Q: Which of the following limits is an indeterminate form of type $$\frac{0}{0}$$?

$$\lim_{x \to 0} \frac{\sin x}{x}$$

Question 1

Let $$f(x) = x^3$ - 3x + 2. On $$the interval $$[0,2]$$, does $$f$$ satisfy the hypotheses of the Mean Value Theorem? If so, what is the value of $$c in$$ $$(0,2)$$ such that $$f'(c) = \frac{f(2) - f(0)}{2-0}$$?

Accepted Answer

$$c = 1 + \frac{1}{\sqrt{3}}$$

Question 2

Suppose $$f(x) is $$continuous on $$[a,b]$$ and differentiable on $$(a,b)$$, and $$f(a) = f(b). $$Which theorem guarantees that there exists $$c in$$ $$(a,b)$$ such that $$f'(c) = 0$$?

Accepted Answer

Rolle's Theorem

Question 3

Which of the following functions does NOT satisfy the hypotheses of the Mean Value Theorem on $$[0,1]$$?

Accepted Answer

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

Question 4

If $$f'(x) \u003e 0$$ for all $$x in an $$interval $$I$$, which of the following statements is true about $$f on$$ $$I$$?

Accepted Answer

$$f is $$strictly increasing on $$I$$

Question 5

Given $$f(x) = x^4$ - 4x^2$, find all critical points and determine which are local minima, maxima, or neither using the first derivative test.

Accepted Answer

Critical points at $$x = 0, \pm 2$$; $$x=0 is a $$local maximum, $$x=\pm2$$ are local minima.

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = 5t e$$^{-0.5t}$$, where t$ is in hours. At what time does the maximum concentration occur?

Accepted Answer

$$t = 2$$

Question 7

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

Accepted Answer

The graph is concave up.

Question 8

Find the inflection points of $$f(x) = x^3$ - 6x^2$ + 9x + 1.$$

Accepted Answer

$$x = 2$$

Question 9

Which of the following limits is an indeterminate form of type $$\frac{0}{0}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\sin x}{x}$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}$$ using L'Hôpital's Rule.

Accepted Answer

$$1$$

Question 11

A biologist models the population of a bacteria culture by $$P(t) = 100e$$^{0.03t}$$. Find $$\int P'(t) \, dt$$.

Accepted Answer

$$100e^{0.03t} + C$

Question 12

Which of the following is the general antiderivative of $$f(x) = 2x$$?

Accepted Answer

$$x^2 + C$

Question 13

Suppose $$f'(x) = 0$$ for all $$x in an $$interval. What can you conclude about $$f$$?

Accepted Answer

$$f is $$constant.

Question 14

Evaluate $$\lim_{x 	o 0^+} x \ln x.$$

Accepted Answer

$$0$$

Question 15

Which of the following statements about the function $$f(x) = \frac{1}{x} is $$true?

Accepted Answer

$$f$$ has a vertical asymptote at $$x=0$$ and a horizontal asymptote at $$y=0.$$

Question 16

If $$f(x)$$ and $$g(x)$$ are differentiable on an interval and $$f'(x) = g'(x)$$ for all $$x in $$the interval, what can you conclude?

Accepted Answer

$$f(x) = g(x) + C$$ for some constant $$C.$$

Question 17

Find $$\int (3x^2$ - 4x + 5) \, dx.$$

Accepted Answer

$$x^3$$ - $$2x^2 + 5x + C$

Question 18

A function $$f$$ has $$f''(x) = 0 at$$ $$x = 2. If$$ $$f''(x)$$ changes sign at $$x = 2$$, what does this imply?

Accepted Answer

$$x=2 is a $$point of inflection.

Question 19

Evaluate $$\lim_{x 	o 0} \frac{\ln(1+x)}{x}$$ using L'Hôpital's Rule.

Accepted Answer

$$1$$

Question 20

Which of the following is an indeterminate form that may require rewriting before applying L'Hôpital's Rule?

Accepted Answer

$$0 \cdot \infty$$

Calculus Study Guide: Mean Value Theorem, Monotonicity, Curve Sketching, L'Hôpital's Rule, and Antiderivatives

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