Q: Which of the following limits can be evaluated using l'Hôpital's Rule?

$$\lim_{x \to 0} \frac{\cos x - 1}{x}$$

Q: Which of the following is the correct second derivative of $$f(x) = \sec x$$?

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

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 $$f'(x)$$?

Accepted Answer

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

For the function $$f(x) = \frac{x^2$ + 4}{x}$$, which of the following statements is true about its critical points?

Accepted Answer

There are critical points at $$x = -2$$ and $$x = 2$$ because the derivative is zero.

Question 5

Which of the following best describes the end behavior of $$y = \frac{3x^4$}{5x^2$ - 2x + 7} as$$ $$x 	o \infty$$?

Accepted Answer

The function increases without bound.

Question 6

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

Accepted Answer

$$f'(x) = -\csc x \sec x$$

Question 7

Suppose $$f(x) = x^4$ + e$$^{2x}$$ + \ln(x^5$). $$What is the fourth derivative $$f^{(4)}(x$$)$$?

Accepted Answer

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

Question 8

Which of the following graphs best represents a function that is increasing on $$(-\infty, -2)$$ and $$(4, \infty)$$, decreasing on $$(-2, 4)$$, concave up on $$(1, \infty)$$, concave down on $$(-\infty, 1)$$, with a local maximum at $$(-2, 8)$$, a local minimum at $$(4, -10)$$, and a point of inflection at $$(1, -1)$$?

Accepted Answer

Graph 1

Question 9

For the function $$f(x) = x e^x$$, which of the following is true about its intervals of increase and decrease?

Accepted Answer

Increasing on $$(-1, \infty)$$, decreasing on $$(-\infty, -1)$$

Question 10

Which of the following limits can be evaluated using l'Hôpital's Rule?

Accepted Answer

$$\lim_{x 	o 0} \frac{\cos x - 1}{x}$$

Question 11

Given $$f(x) = \arccos x$$, which of the following is true about the relationship between $$f(x)$$ and its first two derivatives?

Accepted Answer

The first derivative is negative and the second derivative changes sign at $$x = 0.$$

Question 12

If $$f(x) = \frac{x - 2}{(x^2$ + 3)^4$}$$, which rule(s) must be used to find $$f'(x)$$?

Accepted Answer

Quotient rule and chain rule

Question 13

Which of the following best describes the behavior of $$y = t e$$^{-t}$$ as t$$ 	o \infty$$?

Accepted Answer

The function approaches a horizontal asymptote at $$y = 0.$$

Question 14

For the function $$f(x) = \ln(\csc x) on $$the interval $$(0, \pi)$$, which of the following is true?

Accepted Answer

The function has a minimum at $$x = \frac{\pi}{2}.$$

Question 15

Which of the following is the correct second derivative of $$f(x) = \sec x$$?

Accepted Answer

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

Question 16

Which of the following best describes the behavior of $$y = (1 - 2x$$)^{1/x}$$ as x$$ 	o 0^+$$?

Accepted Answer

The function approaches $$e^{-2}.$$

Question 17

If $$f(x) = x^4$ \ln(2x^2$ + 5)$$, which rule(s) must be used to find $$f'(x)$$?

Accepted Answer

Product rule and chain rule

Question 18

Which of the following is the correct derivative of $$f(x) = \arccos(3x) - 3 \arccsc(x)$$?

Accepted Answer

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

Curve Sketching and Analysis: Increasing/Decreasing Intervals, Concavity, and Extrema

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