Exam 4: Exponential and Logarithmic Functions

Some amount of money is to be invested today at 7% compounded continuously so that 10 years from now the account will be worth $5,000. The amount is $2,000.

(True/False)

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Question 121

Solve for x: $\log _ { 2 } ( x - 1 ) = 5$ .

(Short Answer)

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Question 122

$\frac { 8 ^ { 6 } } { 8 ^ { 9 } } = \frac { 1 } { 512 }$

(True/False)

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Question 123

A radioactive substance decays exponentially. If 900 grams were present initially and 500 grams are present 100 years later, how many grams will be present after 400 years? Round your answer to two decimal places, if necessary.

(Short Answer)

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Question 124

If $7,000 is invested at 6% interest compounded monthly, the balance after 10 years will be $12,735.78.

(True/False)

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Question 125

When a certain industrial machine is t years old, its resale value will be $V ( t ) = 4,000 e ^ { - t / 4 } + 500$ dollars. How much does the resale value change between the 3rd and 4th years?

(Multiple Choice)

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Question 126

Consider the function $f ( x ) = e ^ { - ( x - 4 ) ^ { 2 } / 7 }$ . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

(Short Answer)

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Question 127

A statistical study indicates that the fraction of the can openers manufactured by a certain company that are still in working condition after t years of use is approximately $f ( t ) = e ^ { - 0.3 t }$ . The fraction that can be expected to fail during the 2nd year of use is about $\frac { 19 } { 100 }$ .

(True/False)

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Question 128

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

(Multiple Choice)

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Question 129

Evaluation of $e ^ { 1 / 2 * \mathrm { n } 4.1 / 3 { *} \mathbf { n } 27 }$ without using tables or a calculator will show it is 6.

(True/False)

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Question 130

Differentiate the given function. $f ( x ) = 25 - 6 e ^ { - 0.07 x }$

(Multiple Choice)

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Question 131

Suppose the number of bacteria grows from 2,000 to 6,000 in the first 15 minutes of an experiment. Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 1 hour?Round your answer to the nearest whole number, if necessary.

(Short Answer)

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Question 132

If $3 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 3 } { 50 }$ .

(True/False)

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Question 133

Solve for x: $a ^ { 7 x - 3 } = b$

(Short Answer)

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Question 134

Simplify $\frac { 3 ^ { 5 } } { 3 ^ { 4 } }$ .

(Multiple Choice)

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Question 135

The graph of $f ( x ) = \ln x ^ { 8 }$ has

(Multiple Choice)

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Question 136

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

(Multiple Choice)

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Question 137

A bank compounds interest continuously. What nominal interest rate does it offer if $1,500 grows to $2,500 in 10 years? Round to two decimal places.

(Multiple Choice)

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Question 138

Simplify $64 ^ { 2 / 3 }$ .

(Short Answer)

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Question 139

Solve for x: $a ^ { 2 x - 1 } = b$

(Multiple Choice)

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Question 140

Some amount of money is to be invested today at 7% compounded continuously so that 10 years from now the account will be worth $5,000. The amount is $2,000.

Solve for x: $\log _ { 2 } ( x - 1 ) = 5$ .

$\frac { 8 ^ { 6 } } { 8 ^ { 9 } } = \frac { 1 } { 512 }$

A radioactive substance decays exponentially. If 900 grams were present initially and 500 grams are present 100 years later, how many grams will be present after 400 years? Round your answer to two decimal places, if necessary.

If $7,000 is invested at 6% interest compounded monthly, the balance after 10 years will be $12,735.78.

When a certain industrial machine is t years old, its resale value will be $V ( t ) = 4,000 e ^ { - t / 4 } + 500$ dollars. How much does the resale value change between the 3rd and 4th years?

Consider the function $f ( x ) = e ^ { - ( x - 4 ) ^ { 2 } / 7 }$ . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

Evaluation of $e ^ { 1 / 2 * \mathrm { n } 4.1 / 3 { *} \mathbf { n } 27 }$ without using tables or a calculator will show it is 6.

Differentiate the given function. $f ( x ) = 25 - 6 e ^ { - 0.07 x }$

Suppose the number of bacteria grows from 2,000 to 6,000 in the first 15 minutes of an experiment. Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 1 hour?Round your answer to the nearest whole number, if necessary.

If $3 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 3 } { 50 }$ .

Solve for x: $a ^ { 7 x - 3 } = b$

Simplify $\frac { 3 ^ { 5 } } { 3 ^ { 4 } }$ .

