Question 1

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 2

Given the function $$f(x) = 4\sqrt{x} + 5x$$^{1/4}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = 2x$$^{-1/2}$$ + \frac{5}{4}x$$^{-3/4}$$

Question 3

Suppose the demand for a product is given by $$D(p) = -2p^2$ - 6p + 300$$, where $$p is $$the price. What is the rate of change of demand with respect to price when $$p = 14$$?

Accepted Answer

$$-62$$

Question 4

Which of the following best describes the meaning of the derivative $$f'(x) at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to $$f(x) at$$ $$x = a$$

Question 5

If $$f(x) = x^3$ - 12x^2$ + 54x + 14$$, at what value of $$x is $$the slope of the tangent line equal to $$6$$?

Accepted Answer

$$x = -4$$

Question 6

Given $$y = \frac{6}{x} - \frac{4}{x^5$} + 6 + \sqrt{3}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$-6x^{-2}$$ + $$20x^{-6}$$

Question 7

Which of the following is the correct process to find the slope of the tangent line to $$y = x^4$ - 3x^3$ + 2 at$$ $$x = 1$$?

Accepted Answer

Substitute $$x = 1$$ into the derivative of the function and evaluate

Question 8

For which of the following functions is the derivative always zero?

Accepted Answer

$$f(x) = 7$$

Question 9

If $$f(x) = x^3$ - 4x^2$ - 4x + 1$$, for which values of $$x is $$the tangent line horizontal?

Accepted Answer

$$x = \frac{4 + 2\sqrt{7}}{3}, \frac{4 - 2\sqrt{7}}{3}$$

Question 10

Which of the following statements is TRUE about differentiability?

Accepted Answer

If a function is differentiable at $$x = a$$, then it must also be continuous at $$x = a$$

Question 11

Given the graph of a function, how can you estimate the slope of the tangent line at a point?

Accepted Answer

Draw a line that just touches the curve at the point and estimate rise over run

Question 12

Which of the following is the correct derivative of $$f(x) = x^6$ + 3x^2$ - 7x$$?

Accepted Answer

$$f'(x) = 6x^5$ + 6x - 7$$

Question 13

If $$f(x) = 2x^3$ - 5x^2$ + 4x - 9$$, what is $$f'(x)$$?

Accepted Answer

$$6x^2 - 10x + 4$

Question 14

Which of the following is the correct interpretation of the units of a derivative if $$f(x) is $$measured in dollars and $$x in $$items?

Accepted Answer

Dollars per item

Question 15

If $$f(x) = x^n$$, what is $$f'(x)$$ according to the power rule?

Accepted Answer

$$n x$$^{n-1}$$

Question 16

Which of the following is NOT a correct notation for the derivative of $$f(x)$$?

Accepted Answer

$$f(x)$$

Question 17

If $$f(x) = e$$^{3x}$$, what is f$$'(x)$$?

Accepted Answer

$$3e^{3x}$$

Question 18

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

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

Question 19

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

Accepted Answer

$$\frac{1}{x}$$

Business Calculus: Derivatives, Tangent Lines, and Marginal Analysis Study Guide

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