Question 1

A company’s revenue function is given by $$R(x) = x \cdot p(x)$$, where $$x is $$the number of units sold and $$p(x) is $$the price per unit. If $$p(x) = 100 - 2x$$, what is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 2

Suppose you invest $$\5$$,000$$ in an account that pays 6$$\%$$ annual interest, compounded quarterly. What will the account balance be after 10 years? Use A = P$$ \left(1 + \frac{r}{m}\$$right)^{mt}$.

Accepted Answer

$$A = 5000 \left(1 + \frac{0.06}{4}\$$right)^{40}$$ \approx \9$$,037.04$

Question 3

Which of the following is the correct derivative of $$f(x) = 3x^4$ - 5x + 7$$?

Accepted Answer

$$f'(x) = 12x^3$ - 5$$

Question 4

A company’s cost function is $$C(x) = 2000 + 10x + 0.5x^2$. What is the marginal cost when x = 100$ units?

Accepted Answer

$$C'(x) = 10 + x$$,\ C'(100) = 110$

Question 5

Find the equation of the tangent line to $$f(x) = x^2$ + 2x at$$ $$x = 1.$$

Accepted Answer

$$y = 4x - 1$$

Question 6

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 7

A pharmaceutical company’s profit function is $$P(x) = -2x^2$ + 40x - 100. At $$what production level $$x$$ does the profit have a horizontal tangent?

Accepted Answer

$$x = 10$$

Question 8

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 9

If $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 10

A company’s average cost function is $$AC(x) = \frac{C(x)}{x}$$, where $$C(x) is $$the total cost. If $$C(x) = 500 + 20x$$, what is $$AC(10)$$?

Accepted Answer

$$AC(10) = \frac{500 + 200}{10} = 70$$

Question 11

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 12

A company’s revenue function is $$R(x) = 50x - x^2$. What is the marginal revenue when x = 10$?

Accepted Answer

$$R'(x) = 50 - 2x$$,\ R'(10) = 30$

Question 13

If $$f(x) = \frac{3x^2$ + 2x}{x}$$, what is $$f'(x)$$ using the quotient rule?

Accepted Answer

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

Question 14

A company’s profit function is $$P(x) = 100x - 0.5x^2$ - 200. $$What is the marginal profit when $$x = 20$$?

Accepted Answer

$$P'(x) = 100 - x$$,\ P'(20) = 80$

Question 15

If $$f(x) = \ln(x)$$, what is the slope of the tangent line to $$f(x) at$$ $$x = e$$?

Accepted Answer

$$\frac{1}{e}$$

Question 16

A company’s cost function is $$C(x) = 1000 + 25x. $$What is the average cost per unit when producing 50 units?

Accepted Answer

$$C_{avg}(50) = \frac{1000 + 25 \cdot 50}{50} = 45$$

Question 17

If $$f(x) = x^3$ - 6x^2$ + 9x$$, find all $$x$$ where the tangent line is horizontal.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 18

A company’s revenue function is $$R(x) = 200e$$^{0.01x}$$. What is the marginal revenue when x = 0$?

Accepted Answer

$$R'(x) = 2e$$^{0.01x}$$,\ R'(0) = 2$

Question 19

Suppose the demand function for a product is $$p(x) = 80 - 0.5x. $$What is the price when 60 units are sold?

Accepted Answer

$$p(60) = 80 - 0.5 \cdot 60 = 50$$

Business Calculus Exam 2 Study Guide: Functions, Limits, Derivatives, and Applications

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