Q: Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

$$\lim_{x \to 2} f(x)$$ does not exist or is infinite.

Q: Which of the following is the correct application of L'Hôpital's Rule for $$\lim_{x \to a} \dfrac{f(x)}{g(x)}$$ when both $$f(x)$$ and $$g(x)$$ approach $$0 as$$ $$x \to a$$?

$$\lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)} if $$the latter limit exists.

Question 1

Which of the following is a correct statement of the Mean Value Theorem for a function $$f$$ that is continuous on $$[a, b]$$ and differentiable on $$(a, b)$$?

Accepted Answer

$$f'(c) = \dfrac{f(b) - f(a)}{b - a}$$ for some $$c in$$ $$(a, b)$$

Question 2

Suppose $$f(x) = \sqrt{x}$$ and you want to approximate $$\sqrt{1.1}$$ using linear approximation at $$x = 1. $$What is the linear approximation $$L(x)$$?

Accepted Answer

$$L(x) = 1 + \dfrac{1}{2}(x - 1)$$

Question 3

Evaluate $$\displaystyle \lim_{x 	o 1} \dfrac{x^3$ + x^2$ - 2x}{x - 1}$$ using L'Hôpital's Rule.

Accepted Answer

$$3$$

Question 4

Which of the following is an indeterminate form that can be resolved using L'Hôpital's Rule?

Accepted Answer

$$\dfrac{0}{0}$$

Question 5

Given $$f(x) = 2x^3$ - 3x + 1 on$$ $$[-2, 2]$$, what is the average rate of change of $$f on $$this interval?

Accepted Answer

$$5$$

Question 6

For the function $$f(x) = x^2$ - 4x + 3$$, which of the following $$x-$$values is a critical point?

Accepted Answer

$$x = 2$$

Question 7

Which of the following best describes the steps to find the absolute maximum and minimum of a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

Find critical points in $$(a, b)$$, evaluate $$f at $$critical points and endpoints, compare values.

Question 8

If $$f(x) = \dfrac{2x}{x^2$ + 1}$$, what is the domain of $$f$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 9

Which of the following is the correct formula for the linear approximation $$L(x) of$$ $$f(x) at$$ $$x = a$$?

Accepted Answer

$$L(x) = f(a) + f'(a)(x - a)$$

Question 10

Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ does not exist or is infinite.

Question 11

Which of the following is the correct antiderivative of $$\cos(x)$$?

Accepted Answer

$$\sin(x) + C$$

Question 12

If $$f'(x) \u003e 0$$ for all $$x in an $$interval $$I$$, what can you conclude about $$f on$$ $$I$$?

Accepted Answer

$$f is $$increasing on $$I.$$

Question 13

Which of the following is the correct application of L'Hôpital's Rule for $$\lim_{x 	o a} \dfrac{f(x)}{g(x)}$$ when both $$f(x)$$ and $$g(x)$$ approach $$0 as$$ $$x 	o a$$?

Accepted Answer

$$\lim_{x 	o a} \dfrac{f(x)}{g(x)} = \lim_{x 	o a} \dfrac{f'(x)}{g'(x)} if $$the latter limit exists.

Question 14

Which of the following is a necessary hypothesis for the Mean Value Theorem to apply to $$f on$$ $$[a, b]$$?

Accepted Answer

$$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$

Question 15

Suppose $$f(x) = x^3$ - 3x^2$ + 4 on$$ $$[0, 3]. $$What is the absolute maximum value of $$f on $$this interval?

Accepted Answer

$$f(3) = 4$$

Question 16

Which of the following is the correct formula for the area of a circle?

Accepted Answer

$$A = \pi r^2$

Question 17

If $$f(x) = \ln x / x^2$, what is the domain of f$?

Accepted Answer

$$(0, \infty)$$

Question 18

Which of the following is the correct formula for the volume of a cylinder with radius $$r$$ and height $$h$$?

Accepted Answer

$$V = \pi r^2$ h$$

Question 19

Which of the following is the correct formula for the Pythagorean Theorem?

Accepted Answer

$$a^2$$ + $$b^2$$ = $$c^2$$

Calculus I: Applications of the Derivative, Curve Sketching, and L'Hôpital's Rule

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