Q: Which of the following is a correct proof of the identity $$\frac{1+\sin x}{1-\sin x} = 1 + 2\tan x \sec x + 2\tan^2$ x \sec^2$ x$$?

Express in terms of $$\tan x$$ and $$\sec x$$ and expand.

Q: Which of the following limits does NOT exist?

$$\lim\limits_{x \to 0} \frac{1}{x}$$

Question 1

If $$\cot A = -\frac{3}{4}$$ and $$\sec B = -5$$, where $$A$$ and $$B$$ are in $$[\pi, 2\pi]$$, what is $$\sin(A+B)$$?

Accepted Answer

$$-\frac{24}{25}$$

Question 2

Find all $$x in$$ $$[0, 2\pi]$$ that satisfy $$	an x \cos x + \cos x + 2 	an x + 2 = 0.$$

Accepted Answer

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

Question 3

What is the inverse function of $$f(x) = \frac{2x+3}{x-5}$$?

Accepted Answer

$$f^{-1}(y$$) = \frac{5y+3}{y-2}$$

Question 4

Which of the following is a correct proof of the identity $$\frac{1+\sin x}{1-\sin x} = 1 + 2	an x \sec x + 2	an^2$ x \sec^2$ x$$?

Accepted Answer

Express in terms of $$	an x$$ and $$\sec x$$ and expand.

Question 5

Solve for $$x$$: $$e^{2x} + e^x - 6 = 0$.

Accepted Answer

$$x = \ln 2,\ x = \ln(-3) (no $$real solution for $$\ln(-3))$$

Question 6

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin 3x}{x}.$$

Accepted Answer

$$3$$

Question 7

Evaluate $$\lim\limits_{x 	o 1} \frac{\ln x + 	an(x-1) + e$$^{x-1}$$}{2x^2$ + 3x - 5}.$$

Accepted Answer

$$0$$

Question 8

Find the horizontal asymptotes of $$y = \frac{3x-3e^x}{4+4e^x}.$$

Accepted Answer

$$y = -3$$ and $$y = \frac{3}{4}$$

Question 9

Use the Intermediate Value Theorem to show that $$10\cos x + e^x + 3 = 0$$ has at least one real root.

Accepted Answer

Because the function is continuous and changes sign between $$x=0$$ and $$x=\pi$$, there is a root.

Question 10

Using the definition of derivative, find $$f'(x)$$ for $$f(x) = 2x^2$ - 8.$$

Accepted Answer

$$f'(x) = 4x$$

Question 11

Find $$y'$$ for $$y = (\ln x + 5x^2$)(8x^7$$ + \sec x)$ (do not simplify).

Accepted Answer

$$(\frac{1}{x} + 10x)(8x^7$ + \sec x) + (\ln x + 5x^2$)(56x^6$$ + \sec x 	an x)$

Question 12

Find $$y'$$ for $$y = 	an$$^{-1}$$(x^2$$) + \sqrt{x}$.

Accepted Answer

$$\frac{2x}{1 + x^4$} + \frac{1}{2\sqrt{x}}$$

Question 13

Find $$y'$$ for $$y = \cot(e^x) + \log_9$(x^2 + 7x$$)$.

Accepted Answer

$$-\csc^2$(e^x) e^x + \frac{2x+7}{(x^2$+7x) \ln 9}$$

Question 14

Which of the following limits does NOT exist?

Accepted Answer

$$\lim\limits_{x 	o 0} \frac{1}{x}$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = 10\cos t + e^t + 3. $$Use the Intermediate Value Theorem to argue if there is a time $$t$$ where $$R(t) = 0$$ for $$t \in [0, \pi].$$

Accepted Answer

Yes, because $$R(0) \u003e 0$$ and $$R(\pi) \u003c 0$$, so by IVT there is a root.

Question 16

Suppose $$f(x) = \frac{2x+3}{x-5}. $$What is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$x 
eq 2$$

Question 17

Which of the following is the correct derivative using the definition for $$f(x) = 2x^2$ - 8$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{2(x+h)^2$ - 8 - (2x^2$ - 8)}{h}$$

Question 18

If $$\lim\limits_{x 	o a} f(x) = L$$ and $$\lim\limits_{x 	o a} g(x) = M$$, what is $$\lim\limits_{x 	o a} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

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