Question 1

What percent of a present amount of radioactive radium $\left( { } ^ { 226 } R a \right)$ will remain after 900 years?

Accepted Answer

A) 45% 
B) 25% 
C) 65% 
D) 68%. 
A) 45% 
B) 25% 
C) 65% 
D) 68%.

Question 2

How long (in years) would $450 have to be invested at an annual rate of 12%, compounded continuously, to amount to $790?

Accepted Answer

A) 6.30 years 
B) 4.97 years 
C) 1.13 years 
D) 5.68 years 
E) 4.69 years 
A) 6.30 years 
B) 4.97 years 
C) 1.13 years 
D) 5.68 years 
E) 4.69 years

Question 3

Write the equation of the line tangent to the graph of $y = 2 x e ^ { - x } + 6$ at $x = 1$

Accepted Answer

A)  $y = 2 e ^ { - 1 } x + 6$ 
B)  $y = 2 e ^ { - 1 } x - 6$ 
C)  $y = 2 e ^ { - 1 } x$ 
D)  $y = 2 e ^ { - 1 } + 6$ 
E)  $y = 2 e ^ { - 1 } - 6$ 
A)  $y = 2 e ^ { - 1 } x + 6$ 
B)  $y = 2 e ^ { - 1 } x - 6$ 
C)  $y = 2 e ^ { - 1 } x$ 
D)  $y = 2 e ^ { - 1 } + 6$ 
E)  $y = 2 e ^ { - 1 } - 6$

Question 4

Find the derivative of $y = 2 - 15 \ln x$

Accepted Answer

A)  $\frac { d y } { d x } = - \frac { 1 } { 15 x }$ 
B)  $\frac { d y } { d x } = - \frac { 15 } { x }$ 
C)  $\frac { d y } { d x } = - \frac { 13 } { x }$ 
D)  $\frac { d y } { d x } = \frac { 15 } { x }$ 
E)  $\frac { d y } { d x } = \frac { 13 } { x }$ 
A)  $\frac { d y } { d x } = - \frac { 1 } { 15 x }$ 
B)  $\frac { d y } { d x } = - \frac { 15 } { x }$ 
C)  $\frac { d y } { d x } = - \frac { 13 } { x }$ 
D)  $\frac { d y } { d x } = \frac { 15 } { x }$ 
E)  $\frac { d y } { d x } = \frac { 13 } { x }$

Question 5

Solve the exponential equation. Give the answer correct to 3 decimal places. $72 = 300 - 300 e ^ { - 0.07 x }$

Accepted Answer

A) -0.274 
B) 20.387 
C) 3.921 
D) -20.387 
E) -3.073 
A) -0.274 
B) 20.387 
C) 3.921 
D) -20.387 
E) -3.073

Question 6

Use the properties of logarithms to approximate $\ln \frac { 5 } { 31 },$ given that $\ln 5 \approx 1.6094$ and $\ln 31 \approx 3.4340$

Accepted Answer

A) -5.0434 
B) -1.8246 
C) 0.4687 
D) 5.5267 
E) 5.0434 
A) -5.0434 
B) -1.8246 
C) 0.4687 
D) 5.5267 
E) 5.0434

Question 7

What is the annual percentage yield (or effective annual rate) for a nominal rate of 9% compounded quarterly?

Accepted Answer

A) 9.00% 
B) 9.38% 
C) 9.42% 
D) 9.31% 
E) 9.20% 
A) 9.00% 
B) 9.38% 
C) 9.42% 
D) 9.31% 
E) 9.20%

Question 8

Write the exponential equation $e ^ { 10 } = 22026.4658 \mathrm {~K}$ as a logarithmic equation.

