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For problems 1 to 3, take the derivative.

1.  \(f(x) = \sqrt{x} \sin x\)

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

2.  \(g(x) = \frac{2^x + 5}{x}\)

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

3.  \(g(x) = x^e\)

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.  \( g'(x) = (\ln e) x^e\)

4.  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.  \(f^{(10)}(x)= (\ln 2)^{9} 2^x\)

In Problems 5 - 8 assume that the height, \(h\), of a hot-air balloon in meters, \(t\) minutes after it is launched is given by the function \( h(t) = t^3 - 8t^2 + 16 t\)

5.  What is the height of the balloon after 2 minutes?

a.  2 m

b.  4 m

c.  8 m

d.  16 m

e.  32 m