Q: A company’s total cost function is $$C(x) = 500 + 20x + 0.1x^2$. What is the marginal cost when x = 100$ units?

$$C'(100) = 20 + 0.2 \times 100 = 40$$

Question 1

Which of the following correctly applies the Power Rule to find the derivative of $$f(x) = 7x^5$?

Accepted Answer

$$f'(x) = 35x^4$

Question 2

A company’s total cost function is $$C(x) = 500 + 20x + 0.1x^2$. What is the marginal cost when x = 100$ units?

Accepted Answer

$$C'(100) = 20 + 0.2 	imes 100 = 40$$

Question 3

If $$f(x) = 3x^2$ + 2x$$ and $$g(x) = x^3$, what is the derivative of h(x) = f(x) + g(x$$)$$?

Accepted Answer

$$h'(x) = 6x + 2 + 3x^2$

Question 4

Which of the following is the correct application of the Product Rule for $$y = x^2$ e^x$$?

Accepted Answer

$$y' = 2x e^x + x^2$ e^x$$

Question 5

Given $$f(x) = \frac{5x^3$}{x^2$ + 1}$$, use the Quotient Rule to find $$f'(x).$$

Accepted Answer

$$f'(x) = \frac{(15x^2$)(x^2 + 1$$) - $$5x^3(2x$$)}{$$(x^2$$ + $$1)^2$$}$

Question 6

If $$y = (3x^2$ + 2)^4$, which rule is most appropriate to find $$\frac{dy}{dx}$$ and what is the derivative?

Accepted Answer

Chain Rule; $$\frac{dy}{dx} = 4(3x^2$ + 2)^3$ \cdot 6x$$

Question 7

A company’s revenue function is $$R(q) = 2q - \frac{1}{25,000}q^2$. What is the marginal revenue when q = 10$$,000$$?

Accepted Answer

$$R'(q) = 2 - \frac{2}{25,000}q$$, so R'(10,000) = 2 - 0.8 = 1.2$

Question 8

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

Accepted Answer

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

Question 9

Which of the following best describes the derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 10

Suppose the average cost per item is $$A(x) = \frac{C(x)}{x}$$, where $$C(x) is $$the total cost. What is the marginal average cost?

Accepted Answer

$$A'(x) = \frac{xC'(x) - C(x)}{x^2$}$$

Question 11

If $$y = 5^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 5^x \ln 5$$

Question 12

A bank offers a compound interest account with $$A = 500\left(1 + \frac{r}{1200}\$$right)^{120}$$. What calculus concept would you use to find how fast the amount A$ changes with respect to the interest rate r$?

Accepted Answer

The Chain Rule

Question 13

If $$f(x) = \ln|x|$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 14

Given $$y = \frac{e$$^{3x}$$}{x^2$}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{3e$$^{3x}$$x^2 - 2x$$ $$e^{3x}$$}{$$x^4$$}$

Question 15

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

Accepted Answer

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

Question 16

A company’s profit function is $$P(q) = 1.75q - \frac{1}{25,000}q^2$ - 2100. $$What is the marginal profit when $$q = 10,000$$?

Accepted Answer

$$P'(q) = 1.75 - \frac{2}{25,000}q$$, so P'(10,000) = 1.75 - 0.8 = 0.95$

Question 17

Which of the following best describes the Chain Rule?

Accepted Answer

The derivative of a composite function $$y = f(g(x)) is$$ $$f'(g(x)) \cdot g'(x)$$

Question 18

If $$f(x) = x^2$ \ln x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x \ln x + x$$

Question 19

Suppose $$y = \ln(e$$^{2x}$$ + 1). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2e$$^{2x}$$}{e$$^{2x}$$ + 1}$$

Business Calculus Study Guide: Derivative Techniques and Applications

Study Guide - Practice Questions

Study Guide - Flashcards