Question 1

Find all real numbers x that satisfy the given equation. 3x22x = 1,728

Accepted Answer

A)  6 
B)  3 
C)  9 
D)  12 
A)  6 
B)  3 
C)  9 
D)  12

Question 2

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and the interest is compounded continuously. After 7 years, the money has doubled. At the end of 14 years the value is $2 A _ { 0 }$ .

Accepted Answer

A) True 
 B)False

Question 3

An archaeologist has found a fossil in which the ratio of ${ } ^ { 14 } \mathrm { C }$ to ${ } ^ { 12 } \mathrm { C }$ is $\frac { 1 } { 9 }$ the ratio in the atmosphere. Approximately how old is the fossil? The half-life of ${ } ^ { 14 } \mathrm { C }$ is 5,730 years. Round to the nearest whole year.

Accepted Answer

A)  18,164 years 
B)  18,194 years 
C)  25,785 years 
D)  51,570 years 
A)  18,164 years 
B)  18,194 years 
C)  25,785 years 
D)  51,570 years

Question 4

A radioactive substance decays exponentially. If 800 grams were present initially and 600 grams are present 100 years later, how many grams will be present after 400 years? Round your answer to two decimal places, if necessary.

Accepted Answer

A)  253.13 grams 
B)  0.00 grams 
C)  251.88 grams 
D)  250.63 grams 
A)  253.13 grams 
B)  0.00 grams 
C)  251.88 grams 
D)  250.63 grams

Question 5

If $2 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 2 } { 500 }$ .

Accepted Answer

A) True 
 B)False

Question 6

The equation of the tangent line to $f ( x ) = e ^ { x } - e ^ { - x }$ at x = 0 is y = 2x.

Accepted Answer

A) True 
 B)False

Question 7

Differentiate the given function. $f ( x ) = e ^ { - 5 x }$

Accepted Answer

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

Question 8

If $7,000 is invested at 7% interest for 20 years, there will be $400 more if the compounding is continuous rather than quarterly.

Accepted Answer

A) True 
 B)False

Question 9

If $2,000 is invested at 7 percent compounded monthly, what is the balance after 8 years?

Accepted Answer

A)  $3,490.51 
B)  $3,495.65 
C)  $3,521.62 
D)  $3,601.84 
A)  $3,490.51 
B)  $3,495.65 
C)  $3,521.62 
D)  $3,601.84

Question 10

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 6 ] { \frac { 7 x + 1 } { 4 + 7 x } }$

Accepted Answer

The answer of Use logarithmic differentiation to find \(f ^...

Question 11

Let $f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }$ . Which value of t corresponds to a possible inflection point for f (t)?

Accepted Answer

A)  $\frac { 5 } { 8 } \ln ( 200 )$ 
B)  $\ln \left( \frac { 5 } { 8 } \right)$ 
C)  $200 \ln \left( \frac { 8 } { 5 } \right)$ 
D)  There is no inflection point. 
A)  $\frac { 5 } { 8 } \ln ( 200 )$ 
B)  $\ln \left( \frac { 5 } { 8 } \right)$ 
C)  $200 \ln \left( \frac { 8 } { 5 } \right)$ 
D)  There is no inflection point.

Question 12

Consider the function $f ( x ) = e ^ { - ( x - 6 ) ^ { 2 } / 5 }$ 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.

Accepted Answer

A)  x = 6, f (6) = 1 
B)  $x = \frac { 3 } { 5 } , f \left( \frac { 3 } { 5 } \right) \approx 0.00$ 
C)  There is no maximum. 
D)  $x = \frac { 6 } { 5 } , f \left( \frac { 6 } { 5 } \right) \approx 0.01$ 
A)  x = 6, f (6) = 1 
B)  $x = \frac { 3 } { 5 } , f \left( \frac { 3 } { 5 } \right) \approx 0.00$ 
C)  There is no maximum. 
D)  $x = \frac { 6 } { 5 } , f \left( \frac { 6 } { 5 } \right) \approx 0.01$

Question 13

An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: $Q ( t ) = 600 - A e ^ { - k t }$ Experience (months) 0 9 Output (units per hour) 200 580 The expert believes that the output Q is related to experience t by a function of the form. Find the function of this form that fits the data. Round numbers to four decimal places, if necessary.

