Question 1

Solve for $$x$$: $$\ln x + \ln(x-3) = 0.$$

Accepted Answer

$$x = \frac{3 + \sqrt{21}}{2}$$

Question 2

Find the exact value of $$\sin$$^{-1}$$\left( -\frac{\sqrt{3}}{2} \right).$$

Accepted Answer

$$-\frac{\pi}{3}$$

Question 3

Which of the following intervals contains a root of the equation $$x^3 - 3x = 5$ according to the Intermediate Value Theorem?

Accepted Answer

$$(2,3)$$

Question 4

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

Accepted Answer

$$x \u003e \frac{1}{2}\ln(9)$$

Question 5

Compute $$\lim_{x 	o 0} \frac{\sin(5x)}{2x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 6

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 7

Find the horizontal asymptote(s) of $$y = \frac{7 + e^x}{4 - e^x}.$$

Accepted Answer

$$y = -1$$

Question 8

Let $$f(x) = x^2$ - a$$ for $$x \leq 1$$ and $$f(x) = \frac{3x^2$ + 12x - b}{x^2$ + 2x - 3}$$ for $$x \u003e 1. $$For which values of $$a$$ and $$b is$$ $$f(x)$$ continuous on $$(-\infty, \infty)$$?

Accepted Answer

$$a = 0, b = 12$$

Question 9

Evaluate $$\lim_{h 	o 0} \frac{\ln(1 + 3h)}{7h}.$$

Accepted Answer

$$\frac{3}{7}$$

Question 10

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

Accepted Answer

$$-\frac{1}{15}$$

Question 11

Find all $$x in$$ $$[0,2\pi]$$ such that $$	an^2$ x - 3 = 0.$$

Accepted Answer

$$x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$$

Question 12

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

Accepted Answer

$$\sqrt{1 - (x/5)^2$}$$

Question 13

Let $$f(x) = x \cos(x). $$Where does the tangent line at $$x = \frac{\pi}{2}$$ intersect the $$y-$$axis?

Accepted Answer

$$y = -\frac{\pi}{2} + 1$$

Question 14

For which $$x is $$the function $$f(x) = \frac{x-2}{|x-2|}$$ not differentiable?

Accepted Answer

$$x = 2$$

Question 15

Suppose $$f(x) is $$defined as $$f(x) = x \sin(1/x)$$ for $$x 
eq 0$$ and $$f(0) = 0. $$Which of the following is true?

Accepted Answer

$$f(x) is $$continuous but not differentiable at $$x=0$$

Question 16

Find the limit: $$\lim_{x 	o \infty} (\sqrt{x+1} - \sqrt{x}) \cdot \sqrt{x}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 17

Let $$f(x) = x^2$ \sin(1/x)$$ for $$x 
eq 0$$ and $$f(0) = 0. $$Which of the following is true?

Accepted Answer

$$f(x) is $$continuous and differentiable at $$x=0$$

Question 18

Suppose $$f(x) is a $$function with $$f'(2) = 5. $$What is $$\lim_{h 	o 0} \frac{f(2 + 4h) - f(2)}{3h}$$?

Accepted Answer

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

Question 19

Find the equation of the tangent line to $$y = \sin(x)$$ over $$(0,\pi)$$ that is parallel to $$y = -\frac{1}{2}x + 5.$$

Accepted Answer

$$y = -\frac{1}{2}x + 1 + \frac{1}{2}\arccos(-\frac{1}{2})$$

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