Exam 5: Exponential and Logarithmic Functions

An industrial psychologist has determined that the average percent score for an employee on a test of the employee's knowledge of the company's product is given by $P = \frac { 100 } { 1 + 24 e ^ { - 0.15 t } }$ where t is the number of weeks on the job and P is the percent score. Estimate (to the nearest week) the expected number of weeks of employment that are necessary for an employee to earn a 85% score on the test.

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ f ^ { - 1 } \right) ( - 3 )$

Use a graphing utility to approximate the solution to $\log _ { 4 } x + \log _ { 4 } ( 2 x + 1 ) = 2$ . Round to 3 decimal places.

The present value of money is the principal $P$ you need to invest today so that it will grow to an amount $A$ at the end of specified time. The present value formula $P = A \left( 1 + \frac { r } { n } \right) ^ { - n t }$ is obtained by solving the compound interest formula $A = P \left( 1 + \frac { r } { n } \right) ^ { n t }$ for $P$ . Recall that $t$ is the number of years, $r$ is the interest rate per year, and $n$ is the number of compoundings per year. find the present value of amount $A$ invested at rate $r$ for $t$ years, compounded $n$ times per year. $A = \$ 10,000 , r = 6 \% , t = 5 \text { years } , n = 4$

A hunting club stocks a wildlife preserve with 18 elk. The carrying capacity of the preserve is 180 animals and the growth of the herd, allowing for the effect of controlled hunting of the elk, is expected to be modeled by the logistic curve $p ( t ) = \frac { 180 } { 1 + 9 e ^ { - 0.154 t } }$ , where t is measured in years. After how many years will the population be 153? Round to the nearest year.

Strontinum-90 has a half life of 29.1 years.The amount S of 100 kilograms of Strontinum6990 present after t years is given by $S = 100 e ^ { - 0.0238 t }$ How much of the 100 kilograms will remain after 50 years?

Use a calculator to evaluate the function $f ( x ) = e ^ { x }$ for the given value of $x$ , $x = 4$ . Round your result to three decimal places.

The table below shows the cost $C$ (in millions of dollars) of a 30-second ad during the Super Bowl for several years from 1987 to 2006. Year Cost 1987 0.6 1992 0.9 1997 1.2 2002 2.2 2006 2.5 Use the regression feature of a graphing utility to find an exponential model for the data. Predict the cost of a 30-second ad during the Super Bowl in 2011. Round to the nearest ten thousand.

How long will it take $30,000 to quadruple if it is invested in a savings account that pays 3.5% annual interest compounded continuously? Round to the nearest year.

Estimate the percentage of radioactive material that remains in a sample after 2 years if the half-life of the material is 211 days. Round to the nearest hundredth of a percent.

Use the graph of $f$ to determine whether the function has an inverse function.

Use the graph of $f$ to complete the table and to sketch the graph of $f ^ { - 1 }$ x 0 1 2 3 4 (x)

Sketch the graphs of inverse functions $f ( x ) = \frac { x } { 3 } , f ^ { - 1 } ( x ) = 3 x$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$

Use a calculator to evaluate $( 2.6 )$ . Round your result to three decimal places.

Solve for x: $9 \left( 10 ^ { x - 3 } \right) = 23$ . Round to 3 decimal places.

Find the balance if $21,000 is invested at an annual rate of 7% for 2 years, compounded continuously.

Determine which of the following functions is graphed below. $2 ^ { x } - 1,2 ^ { - x } - 1,2 ^ { - x } + 1,2 ^ { - x } + 2,2 ^ { x } + 2$

You take a three-pound package of steaks out of the freezer at 11 A.M. and place it in the refrigerator. Assume that the refrigerator temperature is $50 ^ { \circ } F$ and that the freezer temperature is $5 ^ { \circ } F$ Use the formula for Newton's Law of Cooling $t = - 5.05 \ln \frac { T - 50 } { 5 - 50 }$ where $t$ is the time in hours (with $t = 0$ corresponding to 12 P.M.) and $T$ is the temperature of the package of steaks (in degrees Fahrenheit). Find the time when the steaks will thaw (or reach $32 ^ { \circ } F$ ).

Use algebraic procedures to find the exact solution(s) of the equation below. $10 ^ { 3 - x } = 370$

Sketch the graphs of inverse functions $f ( x ) = x ^ { 5 } - 2 , f ^ { - 1 } ( x ) = \sqrt [ 5 ] { x + 2 }$ in the same coordinate plane and show that the graphs are reflections of each other in the line $y = x$