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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