Accepted Answer

A)  $Q ( t ) = 600 - 400 e ^ { 0.3329 t }$ 
B)  $Q ( t ) = 600 - 400 e ^ { - 0.3329 t }$ 
C)  $Q ( t ) = 600 - 580 e ^ { 0.3329 t }$ 
D)  $Q ( t ) = 600 - 580 e ^ { - 0.3329 t }$ 
A)  $Q ( t ) = 600 - 400 e ^ { 0.3329 t }$ 
B)  $Q ( t ) = 600 - 400 e ^ { - 0.3329 t }$ 
C)  $Q ( t ) = 600 - 580 e ^ { 0.3329 t }$ 
D)  $Q ( t ) = 600 - 580 e ^ { - 0.3329 t }$

Question 14

Solve the given equation for x. $$- 5 = - 6 + 4 e ^ { 7 x }$$

Accepted Answer

A)  $7 e ^ { 4 }$ 
B)  $\frac { e ^ { 4 } } { 7 }$ 
C)  $\frac { \ln 4 } { 7 }$ 
D)  $- \frac { \ln 4 } { 7 }$ 
A)  $7 e ^ { 4 }$ 
B)  $\frac { e ^ { 4 } } { 7 }$ 
C)  $\frac { \ln 4 } { 7 }$ 
D)  $- \frac { \ln 4 } { 7 }$

Question 15

Solve for x: $$2 \ln x - \frac { 1 } { 3 } \ln x ^ { 2 } = 4$$

Accepted Answer

A)  x = e 
B)  $x = e ^ { 3 }$ 
C)  $x = e ^ { 4 }$ 
D)  $x = e ^ { 2 }$ 
A)  x = e 
B)  $x = e ^ { 3 }$ 
C)  $x = e ^ { 4 }$ 
D)  $x = e ^ { 2 }$

Question 16

Solve the given equation for x. $$0 = - 1 + 5 e ^ { - 7 x }$$

Accepted Answer

A)  $- 7 e ^ { 5 }$ 
B)  $- \frac { e ^ { 5 } } { 7 }$ 
C)  $- \frac { \ln 5 } { 7 }$ 
D)  $\frac { \ln 5 } { 7 }$ 
A)  $- 7 e ^ { 5 }$ 
B)  $- \frac { e ^ { 5 } } { 7 }$ 
C)  $- \frac { \ln 5 } { 7 }$ 
D)  $\frac { \ln 5 } { 7 }$

Question 17

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

Accepted Answer

A)  x = 32 
B)  x = 31 
C)  x = 33 
D)  x = 3,322 
A)  x = 32 
B)  x = 31 
C)  x = 33 
D)  x = 3,322

Question 18

An efficiency expert hired by a manufacturing firm has compiled the following data relating workers' output to their experience: $$\begin{array} { l l l } 
\text { Experience (months) } & 0 & 6 \
\text { Output (units per hour) } & 300 & 330
\end{array}$$ The expert believes that the output Q is related to experience t by a function of the form $Q ( t ) = 400 - A e ^ { - k t }$ . Find the function of this form that fits the data. Round numbers to four decimal places, if necessary.

Accepted Answer

The answer of An efficiency expert hired by a manufacturing...

Question 19

Find $f ^ { \prime \prime } ( x )$ : $f ( x ) = 3 \ln \left( 2 x ^ { 3 } - 5 x \right)$

Accepted Answer

The answer of Find \(f ^ { \prime \prime }...

Question 20

If money is compounded continuously, it will double in about 6.3 years if it triples in 10 years.

Accepted Answer

A) True 
 B)False

Find all real numbers x that satisfy the given equation. 3^x2²^x = 1,728

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and the interest is compounded continuously. After 7 years, the money has doubled. At the end of 14 years the value is $2 A _ { 0 }$ .

A radioactive substance decays exponentially. If 800 grams were present initially and 600 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 $2 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 2 } { 500 }$ .

The equation of the tangent line to $f ( x ) = e ^ { x } - e ^ { - x }$ at x = 0 is y = 2x.

Differentiate the given function. $f ( x ) = e ^ { - 5 x }$

If $7,000 is invested at 7% interest for 20 years, there will be $400 more if the compounding is continuous rather than quarterly.

If $2,000 is invested at 7 percent compounded monthly, what is the balance after 8 years?

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 6 ] { \frac { 7 x + 1 } { 4 + 7 x } }$

Let $f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }$ . Which value of t corresponds to a possible inflection point for f (t)?

