Q: Solve for $$x in$$ $$[0, 2\pi]$$: $$\tan^2$ x - 3 = 0.$$

$$x = \tan$$^{-1}$$(\sqrt{3}), \tan$$^{-1}$$(\sqrt{3}) + \pi$$

Question 1

Solve for $$x$$: $$\ln x + \ln(x-3) = \ln 4.$$

Accepted Answer

$$x = 4 or$$ $$x = -1 ($$but only $$x=4 is $$valid)$

Question 2

Which of the following is the domain of the function $$f(x) = \arcsin(x)$$?

Accepted Answer

$$[-1, 1]$$

Question 3

Find the exact value: $$\sin$$^{-1}$$\left(\sin\left(\frac{8\pi}{7}\right)\right).$$

Accepted Answer

$$\frac{8\pi}{7} - \pi$$

Question 4

Solve for $$x in$$ $$[0, 2\pi]$$: $$	an^2$ x - 3 = 0.$$

Accepted Answer

$$x = 	an$$^{-1}$$(\sqrt{3}), 	an$$^{-1}$$(\sqrt{3}) + \pi$$

Question 5

Express $$\cos\left(\sin$$^{-1}$$\left(\frac{x}{5}\right)\right) as an $$algebraic function of $$x$$ for $$|x| \leq 5.$$

Accepted Answer

$$\frac{\sqrt{25 - x^2$}}{5}$$

Question 6

Let $$f(x) = \ln(\frac{1}{3}e$$^{2x}$$ - 3). $$What is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$(\ln 3, \ln 6)$$

Question 7

Compute $$\displaystyle \lim_{x 	o 5^-} \frac{x^2$ - 3x - 10}{|x-5|}.$$

Accepted Answer

$$7$$

Question 8

Suppose $$f(x) is $$continuous on $$[1,6]$$ and $$f(1) = 1.5$$, $$f(2) = 2.2$$, $$f(3) = 2.9$$, $$f(4) = 3.5$$, $$f(5) = 5.5$$, $$f(6) = 5.9. $$How many solutions does $$f(x) = x$$ have on $$[1,6] by $$the Intermediate Value Theorem?

Accepted Answer

2

Question 9

Let $$f(x) be $$differentiable and $$f'(2) = 5. $$Compute $$\displaystyle \lim_{h 	o 0} \frac{f(2 + 4h) - f(2)}{3h}.$$

Accepted Answer

$$\frac{20}{3}$$

Question 10

Find the values of $$a$$ and $$b so $$that the function $$f(x) = \begin{cases} x^2$ - a & x \leq 1 \ \frac{3x^2$ + 12x - b}{x^2$ + 2x - 3} & x \u003e 1 \end{cases} is $$continuous on $$(-\infty, \infty).$$

Accepted Answer

$$a = 1, b = 16$$

Question 11

Find the horizontal asymptote(s) of $$f(x) = \frac{x^3$ + x^2$}{3x^2$ + \sqrt{25x^6$ + x^2$}}.$$

Accepted Answer

$$y = \frac{1}{6}$$

Question 12

Find the equation of the tangent line to $$y = \sin(x) at $$the point where the tangent is parallel to $$y = -\frac{1}{2}x + 5.$$

Accepted Answer

$$y = -\frac{1}{2}x + \frac{1}{2} + \sin\left(\frac{2\pi}{3}\right)$$

Question 13

Which of the following statements is true about the function $$f(x) = x \sin\left(\frac{1}{x}ight)$$ for $$x 
eq 0$$, $$f(0) = 0$$?

Accepted Answer

It is continuous but not differentiable at $$x=0.$$

Question 14

Suppose $$f(2) = 3$$ and $$f'(2) = 5. $$Evaluate $$\left.\frac{d}{dx}\left(\frac{1}{f$$^{-1}$$(x)}\right)\right|_{x=3}.$$

Accepted Answer

$$-\frac{5}{9}$$

Question 15

Solve for $$x in$$ $$[0, 2\pi)$$: $$\ln(\sin x) - \ln(\cos x) = \frac{1}{2} \ln 3.$$

Accepted Answer

$$x = \frac{\pi}{3}, \frac{2\pi}{3}$$

Question 16

Let $$f(x) = x^2$ \sin\left(\frac{1}{x}ight)$$ for $$x 
eq 0$$, $$f(0) = 0. $$Which statement is true?

Accepted Answer

It is continuous and differentiable at $$x=0.$$

Question 17

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one $$c \in (a, b)$$ such that $$f(c) = 0.$$

Question 18

Let $$f(x) = x + 5$$ for $$x \u003e 2. $$Find the formula for $$f^{-1}(x$$)$$ and its domain.

Accepted Answer

$$f^{-1}(x) = x - 5$, domain: (7$$, \infty)$$

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