Question 1

Find $\frac { d y } { d x } \text { if } y = \ln \left( \frac { 5 x + 7 } { x ^ { 2 } - 4 } \right) ^ { 1 / 8 }$ .

Accepted Answer

A)  $\frac { 5 x ^ { 2 } + 14 x + 5 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
B)  $- \frac { 5 x ^ { 2 } + 7 x + 20 } { ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
C)  $\frac { 5 x ^ { 2 } + 7 x + 20 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
D)  $\frac { 5 x ^ { 2 } + 14 x + 20 } { ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
E)  $- \frac { 5 x ^ { 2 } + 14 x + 20 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
A)  $\frac { 5 x ^ { 2 } + 14 x + 5 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
B)  $- \frac { 5 x ^ { 2 } + 7 x + 20 } { ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
C)  $\frac { 5 x ^ { 2 } + 7 x + 20 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
D)  $\frac { 5 x ^ { 2 } + 14 x + 20 } { ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$ 
E)  $- \frac { 5 x ^ { 2 } + 14 x + 20 } { 8 ( 5 x + 7 ) \left( x ^ { 2 } - 4 \right) }$

Question 2

Assume the population P (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function $P ( t ) = 255.82 ( 1.606 ) ^ { t }$ , where t is the time in years, with t = 2 corresponding to1992. Use the model to estimate the population in the year 2006. Round your answer to the nearest million.

Accepted Answer

A) 4389 million 
B) 2733 million 
C) 7049 million 
D) 660 million 
E) 4388 million 
A) 4389 million 
B) 2733 million 
C) 7049 million 
D) 660 million 
E) 4388 million

Question 3

Find $\frac { d y } { d x }$ if $y = \ln \left( x ^ { 9 } ( x + 5 ) ^ { \frac { 1 } { 2 } } \right)$

Accepted Answer

A)  $\frac { 9 ( 2 x + 5 ) } { 2 x ( x + 5 ) }$ 
B)  $\frac { 19 x + 90 } { 2 x ( x + 5 ) }$ 
C)  $\frac { 9 ( 2 x + 10 ) } { x ( x + 5 ) }$ 
D)  $\frac { 19 x + 45 } { x ( x + 5 ) }$ 
E)  $\frac { 19 x + 18 } { x ( x + 5 ) }$ 
A)  $\frac { 9 ( 2 x + 5 ) } { 2 x ( x + 5 ) }$ 
B)  $\frac { 19 x + 90 } { 2 x ( x + 5 ) }$ 
C)  $\frac { 9 ( 2 x + 10 ) } { x ( x + 5 ) }$ 
D)  $\frac { 19 x + 45 } { x ( x + 5 ) }$ 
E)  $\frac { 19 x + 18 } { x ( x + 5 ) }$

Question 4

Find the derivative of the following function. $y = 1 - 6 e ^ { - x ^ { 6 } }$

Accepted Answer

A)  $y ^ { \prime } = 36 x ^ { 5 } e ^ { - x ^ { 6 } }$ 
B)  $y ^ { \prime } = - 36 x ^ { 5 } e ^ { - x ^ { 6 } }$ 
C)  $y ^ { \prime } = 6 e ^ { - x ^ { 6 } }$ 
D)  $y ^ { \prime } = 6 x ^ { 6 } e ^ { - x ^ { 6 } }$ 
E)  $y ^ { \prime } = - 6 x ^ { 6 } e ^ { - x ^ { 6 } }$ 
A)  $y ^ { \prime } = 36 x ^ { 5 } e ^ { - x ^ { 6 } }$ 
B)  $y ^ { \prime } = - 36 x ^ { 5 } e ^ { - x ^ { 6 } }$ 
C)  $y ^ { \prime } = 6 e ^ { - x ^ { 6 } }$ 
D)  $y ^ { \prime } = 6 x ^ { 6 } e ^ { - x ^ { 6 } }$ 
E)  $y ^ { \prime } = - 6 x ^ { 6 } e ^ { - x ^ { 6 } }$

Question 5

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } + 3 }$ .

