Exam 4: Exponential and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. -The surface area $S$ (in square inches) of a cylindrical pipe with length 12 inches is given by $S ( r ) = 2 \pi r ^ { 2 } + 24 \pi r$ , where $\mathrm { r }$ is the radius of the piston (in inches). If the radius is increasing with time $t$ (in minutes) according to the formula $r ( t ) = \frac { 1 } { 6 } t ^ { 2 } , t \geq 0$ , find the surface area $S$ of the pipe as a function of the time $t$ .

Find the present value. Round to the nearest cent. -To get $25,000 after 6 years at 10% compounded semiannually

Write the word or phrase that best completes each statement or answers the question. -The population (in hundred thousands) for the Colonial United States in ten-year increments for the years $1700 - 1780$ is given in the table. (Source: 1998 Information Please Almanac) State whether the data can be more accurately modeled using an exponential function or a logarithmic function. Using a graphing utility, find a model for population (in hundred thousands) as a function of decades since 1700.

Use a graphing calculator to solve the equation. Round your answer to two decimal places. - $e ^ { x } = x ^ { 2 } - 1$

Decide whether or not the functions are inverses of each other. - $f ( x ) = \frac { 1 } { x + 6 } , g ( x ) = \frac { 6 x + 1 } { x }$

Solve the problem. -The surface area of a balloon is given by $S ( r ) = 4 \pi r ^ { 2 }$ , where $r$ is the radius of the balloon. If the radius is increasing with time $t$ , as the balloon is being blown up, according to the formula $r ( t ) = \frac { 2 } { 3 } t ^ { 3 }$ , $t \geq 0$ , find the surface area $S$ as a function of the time $t$ .

Solve the given exponential equation. Round answer to three decimal places. - $2 ^ { x } - 3 x - 3 = 0$

Graph the function. - $f ( x ) = 5 ( x - 1 ) - 2$

Graph the function. - ()=

The function f is one-to-one. Find its inverse. - $f ( x ) = \frac { 3 x - 6 } { 8 x - 8 }$

For the given functions f and g, find the requested composite function value. - $f ( x ) = \sqrt { x + 2 } , \quad g ( x ) = 3 x ; \quad$ Find $( f \circ g ) ( 0 )$ .

Choose the one alternative that best completes the statement or answers the question. The graph of a logarithmic function is shown. Select the function which matches the graph. -

Use the horizontal line test to determine whether the function is one-to-one. -

Choose the one alternative that best completes the statement or answers the question. Find the domain of the composite function f ° g. - $f ( x ) = \frac { 1 } { x - 7 } ; \quad g ( x ) = \sqrt { x - 1 }$

Express as a single logarithm. - $3 \log _ { a } ( 2 x + 1 ) - 2 \log _ { a } ( 2 x - 1 ) + 2$

Choose the one alternative that best completes the statement or answers the question. -The half-life of a radioactive element is 130 days, but your sample will not be useful to you after 80% of the radioactive nuclei originally present have disintegrated. About how many days can you use the sample?

Express y as a function of x. The constant C is a positive number. - $3 \ln y = \frac { 1 } { 3 } \ln x - \ln \frac { x ^ { 2 } - 1 } { x ^ { 4 / 3 } } + \ln C$

Use a graphing calculator to solve the equation. Round your answer to two decimal places. - $\log _ { 4 } x + \log _ { 3 } x = 3$

Solve the given exponential equation. Round answer to three decimal places. - $2 ^ { x } - 4 x + 2 = 0$

Write the word or phrase that best completes each statement or answers the question. -A cancer patient undergoing chemotherapy is injected with a particular drug. The function $\mathrm { D } ( \mathrm { h } ) = 4 \mathrm { e } ^ { - 0.35 \mathrm {~h} }$ gives the number of milligrams D of this drug that is in the patient's bloodstream hours after the drug has been administered. How many milligrams of the drug were injected? To the nearest milligram, how much of the drug will be present after 2 hours?