The graph of $f ( x ) = \ln x ^ { 8 }$ has

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

A bank compounds interest continuously. What nominal interest rate does it offer if $1,500 grows to $2,500 in 10 years? Round to two decimal places.

Simplify $64 ^ { 2 / 3 }$ .

Solve for x: $a ^ { 2 x - 1 } = b$

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

Some amount of money is to be invested today at 7% compounded continuously so that 10 years from now the account will be worth $5,000. The amount is $2,000.

Solve for x: log⁡2(x−1)=5\log _ { 2 } ( x - 1 ) = 5log2​(x−1)=5 .

8689=1512\frac { 8 ^ { 6 } } { 8 ^ { 9 } } = \frac { 1 } { 512 }8986​=5121​

A radioactive substance decays exponentially. If 900 grams were present initially and 500 grams are present 100 years later, how many grams will be present after 400 years? Round your answer to two decimal places, if necessary.

If $7,000 is invested at 6% interest compounded monthly, the balance after 10 years will be $12,735.78.

When a certain industrial machine is t years old, its resale value will be V(t)=4,000e−t/4+500V ( t ) = 4,000 e ^ { - t / 4 } + 500V(t)=4,000e−t/4+500 dollars. How much does the resale value change between the 3rd and 4th years?

Consider the function f(x)=e−(x−4)2/7f ( x ) = e ^ { - ( x - 4 ) ^ { 2 } / 7 }f(x)=e−(x−4)2/7 . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

Find the derivative of ln⁡[(ln⁡x6)7]\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]ln[(lnx6)7] .

Evaluation of e1/2∗n4.1/3∗n27e ^ { 1 / 2 * \mathrm { n } 4.1 / 3 { *} \mathbf { n } 27 }e1/2∗n4.1/3∗n27 without using tables or a calculator will show it is 6.

Differentiate the given function. f(x)=25−6e−0.07xf ( x ) = 25 - 6 e ^ { - 0.07 x }f(x)=25−6e−0.07x

Suppose the number of bacteria grows from 2,000 to 6,000 in the first 15 minutes of an experiment. Assuming that the number of bacteria grows exponentially, how many bacteria will be present after 1 hour?Round your answer to the nearest whole number, if necessary.

If 3=e−0.002x3 = e ^ { - 0.002 x }3=e−0.002x , then x is ln⁡350\frac { \ln 3 } { 50 }50ln3​ .

Solve for x: a7x−3=ba ^ { 7 x - 3 } = ba7x−3=b

Simplify 3534\frac { 3 ^ { 5 } } { 3 ^ { 4 } }3435​ .

The graph of f(x)=ln⁡x8f ( x ) = \ln x ^ { 8 }f(x)=lnx8 has

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

A bank compounds interest continuously. What nominal interest rate does it offer if $1,500 grows to $2,500 in 10 years? Round to two decimal places.

Simplify 642/364 ^ { 2 / 3 }642/3 .

Solve for x: a2x−1=ba ^ { 2 x - 1 } = ba2x−1=b

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

Solve for x: $\log _ { 2 } ( x - 1 ) = 5$ .

$\frac { 8 ^ { 6 } } { 8 ^ { 9 } } = \frac { 1 } { 512 }$

When a certain industrial machine is t years old, its resale value will be $V ( t ) = 4,000 e ^ { - t / 4 } + 500$ dollars. How much does the resale value change between the 3rd and 4th years?

Consider the function $f ( x ) = e ^ { - ( x - 4 ) ^ { 2 } / 7 }$ . For what value of x does this function attain its maximum value, and what is the maximum function value? Round maximum function value to two decimal places, if necessary.

Find the derivative of $\ln \left[ \left( \ln x ^ { 6 } \right) ^ { 7 } \right]$ .

Evaluation of $e ^ { 1 / 2 * \mathrm { n } 4.1 / 3 { *} \mathbf { n } 27 }$ without using tables or a calculator will show it is 6.

Differentiate the given function. $f ( x ) = 25 - 6 e ^ { - 0.07 x }$

If $3 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 3 } { 50 }$ .

Solve for x: $a ^ { 7 x - 3 } = b$

Simplify $\frac { 3 ^ { 5 } } { 3 ^ { 4 } }$ .

The graph of $f ( x ) = \ln x ^ { 8 }$ has

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

Simplify $64 ^ { 2 / 3 }$ .

Solve for x: $a ^ { 2 x - 1 } = b$