Question 1

A baseball is thrown and its height at time $$t$$ seconds is given by $$s(t) = -16t^2$ + 60t + 6. $$What is the average velocity of the baseball over the interval $$[1, 1.5]$$?

Accepted Answer

$$22 ft/s$$

Question 2

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin 2x}{\sin x}$$ analytically using the identity $$\sin 2x = 2 \sin x \cos x.$$

Accepted Answer

$$2$$

Question 3

Find $$\displaystyle \lim_{x 	o 3} \frac{x^4$ - 81}{x - 3}.$$

Accepted Answer

$$108$$

Question 4

Which of the following describes the end behavior of $$f(x) = 1 - e$$^{-2x}$$ as x$$ 	o \infty$$?

Accepted Answer

$$f(x) 	o 1$$

Question 5

Find all vertical asymptotes of $$f(x) = \frac{x^2$ - 5x + 6}{x^2$ - 2x}.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 6

Determine if $$h(x) = \sqrt{x^2$ - 9} is $$left-continuous at $$x = 3.$$

Accepted Answer

No, because $$h(x) is $$not defined for $$x \u003c 3$$

Question 7

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{\cos x}{e$$^{3x}$$}.$$

Accepted Answer

$$0$$

Question 8

For $$f(x) = e$$^{\sqrt{x}$$ - 2}$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

Question 9

Use the Intermediate Value Theorem to determine if $$g(x) = x^5$ + 7x + 5$$ has a root in $$(-1, 0). $$Which statement is correct?

Accepted Answer

$$g(-1)  0$$, so there is a root in $$(-1, 0)$$

Question 10

Evaluate $$\displaystyle \lim_{h 	o 0} \frac{\sqrt{5x + h} - \sqrt{5x}}{h}$$ for $$x \u003e 0.$$

Accepted Answer

$$\frac{1}{2\sqrt{5x}}$$

Question 11

Find $$\displaystyle \lim_{p 	o 1} \frac{p^5$ - 1}{p - 1}.$$

Accepted Answer

$$5$$

Question 12

Does the function $$h(x) = \begin{cases} -2x - 14 & x \leq 5 \ \sqrt{x^2$ - 9} & x \u003e 5 \end{cases}$$ have continuity at $$x = 5$$?

Accepted Answer

No, because the left and right limits are not equal

Question 13

Find $$\displaystyle \lim_{x 	o \infty} 3 	an$$^{-1}$$ x + 2.$$

Accepted Answer

$$\frac{3\pi}{2} + 2$$

Question 14

Find all asymptotes of $$f(x) = \frac{x^2$ + x + 3}{|x|}.$$

Accepted Answer

Vertical asymptote at $$x = 0$$; no horizontal asymptote

Question 15

Evaluate $$\displaystyle \lim_{x 	o -\infty} \frac{1}{2 + e^x}.$$

Accepted Answer

$$1/2$$

Question 16

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{1}{2 + e^x}.$$

Accepted Answer

$$0$$

Question 17

Find $$\displaystyle \lim_{x 	o 3^-} \frac{x - 4}{x^2$ - 3x}.$$

Accepted Answer

$$-1$$

Question 18

Which of the following is the correct limit definition proof for $$\displaystyle \lim_{x 	o 1} (5x - 2) = 3$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, choose $$\delta = \epsilon / 5$$

Question 19

Does $$f(x) = \frac{x^2$ - 5x + 6}{x^2$ - 2x}$$ have a slant (oblique) asymptote?

Accepted Answer

No, because the degree of the numerator is two and the denominator is also two

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