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

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

Q: Which of the following is a correct application of L'Hôpital's Rule?

$$\lim_{x \to 0} \frac{\sin x}{x}$$, since it is $$\frac{0}{0}$$

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 Rolle's Theorem on $$[0,2]$$?

Accepted Answer

$$f(x) = |x-1|$$

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^3$ + 6x^2$, find all critical points and determine which are local minima using the first derivative test.

Accepted Answer

Critical points at $$x=0,2,3$$; local minimum at $$x=3$$

Question 6

For the function $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at which $$x-$$value does the function have a point of inflection?

Accepted Answer

$$x = 2$$

Question 7

Which of the following statements about concavity is correct?

Accepted Answer

If $$f''(x) = 0 at$$ $$x=a$$ and $$f''$$ changes sign at $$a$$, then $$x=a is a $$point of inflection.

Question 8

A function $$f(x)$$ has $$f''(x) = 0 at$$ $$x=2. If$$ $$f''(x)$$ changes from positive to negative as $$x$$ increases through $$2$$, what can you conclude?

Accepted Answer

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

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 patient’s body temperature $$T (in$$ $$^\circC) is $$modeled by $$T(t) = 37 + 2e$$^{-0.1t}$$, where t$ is hours after medication. What is the rate of change of temperature at t=0$?

Accepted Answer

Question 12

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

Accepted Answer

$$0$$

Question 13

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

Accepted Answer

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

Question 14

If $$F(x)$$ and $$G(x)$$ are both antiderivatives of $$f(x)$$, what can you say about $$F(x) - G(x)$$?

Accepted Answer

$$F(x) - G(x) is a $$constant

Question 15

Which of the following is a correct application of L'Hôpital's Rule?

Accepted Answer

$$\lim_{x 	o 0} \frac{\sin x}{x}$$, since it is $$\frac{0}{0}$$

Question 16

A chemical reaction's concentration is modeled by $$C(t) = 5e$$^{-2t}$$. What is the indefinite integral $$\int C(t) \, dt$$?

Accepted Answer

$$-\frac{5}{2} e$$^{-2t}$$ + C$$

Question 17

Suppose $$f(x) is $$continuous on $$[a,b]$$ and differentiable on $$(a,b)$$, and $$f'(x) = 0$$ for all $$x in$$ $$(a,b). $$What can you conclude about $$f$$?

Accepted Answer

$$f is $$constant on $$[a,b]$$

Question 18

Which of the following describes a function with a vertical asymptote at $$x=2$$?

Accepted Answer

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

Question 19

Let $$f(x) = \ln(x). $$What is the second derivative $$f''(x)$$?

Accepted Answer

$$-\frac{1}{x^2$}$$

Calculus I: Study Guide for Exam 1 (Sections 4.2–4.5, 4.8)

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