Question 1

Which of the following is the correct formula for the amount $$P(t) in an $$account after $$t$$ years if $$P_0 is $$invested at an annual interest rate $$r$$ compounded $$n$$ times per year?

Accepted Answer

$$P(t) = P_0$ \left(1 + \frac{r}{n}\$$right)^{nt}$$

Question 2

Suppose you invest $$1000 at an annual interest rate of 6$$\%$ compounded monthly. What is the value of the investment after 10 years? (Round to the nearest cent.)

Accepted Answer

$$1819.40$$

Question 3

Which transformation has been applied to the function $$f(x) = \log_2$(x) to $$obtain $$g(x) = -5\log_2$(x+4) - 7$$?

Accepted Answer

Reflected over the x-axis, vertically stretched by 5, shifted left 4 units, shifted down 7 units

Question 4

A radioactive material decays according to the formula $$m(t) = 400e$$^{-0.06t}$$. How much of the material remains after 10 years? (Round to the nearest whole number.)

Accepted Answer

$$218$$ grams

Question 5

If $$P(t) = 1000e$$^{0.08t}$$, how long will it take for the investment to double?

Accepted Answer

$$t = \frac{\ln 2}{0.08}$$

Question 6

A population of bacteria doubles every 3 hours. If the initial population is 100, how long will it take to reach 200,000?

Accepted Answer

$$t = 3 \cdot \frac{\ln(2000)}{\ln 2}$$

Question 7

Which of the following best describes the domain of the function $$f(x) = \log_2$(x-3)$$?

Accepted Answer

$$x \u003e 3$$

Question 8

If $$f(x) = a b^x is an $$exponential decay function, which of the following must be true?

Accepted Answer

$$0 \u003c b \u003c 1$$

Question 9

Suppose the minimum wage in 1990 was $$3.80 and in 2000 it was 5.15. $$Assuming exponential growth, what is the exponential model for the minimum wage $$t$$ years after 1990?

Accepted Answer

$$f(t) = 3.80 \cdot (1.03)^t$$

Question 10

If $$P(t) = 1000e$$^{0.06t}$$, what is the annual percentage rate (APR) for this continuously compounded account?

Accepted Answer

$$6\%$$

Question 11

A certain type of bacteria triples every 5 hours. Which equation models the population $$P$$ after $$t$$ hours if the initial population is $$P_0$$?

Accepted Answer

$$P(t) = P_0$ \cdot 3$$^{t/5}$$

Question 12

If $$P(t) = 1000 \left(1 + \frac{0.06}{n}\$$right)^{nt}$$, what happens to P(t$$)$$ as n$ increases without bound?

Accepted Answer

$$P(t)$$ approaches $$1000e^{0.06t}$$

Question 13

A substance decays exponentially so that after 6 hours, only 60% of the original amount remains. What is the decay rate $$k in $$the model $$A(t) = A_0$ e$$^{kt}$$?

Accepted Answer

$$k = \frac{\ln 0.6}{6}$$

Question 14

If $$f(x) = \log_2$(x+5)$$, what is the equation of the vertical asymptote?

Accepted Answer

$$x = -5$$

Question 15

You invest $$5000 in an $$account that earns $$8\%$$ interest compounded continuously. How long will it take for your investment to reach $$10,000$$?

Accepted Answer

$$t = \frac{\ln 2}{0.08}$$

Question 16

Which of the following best describes the range of any exponential function $$f(x) = ab^x$$ where $$a \u003e 0$$ and $$b \u003e 0$$?

Accepted Answer

$$(0, \infty)$$

Question 17

If $$f(x) = \log_2$(x-2)$$, what is the $$x-$$intercept of the graph?

Accepted Answer

$$x = 3$$

Question 18

A painting is found to contain $$14$$ grams of a radioactive isotope. If the isotope decays at a rate of $$5\%$$ per year, how old is the painting if the original amount was $$20$$ grams? (Use $$A = A_0$ e$$^{kt}$$, k$$ \u003c 0$$.)

Accepted Answer

$$t = \frac{\ln(14/20)}{-0.05}$$

C6

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