$$f''(x) = \sec x \tan^2$ x + \sec^3$ x$$

$$\lim_{x \to 0} \frac{-\sin x}{1} = 0$$

$$\lim_{t \to \infty} te$$^{-t}$$ = 0$$

Question 1

Which of the following is the correct derivative of $$f(x) = (x^2$ + 1)(3x^3$ - 2x)$$?

Accepted Answer

$$f'(x) = 15x^4$ + 3x^2$ - 2$$

Question 2

Given $$f(x) = x^3$ + 6x^2$ - 1$$, what is the value of $$f'(x)$$ when $$x = 2$$?

Accepted Answer

$$f'(2) = 25$$

Question 3

Which rule(s) must be used to find the derivative of $$f(x) = x^2$ e^x$$?

Accepted Answer

Product rule only

Question 4

Find the second derivative of $$f(x) = \sec x.$$

Accepted Answer

$$f''(x) = \sec x 	an^2$ x + \sec^3$ x$$

Question 5

A function $$f(x)$$ has a critical point at $$x = 2. $$Which of the following must be true?

Accepted Answer

$$f'(2) = 0 or$$ $$f'(2) is $$undefined

Question 6

Given the graph in Image 1, which of the following best describes the intervals where the function is increasing?

Accepted Answer

Increasing on $$(-\infty, -2)$$ and $$(4, \infty)$$

Question 7

For the function $$f(x) = \frac{x}{\sqrt{x^2$ + 2}}$$, what is the horizontal asymptote as $$x 	o \infty$$?

Accepted Answer

$$y = 1$$

Question 8

Which of the following is the correct application of l'Hôpital's Rule to $$\lim_{x 	o 0} \frac{\cos x - 1}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{-\sin x}{1} = 0$$

Question 9

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

Accepted Answer

$$f'(x) = -\csc^2$ x / \cot x$$

Question 10

Which of the following graphs (by image number) best represents a function with a vertical asymptote at $$x = 0$$ and $$x = \pi$$?

Accepted Answer

Image 5

Question 11

For $$f(x) = x^4$ + e$$^{2x}$$ + \ln(x^5$)$$, what is the fourth derivative $$f^{(4)}(x$$)$$?

Accepted Answer

$$f^{(4)}(x) = 24$$ + $$16e^{2x}$$ - $$30x^{-4}$$

Question 12

Which of the following best describes the concavity of the function in Image 6?

Accepted Answer

Concave up on $$(-\infty, 0)$$, concave down on $$(0, \infty)$$

Question 13

If $$f(x) = \arccos(3x) - 3\arccsc(x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{-3}{\sqrt{1-9x^2$}} + \frac{3}{|x|\sqrt{x^2$-1}}$$

Question 14

Which of the following limits describes the end behavior of $$y = te$$^{-t}$$ as t$$ 	o \infty$$?

Accepted Answer

$$\lim_{t 	o \infty} te$$^{-t}$$ = 0$$

Question 15

Which of the following is the correct derivative of $$f(x) = x^4$ \ln(2x^2$ + 5)$$?

Accepted Answer

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

Question 16

Given $$f(x) = \frac{x^3$ + 6x^2$ - 1}{x^2$}$$, which rule(s) must be used to find $$f'(x)$$?

Accepted Answer

Quotient rule

Question 17

Which of the following best describes the intervals where the function in Image 2 is decreasing?

Accepted Answer

Decreasing on $$(-2, 4)$$

Question 18

For $$f(x) = \frac{3x^4$}{5x^2$ - 2x + 7}$$, what is the end behavior as $$x 	o \infty$$?

Accepted Answer

$$y 	o \infty$$

Comprehensive Calculus II Exam Review: Derivatives, Curve Sketching, and Limits

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