Accepted Answer

A)  $\frac { 2 x } { x ^ { 2 } + 3 }$ 
B)  $\frac { x } { 2 x + 3 }$ 
C)  $\frac { 1 } { \sqrt { x ^ { 2 } + 3 } }$ 
D)  $\frac { 2 x } { \sqrt { x ^ { 2 } + 3 } }$ 
E)  $\frac { x } { x ^ { 2 } + 3 }$ 
A)  $\frac { 2 x } { x ^ { 2 } + 3 }$ 
B)  $\frac { x } { 2 x + 3 }$ 
C)  $\frac { 1 } { \sqrt { x ^ { 2 } + 3 } }$ 
D)  $\frac { 2 x } { \sqrt { x ^ { 2 } + 3 } }$ 
E)  $\frac { x } { x ^ { 2 } + 3 }$

Question 6

The effective yield is the annual rate i that will produce the same interest per year as the nominal rate compounded n times per year. For a rate that is compounded n times per year, the formula for effective yield is given as $i = \left( 1 + \frac { r } { n } \right) ^ { n } - 1$ . Find the effective yield for a nominal rate of 10%, compounded monthly. Round your answer to two decimal places.

Accepted Answer

A) 1.05% 
B) 10.71% 
C) 11.10% 
D) 1.61% 
E) 10.47% 
A) 1.05% 
B) 10.71% 
C) 11.10% 
D) 1.61% 
E) 10.47%

Question 7

Use a graphing utility to graph the function $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } = 2 ^ { - x }$ .

Accepted Answer

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

Question 8

Find the second derivative of the function $f ( x ) = 7 ^ { x }$ .

Accepted Answer

A)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) 7 ^ { x }$ 
B)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { x }$ 
C)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { x }$ 
D)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { 2 x }$ 
E)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { 3 x }$ 
A)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) 7 ^ { x }$ 
B)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { x }$ 
C)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { x }$ 
D)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 2 } 7 ^ { 2 x }$ 
E)  $f ^ { \prime \prime } ( x ) = ( \ln 7 ) ^ { 3 } 7 ^ { 3 x }$

Question 9

Solve the exponential equation. Give the answer correct to 3 decimal places. $57 = \frac { 77 } { 1 + 4 e ^ { - 0.1 x } }$

Accepted Answer

A) -2.434 
B) -0.877 
C) 5.315 
D) -0.532 
E) 24.336 
A) -2.434 
B) -0.877 
C) 5.315 
D) -0.532 
E) 24.336

Question 10

Solve the exponential equation. Give the answer correct to 3 decimal places. $10,000 = 3500 e ^ { 0.3 x }$

Accepted Answer

A) 0.886 
B) 2.953 
C) 3.762 
D) 0.315 
E) 3.499 
A) 0.886 
B) 2.953 
C) 3.762 
D) 0.315 
E) 3.499

Question 11

The cost of producing x units of a product is modeled by $C = 900 + 300 x - 300 \ln x,$ $x \geq 1$ . Find the minimum average cost analytically. Round your answer to two decimal places.

Accepted Answer

A) 300.00 dollars per unit 
B) 295.96 dollars per unit 
C) 294.51 dollars per unit 
D) 296.74 dollars per unit 
E) 294.61 dollars per unit 
A) 300.00 dollars per unit 
B) 295.96 dollars per unit 
C) 294.51 dollars per unit 
D) 296.74 dollars per unit 
E) 294.61 dollars per unit

Question 12

Use a calculator to evaluate the logarithm $\log _ { 5 } 45$ . Round your answer to three decimal places.

Accepted Answer

A) 0.423 
B) 6.127 
C) 3.638 
D) 7.400 
E) 2.365 
A) 0.423 
B) 6.127 
C) 3.638 
D) 7.400 
E) 2.365

Question 13

The average typing speed N (in words per minute) after t weeks of lessons is modeled by $N = \frac { 91 } { 1 + 7.5 e ^ { - 0.15 t } }$ . Find the rate at which the typing speed is changing when t = 20 weeks. Round your answer to two decimal places.