Consider the function $f ( x ) = e ^ { - ( x - 6 ) ^ { 2 } / 5 }$ 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.

Solve the given equation for x. $- 5 = - 6 + 4 e ^ { 7 x }$

Solve for x: $2 \ln x - \frac { 1 } { 3 } \ln x ^ { 2 } = 4$

Solve the given equation for x. $0 = - 1 + 5 e ^ { - 7 x }$

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

Find $f ^ { \prime \prime } ( x )$ : $f ( x ) = 3 \ln \left( 2 x ^ { 3 } - 5 x \right)$

If money is compounded continuously, it will double in about 6.3 years if it triples in 10 years.

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. 3x22x = 1,728

A sum of money, A0A _ { 0 }A0​ , is invested at a certain fixed interest rate, and the interest is compounded continuously. After 7 years, the money has doubled. At the end of 14 years the value is 2A02 A _ { 0 }2A0​ .

A radioactive substance decays exponentially. If 800 grams were present initially and 600 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 2=e−0.002x2 = e ^ { - 0.002 x }2=e−0.002x , then x is ln⁡2500\frac { \ln 2 } { 500 }500ln2​ .

The equation of the tangent line to f(x)=ex−e−xf ( x ) = e ^ { x } - e ^ { - x }f(x)=ex−e−x at x = 0 is y = 2x.

Differentiate the given function. f(x)=e−5xf ( x ) = e ^ { - 5 x }f(x)=e−5x

If $7,000 is invested at 7% interest for 20 years, there will be $400 more if the compounding is continuous rather than quarterly.

If $2,000 is invested at 7 percent compounded monthly, what is the balance after 8 years?

Use logarithmic differentiation to find f′(x)f ^ { \prime } ( x )f′(x) . f(x)=7x+14+7x6f ( x ) = \sqrt [ 6 ] { \frac { 7 x + 1 } { 4 + 7 x } }f(x)=64+7x7x+1​​

Let f(t)=125+8e−t/200f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }f(t)=5+8e−t/20012​ . Which value of t corresponds to a possible inflection point for f (t)?

Consider the function f(x)=e−(x−6)2/5f ( x ) = e ^ { - ( x - 6 ) ^ { 2 } / 5 }f(x)=e−(x−6)2/5 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.

Solve the given equation for x. −5=−6+4e7x- 5 = - 6 + 4 e ^ { 7 x }−5=−6+4e7x

Solve for x: 2ln⁡x−13ln⁡x2=42 \ln x - \frac { 1 } { 3 } \ln x ^ { 2 } = 42lnx−31​lnx2=4

Solve the given equation for x. 0=−1+5e−7x0 = - 1 + 5 e ^ { - 7 x }0=−1+5e−7x

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

Find f′′(x)f ^ { \prime \prime } ( x )f′′(x) : f(x)=3ln⁡(2x3−5x)f ( x ) = 3 \ln \left( 2 x ^ { 3 } - 5 x \right)f(x)=3ln(2x3−5x)

If money is compounded continuously, it will double in about 6.3 years if it triples in 10 years.

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^x2²^x = 1,728

A sum of money, $A _ { 0 }$ , is invested at a certain fixed interest rate, and the interest is compounded continuously. After 7 years, the money has doubled. At the end of 14 years the value is $2 A _ { 0 }$ .

If $2 = e ^ { - 0.002 x }$ , then x is $\frac { \ln 2 } { 500 }$ .

The equation of the tangent line to $f ( x ) = e ^ { x } - e ^ { - x }$ at x = 0 is y = 2x.

Differentiate the given function. $f ( x ) = e ^ { - 5 x }$

Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 6 ] { \frac { 7 x + 1 } { 4 + 7 x } }$

Let $f ( t ) = \frac { 12 } { 5 + 8 e ^ { - t / 200 } }$ . Which value of t corresponds to a possible inflection point for f (t)?

Consider the function $f ( x ) = e ^ { - ( x - 6 ) ^ { 2 } / 5 }$ 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.

Solve the given equation for x. $- 5 = - 6 + 4 e ^ { 7 x }$

Solve for x: $2 \ln x - \frac { 1 } { 3 } \ln x ^ { 2 } = 4$

Solve the given equation for x. $0 = - 1 + 5 e ^ { - 7 x }$

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

Find $f ^ { \prime \prime } ( x )$ : $f ( x ) = 3 \ln \left( 2 x ^ { 3 } - 5 x \right)$