } = \frac { 1 } { 2 x - 16 }$ 
B)  $\frac { d y } { d x } = \frac { x } { 2 x - 16 }$ 
C)  $\frac { d y } { d x } = \frac { 1 } { x ^ { 2 } - 16 }$ 
D)  $\frac { d y } { d x } = \frac { x } { x ^ { 2 } - 16 }$ 
E)  $\frac { d y } { d x } = \frac { 2 x } { \sqrt { x ^ { 2 } - 16 } }$

Question 4

Determine the principal P that must be invested at interest rate r compounded continuously, so that $1,000,000 will be available for retirement in years $r = 7.5 \%$ , $t = 40$ .

Accepted Answer

A) $49787.07 
B) $50787.07 
C) $49000.04 
D) $40000.06 
A) $49787.07 
B) $50787.07 
C) $49000.04 
D) $40000.06

Question 5

The cumulative sales (in thousands of units) of a new product after it has been on the market for t years may be modeled by $S = 30 ( 1 - $   ) . During the first year, 5000 units were sold. What is the saturation point for this product? How many units will be sold after 6 years?

Accepted Answer

A) The saturation point for the market is 3000 units and 19,953 units will be sold after 6 years. 
B) The saturation point for the market is 30,000 units and 27,366 units will be sold after 6 years. 
C) The saturation point for the market is 30,000 units and 19,953 units will be sold after 6 years. 
D) The saturation point for the market is 30,000 units and 20,076 units will be sold after 6 years. 
E) The saturation point for the market is 3000 units and 27,366 units will be sold after 6 years. 
A) The saturation point for the market is 3000 units and 19,953 units will be sold after 6 years. 
B) The saturation point for the market is 30,000 units and 27,366 units will be sold after 6 years. 
C) The saturation point for the market is 30,000 units and 19,953 units will be sold after 6 years. 
D) The saturation point for the market is 30,000 units and 20,076 units will be sold after 6 years. 
E) The saturation point for the market is 3000 units and 27,366 units will be sold after 6 years.

Question 6

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

Accepted Answer

A) (0, 1) 
B)  $\left( 11 , e ^ { 11 } \right)$ , (0, 0) 
C)  $( 1,11 ) , \left( 11 , e ^ { - 11 } \right)$ 
D) no relative extrema 
E)  $\left( 11 , e ^ { - 11 } \right)$ 
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 7

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

Accepted Answer

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)  $\frac { 7 } { 4 x + 7 }$ 
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 8

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

Accepted Answer

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)  $\frac { d y } { d x } = 4 x ^ { 3 } e ^ { x ^ { 10 } }$ 
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 9

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

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $\frac { 5 } { x }$ 
C)  $\frac { 1 } { 5 x }$ 
D)  $\frac { 1 } { x ^ { 2 } }$ 
E)  $\frac { 1 } { 5 x ^ { 2 } }$ 
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 10

Use a change-of-base formula to rewrite the logarithm in terms of natural logarithms. $y = \log _ { 4 } x$

Accepted Answer

A)  $y = 4 \ln x$ 
B)  $y = \ln ( x - 4 )$ 
C)  $y = \ln x - \ln 4$ 
D)  $y = 4 ^ { x }$ 
E)  $y = \frac { \ln x } { \ln 4 }$ 
A)  $y = 4 \ln x$ 
B)  $y = \ln ( x - 4 )$ 
C)  $y = \ln x - \ln 4$ 
D)  $y = 4 ^ { x }$ 
E)  $y = \frac { \ln x } { \ln 4 }$

Question 11

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

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

Accepted Answer

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)  $f ^ { \prime \prime } ( x ) = ( 496 + 320 x ) e ^ { - 8 x }$ 
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 13

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

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $\frac { 5 } { x }$ 
C)  $\frac { 1 } { 5 x }$ 
D)  $\frac { 1 } { x ^ { 2 } }$ 
E)  $\frac { 1 } { 5 x ^ { 2 } }$ 
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 14

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

Accepted Answer

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 
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 15

Find the relative maxima, and use a graphing utility to check your results. $y = 4 \ln x - 7 x$

Accepted Answer

A)  $y = - 10.01$ 
B)  $y = - 6.24$ 
C)  $y = - 8.78$ 
D)  $y = 0.00$ 
E) does not exist 
A)  $y = - 10.01$ 
B)  $y = - 6.24$ 
C)  $y = - 8.78$ 
D)  $y = 0.00$ 
E) does not exist

Question 16

The cost of producing x units of a product is modeled by $C = 800 + 700 x - 700 \ln x,$ $x \geq 1$ . Find the average cost function $\overline{ C }$ .

