Q: The Irwins want to make an initial investment $$P$$ that will grow to $$40,000 in 20 $$years, compounded continuously at 3.5%. What should the initial investment be?

$$P = 40000 e$$^{-0.035 \times 20}$$ \approx 20,000$$

Question 1

The population of San Estaban in 2015 was 45,000 and it grows continuously at a rate of 3% per year. Which of the following functions models the population $$A(t)$$ years after 2015?

Accepted Answer

$$A(t) = 45000 e$$^{0.03t}$$

Question 2

Using the model $$A(t) = 45000 e$$^{0.03t}$$, what will the population of San Estaban be in 2025?

Accepted Answer

$$A(10) = 45000 e$$^{0.3}$$ \approx 60,616$$

Question 3

Maria invests $$20,000 in an IRA account whose value after t$ years is modeled by V(t) = 20000$$ $$e^{0.054t}$. What is the value of the account after 7 years?

Accepted Answer

$$V(7) = 20000 e$$^{0.378}$$ \approx 24,600$$

Question 4

Suppose a radioactive sample of carbon-14 has lost 22.3% of its carbon-14. Given the half-life of carbon-14 is 5730 years, how old is the sample? (Use $$A = A_0$ e$$^{kt}$$, k$$ = \frac{\ln(0.5)}{5730}$$)

Accepted Answer

$$t = \frac{\ln(0.777)}{\ln(0.5)/5730} \approx 1,900$$ years

Question 5

If the amount of iodine-125 in a cancer treatment sample decreased by 2% while in storage, and the half-life is 60.1 days, how long was the sample in storage? (Use $$A = A_0$ e$$^{kt}$$, k$$ = \frac{\ln(0.5)}{60.1}$$)

Accepted Answer

$$t = \frac{\ln(0.98)}{\ln(0.5)/60.1} \approx 1.73$$ days

Question 6

Given the function $$f(x) = 2^x - e^x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2^x \ln 2 - e^x$$

Question 7

Find the critical points of $$h(x) = x^3$ - 6x^2$ - 15x + 1.$$

Accepted Answer

$$x = -1$$ and $$x = 5$$

Question 8

Determine the vertical asymptote(s) of $$f(x) = \frac{x+2}{x^2$ + 6x + 8}.$$

Accepted Answer

$$x = -4$$ and $$x = -2$$

Question 9

Determine the horizontal asymptote of $$f(x) = \frac{5x^4$ + 2x^3$ + x}{x^4$ + 8}.$$

Accepted Answer

$$y = 5$$

Question 10

Given the demand function $$p(x) = 35 - 0.01x$$, where $$x is in $$millions of barrels per day and $$p is in $$dollars per barrel, what is the price elasticity of demand $$E(x)$$?

Accepted Answer

$$E(x) = -\frac{x}{35 - 0.01x}$$

Question 11

For the function $$f(x) = \ln(x^2$ + 6)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 6}$$

Question 12

A bank account grows continuously at 5.9% per year. If $$P(0) = 1000$$, what is the balance after 2 years?

Accepted Answer

$$P(2) = 1000 e$$^{0.118}$$ \approx 1,125$$

Question 13

The Irwins want to make an initial investment $$P$$ that will grow to $$40,000 in 20 $$years, compounded continuously at 3.5%. What should the initial investment be?

Accepted Answer

$$P = 40000 e$$^{-0.035 	imes 20}$$ \approx 20,000$$

Question 14

Given $$f(x) = 3x^4$ + 4x^3$ on $$[-2, 0]$$, what is the absolute maximum value?

Accepted Answer

$$f(-2) = -16$$

Question 15

If $$f(x) = \log_2$(3x-1)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{3}{(3x-1) \ln 2}$$

Question 16

For the function $$f(x) = x^3$ + 3x^2$ - 2x + 4 on$$ $$[-2, 0]$$, what is the absolute minimum value?

Accepted Answer

$$f(-2) = 0$$

Question 17

Suppose the demand function for oil is $$p(x) = 35 - 0.01x. At $$what price is the demand unit elastic?

Accepted Answer

$$p = 17.5$$

Question 18

If $$y = 4^x \cdot \log_5$ x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$y' = 4^x \log_5$ x \ln 4 + 4^x \frac{1}{x \ln 5}$$

Business Calculus Study Guide: Exponential Growth, Derivatives, and Extrema

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