Accepted Answer

A)  $\ln$ $22026.4658 \mathrm {~K} = 10$ 
B)  $\ln$ $22026.4658 \mathrm {~K} =$ $20$ 
C)  $\ln$ $10 = 22026.4658 \mathrm {~K}$ 
D)  $\ln$ $10 \mathrm {~K} =$ $44052.9316 \mathrm {~K}$ 
E)  $\ln$ $44052.9316 \mathrm {~K} = 20$ 
A)  $\ln$ $22026.4658 \mathrm {~K} = 10$ 
B)  $\ln$ $22026.4658 \mathrm {~K} =$ $20$ 
C)  $\ln$ $10 = 22026.4658 \mathrm {~K}$ 
D)  $\ln$ $10 \mathrm {~K} =$ $44052.9316 \mathrm {~K}$ 
E)  $\ln$ $44052.9316 \mathrm {~K} = 20$

Question 9

Find $\frac { d s } { d t } \text { if } s = \ln \left[ t ^ { 2 } \left( t ^ { 9 } - 5 \right) \right]$ .

Accepted Answer

A)  $\frac { 9 t ^ { 2 } - 5 } { t \left( t ^ { 9 } - 5 \right) }$ 
B)  $\frac { 2 t ^ { 9 } + 9 t ^ { 8 } - 2 } { t \left( t ^ { 9 } - 5 \right) }$ 
C)  $\frac { 11 t ^ { 8 } - 2 } { t \left( t ^ { 9 } - 5 \right) }$ 
D)  $\frac { 11 t ^ { 9 } - 10 } { t \left( t ^ { 9 } - 5 \right) }$ 
E)  $\frac { 2 t ^ { 9 } + 9 t ^ { 8 } - 10 } { t \left( t ^ { 9 } - 5 \right) }$ 
A)  $\frac { 9 t ^ { 2 } - 5 } { t \left( t ^ { 9 } - 5 \right) }$ 
B)  $\frac { 2 t ^ { 9 } + 9 t ^ { 8 } - 2 } { t \left( t ^ { 9 } - 5 \right) }$ 
C)  $\frac { 11 t ^ { 8 } - 2 } { t \left( t ^ { 9 } - 5 \right) }$ 
D)  $\frac { 11 t ^ { 9 } - 10 } { t \left( t ^ { 9 } - 5 \right) }$ 
E)  $\frac { 2 t ^ { 9 } + 9 t ^ { 8 } - 10 } { t \left( t ^ { 9 } - 5 \right) }$

Question 10

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

Accepted Answer

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

Question 11

Write the equation of the line tangent to the curve $x \ln y + 5 x y = 20 \text { at the point } ( 4,1 )$

Accepted Answer

A)  $24 y - 5 x = 44$ 
B)  $24 y + 5 x = - 44$ 
C)  $5 x - 24 y = 44$ 
D)  $5 x + 24 y = 44$ 
E)  $5 x + 24 y = 0$ 
A)  $24 y - 5 x = 44$ 
B)  $24 y + 5 x = - 44$ 
C)  $5 x - 24 y = 44$ 
D)  $5 x + 24 y = 44$ 
E)  $5 x + 24 y = 0$

Question 12

The demand function for a product is modeled by $p = 4000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0003 x } } \right)$ . What is the limit of the price as x increases without bound? Round your answer to two decimal places where applicable.

Accepted Answer

A) The limit of the price as x increases without bound is -1. 
B) The limit of the price as x increases without bound is 1. 
C) The limit of the price as x increases without bound is 0. 
D) The limit of the price as x increases without bound is $4000$ . 
E) The limit of the price as x increases without bound is $5$ . 
A) The limit of the price as x increases without bound is -1. 
B) The limit of the price as x increases without bound is 1. 
C) The limit of the price as x increases without bound is 0. 
D) The limit of the price as x increases without bound is $4000$ . 
E) The limit of the price as x increases without bound is $5$ .

Question 13

Find the exponential function y=  that passes through the two given points $( 0,3 )$ and $( 7,4 )$ .