Accepted Answer

A) 2.40 words/min/week 
B) 2.70 words/min/week 
C) 3.71 words/min/week 
D) 4.93 words/min/week 
E) 6.24 words/min/week 
A) 2.40 words/min/week 
B) 2.70 words/min/week 
C) 3.71 words/min/week 
D) 4.93 words/min/week 
E) 6.24 words/min/week

Question 14

The average time between incoming calls at a switchboard is 3 minutes. If a call has just come in, the probability that the next call will come within the next t minutes is $P ( t ) = 1 - e ^ { - t / 3 }$ . Find the probability that the next call will come within the next $\frac { 3 } { 6 }$ minute. Round your answer to two decimal places.

Accepted Answer

A) 15.35% 
B) 1.54% 
C) 184.65% 
D) 17.58% 
E) 5.08% 
A) 15.35% 
B) 1.54% 
C) 184.65% 
D) 17.58% 
E) 5.08%

Question 15

How much more interest will be earned if $6000 is invested for 6 years at an annual rate of 9% compounded continuously, instead of at 9% compounded quarterly?

Accepted Answer

A) $20.72 
B) $40.72 
C) $61.44 
D) $994.60 
E) $1035.32 
A) $20.72 
B) $40.72 
C) $61.44 
D) $994.60 
E) $1035.32

Question 16

Find the derivative of the following function. $y = 8 x ^ { 3 } - 6 e ^ { x }$

Accepted Answer

A)  $y ^ { \prime } = 24 x ^ { 2 } - 6 x e ^ { x - 1 }$ 
B)  $y ^ { \prime } = 8 x ^ { 2 } - 6 x e ^ { x - 1 }$ 
C)  $y ^ { \prime } = 8 x ^ { 2 } - 6 e ^ { x }$ 
D)  $y ^ { \prime } = 24 x ^ { 2 } - e ^ { x }$ 
E)  $y ^ { \prime } = 24 x ^ { 2 } - 6 e ^ { x }$ 
A)  $y ^ { \prime } = 24 x ^ { 2 } - 6 x e ^ { x - 1 }$ 
B)  $y ^ { \prime } = 8 x ^ { 2 } - 6 x e ^ { x - 1 }$ 
C)  $y ^ { \prime } = 8 x ^ { 2 } - 6 e ^ { x }$ 
D)  $y ^ { \prime } = 24 x ^ { 2 } - e ^ { x }$ 
E)  $y ^ { \prime } = 24 x ^ { 2 } - 6 e ^ { x }$

Question 17

Sketch the graph of the function $f ( x ) = e ^ { 3 x + 2 }$ .

Accepted Answer

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

Question 18

Find the derivative of the following function. $y = 9 \ln \left( x ^ { 8 } - 4 \right)$

Accepted Answer

A)  $\frac { 9 } { x ^ { 8 } - 4 }$ 
B)  $\frac { 8 x ^ { 7 } } { x ^ { 8 } - 4 }$ 
C)  $\frac { 8 } { x ^ { 8 } - 4 }$ 
D)  $\frac { 72 x ^ { 7 } } { x ^ { 8 } - 4 }$ 
E)  $\frac { 9 x ^ { 7 } } { x ^ {8 } - 4 }$ 
A)  $\frac { 9 } { x ^ { 8 } - 4 }$ 
B)  $\frac { 8 x ^ { 7 } } { x ^ { 8 } - 4 }$ 
C)  $\frac { 8 } { x ^ { 8 } - 4 }$ 
D)  $\frac { 72 x ^ { 7 } } { x ^ { 8 } - 4 }$ 
E)  $\frac { 9 x ^ { 7 } } { x ^ {8 } - 4 }$

Question 19

Locate any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 9 } } { 8 }$ .