Accepted Answer

A)  $C = \frac { 800 } { x } + \frac { 700 } { x } - \frac { 700 \ln x } { x }$ 
B)  $C = \frac { 800 } { x } + 700 - 700 \ln x$ 
C)  $C = \frac { 800 } { x } + 700 - \frac { 700 \ln x } { x }$ 
D)  $C = 800 + 700 - \frac { 700 \ln x } { x }$ 
E)  $C = 800 + \frac { 700 } { x } - 700 \ln x$ 
A)  $C = \frac { 800 } { x } + \frac { 700 } { x } - \frac { 700 \ln x } { x }$ 
B)  $C = \frac { 800 } { x } + 700 - 700 \ln x$ 
C)  $C = \frac { 800 } { x } + 700 - \frac { 700 \ln x } { x }$ 
D)  $C = 800 + 700 - \frac { 700 \ln x } { x }$ 
E)  $C = 800 + \frac { 700 } { x } - 700 \ln x$

Question 17

If $\ln 7 x y = 6 , \text { find } d y / d x$

Accepted Answer

A)  $\frac { d y } { d x } = - \frac { y } { x }$ 
B)  $\frac { d y } { d x } = - \frac { 7 y } { x }$ 
C)  $\frac { d y } { d x } = - \frac { y } { 7 x }$ 
D)  $\frac { d y } { d x } = - \frac { 7 y } { 6 x }$ 
E)  $\frac { d y } { d x } = - \frac { 6 y } { 7 x }$ 
A)  $\frac { d y } { d x } = - \frac { y } { x }$ 
B)  $\frac { d y } { d x } = - \frac { 7 y } { x }$ 
C)  $\frac { d y } { d x } = - \frac { y } { 7 x }$ 
D)  $\frac { d y } { d x } = - \frac { 7 y } { 6 x }$ 
E)  $\frac { d y } { d x } = - \frac { 6 y } { 7 x }$

Question 18

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

Accepted Answer

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)  $p ^ { \prime } = 5 e ^ { q ^ { 4 } } \left( 4 q ^ { 3 } + 5 \right)$ 
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 19

Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 8 } } { 8 } - \ln x$ . Use a graphing utility to confirm your results.

Accepted Answer

A) relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; inflection point at x = 0 
B) relative minimum value $y = 8$ at $x = 1$ ; inflection point at x = 0 
C) relative minimum value $y = 8$ at $x = 1$ ; no inflection points 
D) relative minimum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points 
E) relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points 
A) relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; inflection point at x = 0 
B) relative minimum value $y = 8$ at $x = 1$ ; inflection point at x = 0 
C) relative minimum value $y = 8$ at $x = 1$ ; no inflection points 
D) relative minimum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points 
E) relative maximum value $y = \frac { 1 } { 8 }$ at $x = 1$ ; no inflection points

Question 20

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.

Accepted Answer

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

Find $y ^ { \prime }$ . $y = \frac { 8 + \ln x } { x ^ { 6 } }$

Find $y ^ { \prime }$ . $y = 9 \log _ { 5 } x$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 16 }$ .

Determine the principal P that must be invested at interest rate r compounded continuously, so that $1,000,000 will be available for retirement in years $r = 7.5 \%$ , $t = 40$ .

The cumulative sales (in thousands of units) of a new product after it has been on the market for t years may be modeled by $S = 30 ( 1 -$ ) . During the first year, 5000 units were sold. What is the saturation point for this product? How many units will be sold after 6 years?

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

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

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

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

Use a change-of-base formula to rewrite the logarithm in terms of natural logarithms. $y = \log _ { 4 } x$

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

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

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

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

Find the relative maxima, and use a graphing utility to check your results. $y = 4 \ln x - 7 x$

The cost of producing x units of a product is modeled by $C = 800 + 700 x - 700 \ln x,$ $x \geq 1$ . Find the average cost function $\overline{ C }$ .

If $\ln 7 x y = 6 , \text { find } d y / d x$

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

Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 8 } } { 8 } - \ln x$ . Use a graphing utility to confirm your results.