Accepted Answer

A)  $y = e ^ { 0.0411 t }$ 
B)  $y = 7 e ^ { - 0.0411 t }$ 
C)  $y = e ^ { - 0.0411 t }$ 
D)  $y = 3 e ^ { 0.0411 t }$ 
E)  $y = 3 e ^ { - 0.0411 t }$ 
A)  $y = e ^ { 0.0411 t }$ 
B)  $y = 7 e ^ { - 0.0411 t }$ 
C)  $y = e ^ { - 0.0411 t }$ 
D)  $y = 3 e ^ { 0.0411 t }$ 
E)  $y = 3 e ^ { - 0.0411 t }$

Question 14

Find $y ^ { \prime }$ . $y = \left[ \ln \left( x ^ { 9 } + 5 \right) \right] ^ { 3 }$

Accepted Answer

A)  $\frac { 27 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
B)  $\frac { 3 x ^ { 7 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
C)  $- \frac { 3 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
D)  $- \frac { 9 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
E)  $- \frac { 54 x ^ { 7 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
A)  $\frac { 27 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
B)  $\frac { 3 x ^ { 7 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
C)  $- \frac { 3 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
D)  $- \frac { 9 x ^ { 8 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$ 
E)  $- \frac { 54 x ^ { 7 } \ln \left( x ^ { 9 } + 5 \right) ^ { 2 } } { \left( x ^ { 9 } + 5 \right) }$

Question 15

Use the properties of logarithms to write the expression $\ln \left( x \sqrt [ 5 ] { x ^ { 2 } + 8 } \right)$ as a sum, difference, or multiple of logarithms.

Accepted Answer

A)  $\ln x + \frac { 1 } { 8 } \ln \left( x ^ { 2 } + 5 \right)$ 
B)  $\ln \left( x ^ { 2 } + 8 \right) + \frac { 1 } { 5 } \ln x$ 
C)  $\ln x + \frac { 1 } { 5 } \ln \left( x ^ { 2 } + 8 \right)$ 
D)  $\ln \left( x ^ { 2 } + 5 \right) + \frac { 1 } { 8 } \ln x$ 
E)  $\ln x + \ln \left( x ^ { 2 } + 8 \right)$ 
A)  $\ln x + \frac { 1 } { 8 } \ln \left( x ^ { 2 } + 5 \right)$ 
B)  $\ln \left( x ^ { 2 } + 8 \right) + \frac { 1 } { 5 } \ln x$ 
C)  $\ln x + \frac { 1 } { 5 } \ln \left( x ^ { 2 } + 8 \right)$ 
D)  $\ln \left( x ^ { 2 } + 5 \right) + \frac { 1 } { 8 } \ln x$ 
E)  $\ln x + \ln \left( x ^ { 2 } + 8 \right)$

Question 16

Sketch the graph of the function $y = 5 + \ln x$ .

Accepted Answer

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

Question 17

The relationship between the number of decibels $\beta$ and the intensity of a sound I in watts per square centimeter is given by $\beta = 10 \log _ { 10 } \left( \frac { I } { 10 ^ { - 16 } } \right)$ . Find the rate of change in the number of decibels when the intensity is $10 ^ { - 5 }$ watt per square centimeter. Round your answer to the nearest decibel.

Accepted Answer

A) 43,429 decibels per watt per square cm 
B) 4,342,945 decibels per watt per square cm 
C) 434,294 decibels per watt per square cm 
D) 4,342,944,819 decibels per watt per square cm 
E) 434,296 decibels per watt per square cm 
A) 43,429 decibels per watt per square cm 
B) 4,342,945 decibels per watt per square cm 
C) 434,294 decibels per watt per square cm 
D) 4,342,944,819 decibels per watt per square cm 
E) 434,296 decibels per watt per square cm

Question 18

The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 100 present initially, and 200 present 7 hours later. How many will there be 20 hours after the initial time? Round your answer to the nearest integer.

Accepted Answer

A) 14 bacteria 
B) 620 bacteria 
C) 725 bacteria 
D) 27 bacteria 
E) 22 bacteria 
A) 14 bacteria 
B) 620 bacteria 
C) 725 bacteria 
D) 27 bacteria 
E) 22 bacteria

Question 19

Suppose that the annual rate of inflation averages 4% over the next 10 years. With this rate of inflation, the approximate cost C of goods or services during any year in that decade will be given by C(t) = P(1.04)t, 0 $\leq t \leq$ 10 where t is time in years and P is the present cost. If the price of an oil change for your car is presently $\$$ 24.95, estimate the price 9 years from now. Round your answer to two decimal places.