Accepted Answer

A) no relative extrema; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
B) relative maximum at $x = \left( \frac { 8 } { e } \right) ^ { \frac { 1 } { 9 } }$ ; inflection point at $x = 8 e ^ { \frac { - 1 } { 9 } }$ 
C) relative minimum at $x = 8 e ^ { \frac { 1 } { 9 } }$ ; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
D) no relative extrema; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
E) relative minimum at $x = \left( \frac { 8 } { e } \right) ^ { \frac { 1 } { 9 } }$ ; no inflection points 
A) no relative extrema; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
B) relative maximum at $x = \left( \frac { 8 } { e } \right) ^ { \frac { 1 } { 9 } }$ ; inflection point at $x = 8 e ^ { \frac { - 1 } { 9 } }$ 
C) relative minimum at $x = 8 e ^ { \frac { 1 } { 9 } }$ ; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
D) no relative extrema; inflection point at $x = 8 e ^ { \frac { - 17 } { 72 } }$ 
E) relative minimum at $x = \left( \frac { 8 } { e } \right) ^ { \frac { 1 } { 9 } }$ ; no inflection points

Question 20

Find the future value if $5000 is invested for 2 years at an annual rate of 10% compounded quarterly.

Accepted Answer

A)  $\$$ 4000.00 
B)  $\$$ 6077.53 
C)  $\$$ 5253.12 
D)  $\$$ 6050.00 
E)  $\$$ 6092.01 
A)  $\$$ 4000.00 
B)  $\$$ 6077.53 
C)  $\$$ 5253.12 
D)  $\$$ 6050.00 
E)  $\$$ 6092.01

Find $\frac { d y } { d x } \text { if } y = \ln \left( \frac { 5 x + 7 } { x ^ { 2 } - 4 } \right) ^ { 1 / 8 }$ .

Find $\frac { d y } { d x }$ if $y = \ln \left( x ^ { 9 } ( x + 5 ) ^ { \frac { 1 } { 2 } } \right)$

Find the derivative of the following function. $y = 1 - 6 e ^ { - x ^ { 6 } }$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } + 3 }$ .

Use a graphing utility to graph the function $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } = 2 ^ { - x }$ .

Find the second derivative of the function $f ( x ) = 7 ^ { x }$ .

Solve the exponential equation. Give the answer correct to 3 decimal places. $57 = \frac { 77 } { 1 + 4 e ^ { - 0.1 x } }$

Solve the exponential equation. Give the answer correct to 3 decimal places. $10,000 = 3500 e ^ { 0.3 x }$

The cost of producing x units of a product is modeled by $C = 900 + 300 x - 300 \ln x,$ $x \geq 1$ . Find the minimum average cost analytically. Round your answer to two decimal places.

Use a calculator to evaluate the logarithm $\log _ { 5 } 45$ . Round your answer to three decimal places.

The average typing speed N (in words per minute) after t weeks of lessons is modeled by $N = \frac { 91 } { 1 + 7.5 e ^ { - 0.15 t } }$ . Find the rate at which the typing speed is changing when t = 20 weeks. Round your answer to two decimal places.

How much more interest will be earned if $6000 is invested for 6 years at an annual rate of 9% compounded continuously, instead of at 9% compounded quarterly?

Find the derivative of the following function. $y = 8 x ^ { 3 } - 6 e ^ { x }$

Sketch the graph of the function $f ( x ) = e ^ { 3 x + 2 }$ .

Find the derivative of the following function. $y = 9 \ln \left( x ^ { 8 } - 4 \right)$

Locate any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 9 } } { 8 }$ .

Find the future value if $5000 is invested for 2 years at an annual rate of 10% compounded quarterly.

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 dydx if y=ln⁡(5x+7x2−4)1/8\frac { d y } { d x } \text { if } y = \ln \left( \frac { 5 x + 7 } { x ^ { 2 } - 4 } \right) ^ { 1 / 8 }dxdy​ if y=ln(x2−45x+7​)1/8 .

Find dydx\frac { d y } { d x }dxdy​ if y=ln⁡(x9(x+5)12)y = \ln \left( x ^ { 9 } ( x + 5 ) ^ { \frac { 1 } { 2 } } \right)y=ln(x9(x+5)21​)

Find the derivative of the following function. y=1−6e−x6y = 1 - 6 e ^ { - x ^ { 6 } }y=1−6e−x6

Find the derivative of the function y=ln⁡x2+3y = \ln \sqrt { x ^ { 2 } + 3 }y=lnx2+3​ .

