Question 1

Which of the following best describes a removable discontinuity in a function $$f(x)$$?

Accepted Answer

A point where the limit exists, but the function is either not defined or defined to a different value than the limit.

Question 2

Given $$f(x) = \frac{2x-10}{x^2$-25}$$, which of the following statements is true about its discontinuities?

Accepted Answer

There is a removable discontinuity at $$x = 5$$ and a vertical asymptote at $$x = -5.$$

Question 3

Evaluate $$\lim_{x 	o \infty} \frac{3x^4$ - 2x^3$}{15x^4$ + 10}. $$Which is the correct answer?

Accepted Answer

$$\frac{1}{5}$$

Question 4

For the piecewise function $$f(x) = \begin{cases} \frac{2x^2$ - 4x - 16}{x - 4} & x 
eq 4 \ kx - 8 & x = 4 \end{cases}$$, what value of $$k$$ makes $$f(x)$$ continuous everywhere?

Accepted Answer

$$k = 5$$

Question 5

A company’s cost function is $$C(x) = 2x - 1$$ and its revenue function is $$R(x) = 9x + 5. $$What is the average cost function $$\overline{C}(x)$$?

Accepted Answer

$$\overline{C}(x) = \frac{2x - 1}{x}$$

Question 6

Find the derivative of $$f(x) = \sqrt{x}$$ using the formal definition of the derivative.

Accepted Answer

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

Question 7

If $$f(x) = 3x^2$ - 5x$$, what is the average rate of change from $$x = -2 to$$ $$x = 4$$?

Accepted Answer

$$1$$

Question 8

Which of the following is NOT a case where a function is non-differentiable at a point?

Accepted Answer

A point where the function is continuous and smooth.

Question 9

Given $$f(x) = x^3$ - 9x^2$ - 30x + 12$$, at which $$x-$$values does the tangent line become horizontal?

Accepted Answer

$$x = -2$$ and $$x = 5$$

Question 10

If $$f(x) = \sqrt{x^2$ + 11}$$, what is the equation of the tangent line at $$x = 5$$?

Accepted Answer

$$y = \frac{5}{6}x + \frac{11}{6}$$

Question 11

Find $$\frac{d}{dx}\left[(x^3$ + 4x^2$)(6\sqrt{x} - 5x)\right]$$ using the product rule.

Accepted Answer

$$21x^{5/2}$$ - $$20x^3$$ + $$60x^{3/2}$$ - $$60x^2$$

Question 12

Suppose $$f(x)$$ and $$g(x)$$ are differentiable functions with $$f(3) = 1$$, $$f'(3) = 4$$, $$g(3) = 7$$, $$g'(3) = 5. $$What is $$\frac{d}{dx}[f(3)g(3)]$$?

Accepted Answer

$$33$$

Question 13

Given the cost function $$C(x) = 2x - 1$$ and revenue function $$R(x) = 9x + 5$$, what is the marginal profit function?

Accepted Answer

$$P'(x) = 7$$

Question 14

If $$y = x^3$ - 8x^2$ + 15x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$3x^2 - 16x + 15$

Question 15

Which rule is used to find the derivative of $$f(x) = (2x^2$ + 1)/(x^2$ + 3)$$?

Accepted Answer

Quotient Rule

Question 16

A manufacturer’s cost function is $$C(x) = 50x^2$ + 6x + 2.0. $$What is the marginal cost when $$x = 10$$?

Accepted Answer

Question 17

If $$f(x) = (3x^2$ - 10x + 7$$)^{12}$$, what is $$\frac{d}{dx}f(x)$$ using the chain rule?

Accepted Answer

$$12(3x^2 - 10x$$ + $$7)^{11}(6x - 10$$)$$

Question 18

Which of the following is the correct derivative of $$f(x) = 20x^4$ - 42x^2$ + 40x - 17$$?

Accepted Answer

$$f'(x) = 80x^3$ - 84x + 40$$

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