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For problems 1 - 4, consider the function $$f(x) = \left\{ \begin{array}{ll} x-1 \hspace{.1 in} & \mbox{if } x \leq 2 \\ x+1 & \mbox{if } x>2\end{array}\right.$$

1.  Take the derivative of \(f(x) = \frac{1}{x^2}\).

a.  \(f'(x) = \frac{-2}{x}\)

b.  \(f'(x) = \frac{2}{x}\)

c.  \(f'(x) = \frac{2}{x^3}\)

d.  \(f'(x) = \frac{-2}{x^3}\)

e.  \(f'(x) = 0\)

2.  Take the derivative of \(f(x) = 3x^2 - 5x + 10\).

a.  \(f'(x) = 6x - 5\)

b.  \(f'(x) = 3x - 5\)

c.  \(f'(x) = 2x^2 - 5x\)

d.  \(f'(x) = 2x^2 - 5\)

e.  \(f'(x) = 2x^2 - 4x + 10\)

3.  Take the derivative of \(g(x) = 2^{-100}\).

a.  \(g'(x) = (100) 2^{101}\)

b.  \(g'(x) = (-100) 2^{-101}\)

c.  \(g'(x) = (100) 2^{99}\)

d.  \(g'(x) = (-100) 2^{-99}\)

e.  \(g'(x) = 0\)

4.  Differentiate (that means take the derivative) \(f(x) = \sqrt{x} \sin x\)

a.  \( f'(x) = \frac{\cos x}{2\sqrt{x}}\)

b.  \( f'(x) = \frac{\sin x}{2\sqrt{x}}\)

c.  \( f'(x) = \frac{1}{2\sqrt{x}} + \cos x\)

d.  \( f'(x) = \sqrt{x}\cos x + \frac{1}{2\sqrt{x}} \sin x\)

e.  \( f'(x) = \frac{1}{2\sqrt{x}} \cos x + \sqrt{x}\sin x\)

5.  Differentiate \(g(x) = \frac{2^x + 5}{x}\)

a.  \( g'(x) = x 2^x - 1\)

b.  \( g'(x) = \frac{x^2 2^{x-1}-2^x - 5}{x^2}\)

c.  \( g'(x) = \frac{2^x + 5 - x2^{x-1}}{x^2}\)

d.  \( g'(x) = (\ln 2)2^x\)

e.  \( g'(x) = \frac{(\ln 2)x2^x - 2^x - 5}{x^2}\)

6.  Differentiate\(g(x) = x^e\)

a.  \( g'(x) = (\ln x) x^e\)

b.  \( g'(x) = e x^{e-1}\)

c.  \( g'(x) = x^e\)

d.  \( g'(x) = x^{e+1}\)

e.  \( g'(x) = (\ln e) x^e\)

7.   Which of the following limits represents the slope of the tangent line to the curve \(y = f(x) = x^2\) at \(x = 2\)?

a.  \(\lim_{h \rightarrow 0} \frac{ 2+h^2 - 2^2}{h}\)

b.  \(\lim_{h \rightarrow 0} \frac{ (2+h)^2 - 2^2}{h}\)

c.  \(\lim_{h \rightarrow 0} \frac{ 1+h^2 - 2^2}{h}\)

d.  \(\lim_{h \rightarrow 0} \frac{ (1+h)^2 - 1^2}{h}\)

e.  \(\lim_{h \rightarrow 0} \frac{ (1+h)^2 - h^2}{h}\)

8.  If \(f(x) = 2^x\), find \(f^{(10)}(x)\).

a.  \(f^{(10)}(x)= (\ln 2)^{10} 2^x\)

b.  \(f^{(10)}(x)= x 2^{x-1}\)

c.  \(f^{(10)}(x)= (x)(x-1)\ldots(x-9) 2^{x-10}\)

d.  \(f^{(10)}(x)= (x)(x-1)\ldots(x-10) 2^{x-10}\)

e.  \(f^{(10)}(x)= (\ln 2)^{9} 2^x\)