Accepted Answer

A)  $\$$ 36.93 
B)  $\$$ 37.51 
C)  $\$$ 38.93 
D)  $\$$ 40.51 
E)  $\$$ 35.51 
A)  $\$$ 36.93 
B)  $\$$ 37.51 
C)  $\$$ 38.93 
D)  $\$$ 40.51 
E)  $\$$ 35.51

Question 20

Solve for the equation $e ^ { \sqrt { x } } = e ^ { 16 }$ for $x$ .

Accepted Answer

A)  $x = 16$ 
B)  $x = 4$ 
C)  $x = 4096$ 
D)  $x = 256$ 
E)  $x = 4097$ 
A)  $x = 16$ 
B)  $x = 4$ 
C)  $x = 4096$ 
D)  $x = 256$ 
E)  $x = 4097$

What percent of a present amount of radioactive radium $\left( { } ^ { 226 } R a \right)$ will remain after 900 years?

How long (in years) would $450 have to be invested at an annual rate of 12%, compounded continuously, to amount to $790?

Write the equation of the line tangent to the graph of $y = 2 x e ^ { - x } + 6$ at $x = 1$

Find the derivative of $y = 2 - 15 \ln x$

Solve the exponential equation. Give the answer correct to 3 decimal places. $72 = 300 - 300 e ^ { - 0.07 x }$

Use the properties of logarithms to approximate $\ln \frac { 5 } { 31 },$ given that $\ln 5 \approx 1.6094$ and $\ln 31 \approx 3.4340$

What is the annual percentage yield (or effective annual rate) for a nominal rate of 9% compounded quarterly?

Write the exponential equation $e ^ { 10 } = 22026.4658 \mathrm {~K}$ as a logarithmic equation.

Find $\frac { d s } { d t } \text { if } s = \ln \left[ t ^ { 2 } \left( t ^ { 9 } - 5 \right) \right]$ .

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

Write the equation of the line tangent to the curve $x \ln y + 5 x y = 20 \text { at the point } ( 4,1 )$

The demand function for a product is modeled by $p = 4000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0003 x } } \right)$ . What is the limit of the price as x increases without bound? Round your answer to two decimal places where applicable.

Find the exponential function y= that passes through the two given points $( 0,3 )$ and $( 7,4 )$ .

Find $y ^ { \prime }$ . $y = \left[ \ln \left( x ^ { 9 } + 5 \right) \right] ^ { 3 }$

Use the properties of logarithms to write the expression $\ln \left( x \sqrt [ 5 ] { x ^ { 2 } + 8 } \right)$ as a sum, difference, or multiple of logarithms.

Sketch the graph of the function $y = 5 + \ln x$ .

The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 100 present initially, and 200 present 7 hours later. How many will there be 20 hours after the initial time? Round your answer to the nearest integer.

Solve for the equation $e ^ { \sqrt { x } } = e ^ { 16 }$ for $x$ .

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

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Exam 11: Derivatives of Exponential and Logarithmic Functions

What percent of a present amount of radioactive radium (226Ra)\left( { } ^ { 226 } R a \right)(226Ra) will remain after 900 years?

How long (in years) would $450 have to be invested at an annual rate of 12%, compounded continuously, to amount to $790?

Write the equation of the line tangent to the graph of y=2xe−x+6y = 2 x e ^ { - x } + 6y=2xe−x+6 at x=1x = 1x=1

Find the derivative of y=2−15ln⁡xy = 2 - 15 \ln xy=2−15lnx

Solve the exponential equation. Give the answer correct to 3 decimal places. 72=300−300e−0.07x72 = 300 - 300 e ^ { - 0.07 x }72=300−300e−0.07x

Use the properties of logarithms to approximate ln⁡531,\ln \frac { 5 } { 31 },ln315​, given that ln⁡5≈1.6094\ln 5 \approx 1.6094ln5≈1.6094 and ln⁡31≈3.4340\ln 31 \approx 3.4340ln31≈3.4340

What is the annual percentage yield (or effective annual rate) for a nominal rate of 9% compounded quarterly?