Use a graphing utility to graph the function f(x)=(12)x=2−xf ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } = 2 ^ { - x }f(x)=(21​)x=2−x .

Find the second derivative of the function f(x)=7xf ( x ) = 7 ^ { x }f(x)=7x .

Solve the exponential equation. Give the answer correct to 3 decimal places. 57=771+4e−0.1x57 = \frac { 77 } { 1 + 4 e ^ { - 0.1 x } }57=1+4e−0.1x77​

Solve the exponential equation. Give the answer correct to 3 decimal places. 10,000=3500e0.3x10,000 = 3500 e ^ { 0.3 x }10,000=3500e0.3x

The cost of producing x units of a product is modeled by C=900+300x−300ln⁡x,C = 900 + 300 x - 300 \ln x,C=900+300x−300lnx, x≥1x \geq 1x≥1 . Find the minimum average cost analytically. Round your answer to two decimal places.

Use a calculator to evaluate the logarithm log⁡545\log _ { 5 } 45log5​45 . Round your answer to three decimal places.

The average typing speed N (in words per minute) after t weeks of lessons is modeled by N=911+7.5e−0.15tN = \frac { 91 } { 1 + 7.5 e ^ { - 0.15 t } }N=1+7.5e−0.15t91​ . Find the rate at which the typing speed is changing when t = 20 weeks. Round your answer to two decimal places.

How much more interest will be earned if $6000 is invested for 6 years at an annual rate of 9% compounded continuously, instead of at 9% compounded quarterly?

Find the derivative of the following function. y=8x3−6exy = 8 x ^ { 3 } - 6 e ^ { x }y=8x3−6ex

Sketch the graph of the function f(x)=e3x+2f ( x ) = e ^ { 3 x + 2 }f(x)=e3x+2 .

Find the derivative of the following function. y=9ln⁡(x8−4)y = 9 \ln \left( x ^ { 8 } - 4 \right)y=9ln(x8−4)

Locate any relative extrema and inflection points of the function y=xln⁡x98y = x \ln \frac { x ^ { 9 } } { 8 }y=xln8x9​ .

Find the future value if $5000 is invested for 2 years at an annual rate of 10% compounded quarterly.

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 $\frac { d y } { d x } \text { if } y = \ln \left( \frac { 5 x + 7 } { x ^ { 2 } - 4 } \right) ^ { 1 / 8 }$ .

Find $\frac { d y } { d x }$ if $y = \ln \left( x ^ { 9 } ( x + 5 ) ^ { \frac { 1 } { 2 } } \right)$

Find the derivative of the following function. $y = 1 - 6 e ^ { - x ^ { 6 } }$

Find the derivative of the function $y = \ln \sqrt { x ^ { 2 } + 3 }$ .

Use a graphing utility to graph the function $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x } = 2 ^ { - x }$ .

Find the second derivative of the function $f ( x ) = 7 ^ { x }$ .

Solve the exponential equation. Give the answer correct to 3 decimal places. $57 = \frac { 77 } { 1 + 4 e ^ { - 0.1 x } }$

Solve the exponential equation. Give the answer correct to 3 decimal places. $10,000 = 3500 e ^ { 0.3 x }$

The cost of producing x units of a product is modeled by $C = 900 + 300 x - 300 \ln x,$ $x \geq 1$ . Find the minimum average cost analytically. Round your answer to two decimal places.

Use a calculator to evaluate the logarithm $\log _ { 5 } 45$ . Round your answer to three decimal places.

The average typing speed N (in words per minute) after t weeks of lessons is modeled by $N = \frac { 91 } { 1 + 7.5 e ^ { - 0.15 t } }$ . Find the rate at which the typing speed is changing when t = 20 weeks. Round your answer to two decimal places.

Find the derivative of the following function. $y = 8 x ^ { 3 } - 6 e ^ { x }$

Sketch the graph of the function $f ( x ) = e ^ { 3 x + 2 }$ .

Find the derivative of the following function. $y = 9 \ln \left( x ^ { 8 } - 4 \right)$

Locate any relative extrema and inflection points of the function $y = x \ln \frac { x ^ { 9 } } { 8 }$ .