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.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

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Exam 11: Derivatives of Exponential and Logarithmic Functions

Find y′y ^ { \prime }y′ . y=8+ln⁡xx6y = \frac { 8 + \ln x } { x ^ { 6 } }y=x68+lnx​

Find y′y ^ { \prime }y′ . y=9log⁡5xy = 9 \log _ { 5 } xy=9log5​x

Find the derivative of the function y=ln⁡x2−16y = \ln \sqrt { x ^ { 2 } - 16 }y=lnx2−16​ .

Determine the principal P that must be invested at interest rate r compounded continuously, so that $1,000,000 will be available for retirement in years r=7.5%r = 7.5 \%r=7.5% , t=40t = 40t=40 .

The cumulative sales (in thousands of units) of a new product after it has been on the market for t years may be modeled by S=30(1−S = 30 ( 1 - S=30(1− ) . During the first year, 5000 units were sold. What is the saturation point for this product? How many units will be sold after 6 years?

Find the extrema of the function f(x)=111−e−xf ( x ) = \frac { 1 } { 11 - e ^ { - x } }f(x)=11−e−x1​ .

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

Find dydx\frac { d y } { d x }dxdy​ if y=x4ex10y = x ^ { 4 } e ^ { x ^ { 10 } }y=x4ex10 .

Find the derivative of the following function. y=ln⁡x5y = \ln x ^ { 5 }y=lnx5

Use a change-of-base formula to rewrite the logarithm in terms of natural logarithms. y=log⁡4xy = \log _ { 4 } xy=log4​x

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

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

Find the derivative of the following function. y=8+ln⁡5xy = 8 + \ln 5 xy=8+ln5x

Write the following expression as a logarithm of a single quantity. 5ln⁡x−3ln⁡(x2+3)5 \ln x - 3 \ln \left( x ^ { 2 } + 3 \right)5lnx−3ln(x2+3)

Find the relative maxima, and use a graphing utility to check your results. y=4ln⁡x−7xy = 4 \ln x - 7 xy=4lnx−7x

The cost of producing x units of a product is modeled by C=800+700x−700ln⁡x,C = 800 + 700 x - 700 \ln x,C=800+700x−700lnx, x≥1x \geq 1x≥1 . Find the average cost function C‾\overline{ C }C .

If ln⁡7xy=6, find dy/dx\ln 7 x y = 6 , \text { find } d y / d xln7xy=6, find dy/dx

Find the derivative of the following function. p=5qeq4p = 5 q e ^ { q ^ { 4 } }p=5qeq4

Locate any relative extrema and inflection points of the function y=x88−ln⁡xy = \frac { x ^ { 8 } } { 8 } - \ln xy=8x8​−lnx . Use a graphing utility to confirm your results.

The demand function for a product is modeled by p=3000(1−55+e−0.0002x)p = 3000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0002 x } } \right)p=3000(1−5+e−0.0002x5​) . Find the price of the product if the quantity demanded is x = 100. Round your answer to two decimal places where applicable.

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Integration and Its Applications

Techniques of Integration

Functions of Several Variables

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Find $y ^ { \prime }$ . $y = \frac { 8 + \ln x } { x ^ { 6 } }$

Find $y ^ { \prime }$ . $y = 9 \log _ { 5 } x$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } - 16 }$ .

Determine the principal P that must be invested at interest rate r compounded continuously, so that $1,000,000 will be available for retirement in years $r = 7.5 \%$ , $t = 40$ .

The cumulative sales (in thousands of units) of a new product after it has been on the market for t years may be modeled by $S = 30 ( 1 -$ ) . During the first year, 5000 units were sold. What is the saturation point for this product? How many units will be sold after 6 years?

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

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

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

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

Use a change-of-base formula to rewrite the logarithm in terms of natural logarithms. $y = \log _ { 4 } x$

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

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

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

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

Find the relative maxima, and use a graphing utility to check your results. $y = 4 \ln x - 7 x$

The cost of producing x units of a product is modeled by $C = 800 + 700 x - 700 \ln x,$ $x \geq 1$ . Find the average cost function $\overline{ C }$ .

If $\ln 7 x y = 6 , \text { find } d y / d x$

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

Locate any relative extrema and inflection points of the function $y = \frac { x ^ { 8 } } { 8 } - \ln x$ . Use a graphing utility to confirm your results.

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.