Write the exponential equation e10=22026.4658 Ke ^ { 10 } = 22026.4658 \mathrm {~K}e10=22026.4658 K as a logarithmic equation.

Find dsdt if s=ln⁡[t2(t9−5)]\frac { d s } { d t } \text { if } s = \ln \left[ t ^ { 2 } \left( t ^ { 9 } - 5 \right) \right]dtds​ if s=ln[t2(t9−5)] .

Sketch the graph of the function f(x)=3xf ( x ) = 3 ^ { x }f(x)=3x .

Write the equation of the line tangent to the curve xln⁡y+5xy=20 at the point (4,1)x \ln y + 5 x y = 20 \text { at the point } ( 4,1 )xlny+5xy=20 at the point (4,1)

The demand function for a product is modeled by p=4000(1−55+e−0.0003x)p = 4000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0003 x } } \right)p=4000(1−5+e−0.0003x5​) . What is the limit of the price as x increases without bound? Round your answer to two decimal places where applicable.

Find the exponential function y= that passes through the two given points (0,3)( 0,3 )(0,3) and (7,4)( 7,4 )(7,4) .

Find y′y ^ { \prime }y′ . y=[ln⁡(x9+5)]3y = \left[ \ln \left( x ^ { 9 } + 5 \right) \right] ^ { 3 }y=[ln(x9+5)]3

Use the properties of logarithms to write the expression ln⁡(xx2+85)\ln \left( x \sqrt [ 5 ] { x ^ { 2 } + 8 } \right)ln(x5x2+8​) as a sum, difference, or multiple of logarithms.

Sketch the graph of the function y=5+ln⁡xy = 5 + \ln xy=5+lnx .

The number of a certain type of bacteria increases continuously at a rate proportional to the number present. There are 100 present initially, and 200 present 7 hours later. How many will there be 20 hours after the initial time? Round your answer to the nearest integer.

Solve for the equation ex=e16e ^ { \sqrt { x } } = e ^ { 16 }ex​=e16 for xxx .

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

What percent of a present amount of radioactive radium $\left( { } ^ { 226 } R a \right)$ will remain after 900 years?

Write the equation of the line tangent to the graph of $y = 2 x e ^ { - x } + 6$ at $x = 1$

Find the derivative of $y = 2 - 15 \ln x$

Solve the exponential equation. Give the answer correct to 3 decimal places. $72 = 300 - 300 e ^ { - 0.07 x }$

Use the properties of logarithms to approximate $\ln \frac { 5 } { 31 },$ given that $\ln 5 \approx 1.6094$ and $\ln 31 \approx 3.4340$

Write the exponential equation $e ^ { 10 } = 22026.4658 \mathrm {~K}$ as a logarithmic equation.

Find $\frac { d s } { d t } \text { if } s = \ln \left[ t ^ { 2 } \left( t ^ { 9 } - 5 \right) \right]$ .

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

Write the equation of the line tangent to the curve $x \ln y + 5 x y = 20 \text { at the point } ( 4,1 )$

The demand function for a product is modeled by $p = 4000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.0003 x } } \right)$ . What is the limit of the price as x increases without bound? Round your answer to two decimal places where applicable.

Find the exponential function y= that passes through the two given points $( 0,3 )$ and $( 7,4 )$ .

Find $y ^ { \prime }$ . $y = \left[ \ln \left( x ^ { 9 } + 5 \right) \right] ^ { 3 }$

Use the properties of logarithms to write the expression $\ln \left( x \sqrt [ 5 ] { x ^ { 2 } + 8 } \right)$ as a sum, difference, or multiple of logarithms.

Sketch the graph of the function $y = 5 + \ln x$ .

Solve for the equation $e ^ { \sqrt { x } } = e ^ { 16 }$ for $x$ .