Question 1

Suppose a radioactive substance decays according to $m(t) = 500e^{-t/400}$, where $t$ is in years. After how many years will only 10 grams remain? Round to the nearest year.

Accepted Answer

1,565 years

Question 2

A company’s total cost in millions of dollars is given by $C(t) = 240 - 60e^{-t}$, where $t$ is in years. What is the marginal cost when $t = 2$?

Accepted Answer

$8.12$ million dollars per year

Question 3

The demand function for a certain book is $x = D(p) = 70e^{-0.004p}$. What is the marginal demand $D'(p)$?

Accepted Answer

$D'(p) = -0.28e^{-0.004p}$

Question 4

If $y = e^{6 - 8x}$, what is $\frac{dy}{dx}$?

Accepted Answer

$-8e^{6 - 8x}$

Question 5

A population of rabbits grows exponentially at a rate of $10\%$ per day. How long, to the nearest tenth of a day, will it take for the population to double?

Accepted Answer

6.9 days

Question 6

Suppose $f(x) = \ln(x^4 - 1)/x$. What is $f'(x)$?

Accepted Answer

$\frac{3x^4 + 1}{x(x^4 - 1)}$

Question 7

A savings account earns interest compounded continuously at $5.5\%$ per year. If $\$8000$ is invested, how long will it take for the investment to double?

Accepted Answer

12.6 years

Question 8

The percentage $P$ of doctors who accept a new medicine is given by $P(t) = 100(1 - e^{-0.16t})$, where $t$ is in months. What is $P'(3)$?

Accepted Answer

10\%

Question 9

A pharmaceutical company introduces a new medication. The percentage $P$ of people who buy the product after $t$ weeks is $P(t) = \frac{100}{1 + 50e^{-0.13t}}$. What percentage buy the product after 11 weeks?

Accepted Answer

7.7\%

Question 10

The supply and demand for television sets are $S(p) = \ln p$ and $D(p) = \ln \frac{197,000}{p}$, respectively. What is the equilibrium price?

Accepted Answer

$444$

Question 11

A radioactive substance has a half-life of $402$ years. What is its decay rate $k$ (as a decimal, per year)?

Accepted Answer

$0.00172$

Question 12

If $f'(x) = 7f(x)$, what is the general form of $f(x)$?

Accepted Answer

$f(x) = ce^{7x}$

Question 13

A city’s population was $702,000$ in 1970 and $616,400$ in 1980, following exponential decay. When will the population reach $100,000$?

Accepted Answer

Year 2120

Question 14

Given $y = \ln 2x^2$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{2}{x}$

Question 15

A model for advertising response is $N(a) = 12,000 + 400\ln a$, where $a$ is the amount spent on advertising (in thousands of dollars). What is $N'(10)$?

Accepted Answer

40

Question 16

Suppose $P_0$ is invested at a continuous growth rate of $7\%$ per year. What is the doubling time?

Accepted Answer

9.9 years

Question 17

The number of employees who have heard a rumor after $t$ days is $N(t) = \frac{200}{1 + 51e^{-0.2t}}$. How many employees have heard the rumor after 20 days?

Accepted Answer

103

Question 18

If $A(t) = 866e^{-0.022t}$ models the decay of a radioactive isotope, how much remains after 33 years?

Accepted Answer

419 mg

Question 19

If $y = x^7 \ln x - \frac{1}{3}x^3$, what is $\frac{dy}{dx}$?

Accepted Answer

$x^6 - x^2 + 7x^6\ln x$

Question 20

A quantity $Q_1$ grows exponentially with a tripling time of 1 year, and $Q_2$ with a tripling time of 2 years. If both start with the same amount, after how many years will $Q_1$ be three times $Q_2$?

Accepted Answer

2 years

Business Calculus Unit 2 Review – Step-by-Step Study Guidance

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