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

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

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

$x > 3$

Question 8

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

Accepted Answer

$0 < b < 1$

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 $a > 0$ and $b > 0$?

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 $A = A_0 e^{kt}$, $k < 0$.)

Accepted Answer

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

C6

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