Q: If $$A = A_0$ e$$^{-0.000124t}$$ models the decay of carbon-14, how much of a 500 mg sample remains after 15,000 years? (Round to two decimal places.)

$$A = 500e$$^{-0.000124 \times 15000}$$ \approx 500e$$^{-1.86}$$ \approx 77.60 mg$$

Question 1

Suppose the total cost to produce $$x$$ units of a product is given by $$C(x) = 10,000 + 150x - 0.2x^2$. Use the marginal cost to approximate the cost of producing the 121st unit.

Accepted Answer

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

Question 2

A company’s cost function is $$C(x) = 0.0005x^3$ - 0.38x^2$ + 120x. $$Estimate the marginal cost of adding one more passenger when $$x = 150.$$

Accepted Answer

$$C'(150) = 0.0015(150^2$) - 0.76(150) + 120$$

Question 3

Given the demand function $$p = 50 - \frac{x}{25}$$, find the marginal revenue when $$x = 10.$$

Accepted Answer

$$R'(10) = 50 - \frac{2}{25}(10)$$

Question 4

If $$A = A_0$ e$$^{-0.000124t}$$ models the decay of carbon-14, how much of a 500 mg sample remains after 15,000 years? (Round to two decimal places.)

Accepted Answer

$$A = 500e$$^{-0.000124 	imes 15000}$$ \approx 500e$$^{-1.86}$$ \approx 77.60 mg$$

Question 5

If $$1000 is invested at 10\%$$ interest compounded monthly for 10 years, what is the accumulated amount? (Round to the nearest cent.)

Accepted Answer

$$A = 1000 \left(1 + \frac{0.10}{12}\$$right)^{120}$$ \approx 1000(1.008333$$)^{120}$$ \approx 2707.04$$

Question 6

Rewrite $$\log_4$ 64 = 3 in $$exponential form.

Accepted Answer

$$4^3 = 64$

Question 7

Differentiate $$f(x) = 5x^8$ e^x.$$

Accepted Answer

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

Question 8

Find the derivative of $$y = \frac{2x - 1}{4x + 3}$$ using the quotient rule.

Accepted Answer

$$y' = \frac{2(4x + 3) - 4(2x - 1)}{(4x + 3)^2$}$$

Question 9

Let $$f(x) = (2x - 9)(x^2$ + 6). $$Find the value of $$x$$ where the tangent line is horizontal.

Accepted Answer

$$x = 3$$

Question 10

If $$h(x) = (2x - 1)^4$, use the chain rule to find h$$'(x)$$.

Accepted Answer

$$h'(x) = 4(2x - 1)^3$ \cdot 2$$

Question 11

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

Accepted Answer

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

Question 12

Suppose the price-demand equation is $$x = 2400 - 2p^2$. Determine whether demand is elastic, inelastic, or unit elastic when p = 8$.

Accepted Answer

Elastic, since $$E(8) \u003e 1$$

Question 13

If $$F(t) = 5000e$$^{0.07t}$$ represents the value of an investment, what is the relative rate of change of F$?

Accepted Answer

$$0.07$$

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 $$A = Pe$$^{rt}$$, how long will it take for money to double if invested at 6.5$$\%$$ compounded continuously?

Accepted Answer

$$t = \frac{\ln 2}{0.065}$$

Question 16

Given $$y = \ln\left(\frac{3x^2$ + 4}{e$$^{5x}$$}\right)$$, find $$y'$$ using logarithmic differentiation.

Accepted Answer

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

Question 17

Suppose the demand function for sunglasses is $$x = 9500 - 250p. $$For what value of $$p is $$the demand unit elastic?

Accepted Answer

$$p = 19$$

Question 18

If $$y = \ln(3x^2$ - 4x)$$, find $$y'$$ for $$x \u003e 0.$$

Accepted Answer

$$y' = \frac{6x - 4}{3x^2$ - 4x}$$

Question 19

A model for consumer response to advertising is $$N(x) = 2000 + 500 \ln\sqrt{x^2$ - x + 1}$$ for $$x \geq 1. $$Find $$N'(x).$$

Accepted Answer

$$N'(x) = \frac{250(2x - 1)}{x^2$ - x + 1}$$

Business Calculus: Marginal Analysis, Exponential & Logarithmic Functions, and Derivative Applications

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