Q: Suppose a population of bacteria grows according to $$N = 50e$$^{0.8t}$$, where N$ is the number of bacteria after t$ hours. How many bacteria will there be after 2 hours? (Round to the nearest whole number.)

$$N = 50e$$^{0.8 \times 2}$$ = 50e$$^{1.6}$$ \approx 247$$

Q: A radioactive substance decays according to $$A = 200e$$^{-0.0002t}$$, where A$ is the amount remaining after t$ years. How much remains after 10,000 years? (Round to two decimal places.)

$$A = 200e$$^{-0.0002 \times 10000}$$ = 200e$$^{-2}$$ \approx 27.07$$

Q: A principal of $$P = 5,000 is $$invested at an annual rate $$r = 0.06$$ compounded continuously. What is the amount $$A$$ after 10 years?

$$A = 5000e$$^{0.06 \times 10}$$ = 5000e$$^{0.6}$$ \approx 9110.60$$

Question 1

Which of the following equations defines an exponential function?

Accepted Answer

$$f(x) = 2^x$$

Question 2

Given the exponential function $$f(x) = b^x$$ with $$b \u003e 1$$, which of the following statements is TRUE about its graph?

Accepted Answer

The $$x-$$axis is a horizontal asymptote.

Question 3

Suppose a population of bacteria grows according to $$N = 50e$$^{0.8t}$$, where N$ is the number of bacteria after t$ hours. How many bacteria will there be after 2 hours? (Round to the nearest whole number.)

Accepted Answer

$$N = 50e$$^{0.8 	imes 2}$$ = 50e$$^{1.6}$$ \approx 247$$

Question 4

Which of the following best describes the effect of the base $$b in $$the exponential function $$f(x) = b^x$$ when $$0 \u003c b \u003c 1$$?

Accepted Answer

The function decreases as $$x$$ increases.

Question 5

A radioactive substance decays according to $$A = 200e$$^{-0.0002t}$$, where A$ is the amount remaining after t$ years. How much remains after 10,000 years? (Round to two decimal places.)

Accepted Answer

$$A = 200e$$^{-0.0002 	imes 10000}$$ = 200e$$^{-2}$$ \approx 27.07$$

Question 6

If $$f(x) = 3^x$$ and $$g(x) = x^3$, which of the following statements is correct?

Accepted Answer

$$f(x) is an $$exponential function and $$g(x) is a $$polynomial function.

Question 7

The half-life of a radioactive element is the time it takes for half of the substance to decay. If $$A = 500e$$^{-0.000124t}$$, what equation would you solve to find the half-life?

Accepted Answer

$$250 = 500e$$^{-0.000124t}$$

Question 8

A principal of $$P = 5,000 is $$invested at an annual rate $$r = 0.06$$ compounded continuously. What is the amount $$A$$ after 10 years?

Accepted Answer

$$A = 5000e$$^{0.06 	imes 10}$$ = 5000e$$^{0.6}$$ \approx 9110.60$$

Question 9

Which of the following is the correct formula for the amount $$A in an $$account after $$t$$ years if $$P is $$invested at an annual rate $$r$$ compounded $$m$$ times per year?

Accepted Answer

$$A = P\left(1 + \frac{r}{m}\$$right)^{mt}$$

Question 10

Given the regression equation $$y = 0.4833 \cdot 1.1343^x$$ for the number of internet users (in billions) $$x$$ years after 2000, estimate the number of users in 2020.

Accepted Answer

$$y = 0.4833 \cdot 1.1343$$^{20}$$ \approx 7.13$$

Question 11

Which of the following statements about the number $$e is $$TRUE?

Accepted Answer

$$e is an $$irrational number approximately equal to 2.71828.

Question 12

If $$f(x) = 2$$^{-x}$$, which of the following best describes the behavior of the graph as x$ increases?

Accepted Answer

The graph decreases and levels off toward the positive $$x-$$axis.

Question 13

A sample contains 500 mg of Carbon-14. Using $$A = 500e$$^{-0.000124t}$$, approximately how many years will it take for the sample to decay to 250 mg?

Accepted Answer

$$t \approx 5,590$$ years

Question 14

Which of the following is NOT a property of the graph of $$f(x) = b^x$$ for $$b \u003e 0$$, $$b 
eq 1$$?

Accepted Answer

The graph passes through the origin.

Question 15

If $$y = ce$$^{kt}$$ models population growth, what does the constant c$ represent?

Accepted Answer

The initial population at $$t = 0$$

Question 16

Suppose $$10,000 is $$deposited at $$7.5\%$$ annual interest, compounded monthly. What is the amount after $$20$$ years? (Use $$A = P\left(1 + \frac{r}{m}\$$right)^{mt}$$)

Accepted Answer

$$A = 10000\left(1 + \frac{0.075}{12}\$$right)^{12 	imes 20}$$ \approx 44,608.17$$

Question 17

Which of the following real-world scenarios is best modeled by an exponential decay function?

Accepted Answer

The amount of a radioactive substance over time.

Exponential Functions in Business Calculus: Definitions, Properties, Graphs, and Applications 1.5

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