Q: If $$A = 500e$$^{-0.000124t}$$ represents the amount of radioactive $$^{14}C$$ in a sample, how much will remain after 15$$,000$$ years? (Round to two decimal places.)

$$A = 500e$$^{-0.000124 \times 15000}$$ = 500e$$^{-1.86}$$ \approx 500 \times 0.1556 = 77.80$$

Question 1

Suppose the total cost (in dollars) of producing $$x$$ bicycles is given by $$C(x) = 10,000 + 150x - 0.2x^2$. Use the marginal cost to approximate the cost of producing the 121st bicycle.

Accepted Answer

$$C'(120) = 150 - 0.4(120) = 102$$

Question 2

A company’s cost function is $$C(x) = 0.0005x^3$ - 0.38x^2$ + 120x$$, where $$x is $$the number of passengers on a flight. What is the marginal cost when $$x = 150$$?

Accepted Answer

$$C'(150) = 0.0015(150^2$) - 0.76(150) + 120 = 0.0015(22500) - 114 + 120 = 33.75 - 114 + 120 = 39.75$$

Question 3

Suppose the demand function for a product is $$p = 50 - \frac{x}{25}. $$Find the marginal revenue when $$x = 10.$$

Accepted Answer

$$R'(x) = 50 - \frac{2x}{25}$$,\ R'(10) = 50 - \frac{20}{25} = 49.2$

Question 4

If $$A = 500e$$^{-0.000124t}$$ represents the amount of radioactive $$^{14}C$$ in a sample, how much will remain after 15$$,000$$ years? (Round to two decimal places.)

Accepted Answer

$$A = 500e$$^{-0.000124 	imes 15000}$$ = 500e$$^{-1.86}$$ \approx 500 	imes 0.1556 = 77.80$$

Question 5

A company’s profit function is $$P(x) = R(x) - C(x). If$$ $$P'(x) = 0 at$$ $$x = 200$$, what does this mean in terms of business decisions?

Accepted Answer

Profit is maximized when $$x = 200$$ units are produced.

Question 6

If $$A = Pe$$^{rt}$$, how long will it take for an investment to double if the annual rate is 6.5$$\%$$ compounded continuously?

Accepted Answer

$$t = \frac{\ln 2}{0.065} \approx 10.66$$ years

Question 7

Given $$f(x) = 5x^8$e^x$$, find $$f'(x).$$

Accepted Answer

$$f'(x) = 40x^7$e^x + 5x^8$e^x$$

Question 8

Suppose the price-demand equation for sunglasses is $$x = 9500 - 250p. If $$the current price is $$p = 15$$, will increasing the price increase or decrease revenue?

Accepted Answer

Revenue will decrease because demand is elastic at $$p = 15.$$

Question 9

Differentiate $$f(x) = \ln(3x^2$ + 4) - 5x.$$

Accepted Answer

$$f'(x) = \frac{6x}{3x^2$ + 4} - 5$$

Question 10

If $$y = (3x^4$ + 2x)^5$, use the chain rule to find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{dy}{dx} = 5(3x^4$ + 2x)^4$ (12x^3$ + 2)$$

Question 11

Suppose $$x = 2400 - 2p^2$ is the price-demand equation. What is the elasticity of demand when p = 8$?

Accepted Answer

$$E(8) = -\frac{8 	imes (-4 	imes 8)}{2400 - 2 	imes 8^2$} = \frac{256}{2272} \approx 0.113 ($$inelastic)

Question 12

Expand the logarithmic expression $$\ln\left(\frac{x^2$y^7z^3$}{(7x^2$ + 2)^3$}\right).$$

Accepted Answer

$$2\ln x + 7\ln y + 3\ln z - 3\ln(7x^2$ + 2)$$

Question 13

If $$f(x) = e$$^{4x^2 + 3x + 2}$$, find f$$'(x)$$.

Accepted Answer

$$f'(x) = (8x + 3)e$$^{4x^2 + 3x + 2}$$

Question 14

A company’s market research finds the price-demand equation $$x = 10,000 - 1000p. $$What is the revenue function $$R(x) in $$terms of $$x$$?

Accepted Answer

$$R(x) = x \left(10 - \frac{x}{1000}\right)$$

Question 15

If $$S(t) = \frac{125t^2$}{t^2$ + 100}$$ represents total sales (in thousands) after $$t$$ months, what is $$S'(t)$$?

Accepted Answer

$$S'(t) = \frac{250t(t^2$ + 100) - 125t^2$ 	imes 2t}{(t^2$ + 100)^2$}$$

Question 16

If $$N(x) = 2000 + 500\ln\sqrt{x^2$ - x + 1}$$, what is $$N'(x)$$ for $$x \geq 1$$?

Accepted Answer

$$N'(x) = 500 \cdot \frac{1}{2} \cdot \frac{2x - 1}{x^2$ - x + 1}$$

Question 17

Suppose $$A(t) = 1000e$$^{0.10t}$$ represents an investment with continuous compounding. What is the relative rate of change of A$?

Accepted Answer

$$0.10 (or$$ $$10\%$$ per year)

Question 18

Which of the following best describes the break-even point for a business?

Accepted Answer

The production level where total revenue equals total cost.

Business Calculus Study Guide: Marginal Analysis, Exponential & Logarithmic Functions, Derivative Rules, and Elasticity

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