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For questions 1 through 3, compute the antiderivatives.

1. \(\int 3x + 4 dx\)

a.  \(3 + c\)

b.  \(3x^2 + 4x + c\)

c.  \(\frac{3}{2}x^2 + 4x + c\)

d.  \(3x^2 + 2x + c\)

d.  \(\frac{3}{2}x^2 + 2x + c\)

2. \(\int (x + 1)(x + 2) dx\). (Hint: before you start taking a derivative, try a little algebra.)

a.  \((\frac{1}{2}x^2 + x)(\frac{1}{2}x^2 + 2x) + c\)

b.  \((\frac{1}{2}x^2 + 2x)(\frac{1}{2}x^2 + 4x) + c\)

c.  \(2x + 3 + c\)

d.  \(\frac{1}{3}x^3 + \frac{3}{2}x^2 + 2x + c\)

d.  \(\frac{2}{3}x^3 + 3 x^2 + 2x + c\)

3. \(\int 3\sin x - \sec^2 x dx\).

a.  \(3\cos x - \frac{1}{3} sec^3 x + c\)

b.  \(-3\cos x - \frac{1}{3} sec^3 x + c\)

c.  \(3\cos x - \tan x + c\)

d.  \(-3\cos x - \tan x + c\)

d.  \(cos^3 x - \tan x + c\)

4. If \(\int_0^5 f(x) dx = 10\), and \(\int_2^5 f(x) dx = 4\), find \(\int_0^2 f(x) dx\)

a.  -4

b.  0

c.  4

d.  6

e.  8

5. Use geometry to evaluate the following integral \(\int_{-2}^3 \mid x \mid dx\)

a.  \(\frac{15}{2}\)

b.  \(\frac{13}{2}\)

c.  \(\frac{9}{2}\)

d.  \(\frac{1}{2}\)

e.  \(\frac{-1}{2}\)

For questions 6 and 7, say a hot air balloon starts at an altitude of 100 m and is initially falling at a rate of 2 m/s. At time 0, the balloon operator fires the burner and the balloon accelerates upwards at a rate of 1 m/s\(^2\).

6. Find a formula for the vertical velocity, \(v\) of the balloon in m/s.

a.  \(v(t) = 2t + 100\)

b.  \(v(t) = t + 100\)

c.  \(v(t) = t - 2\)

d.  \(v(t) = 2t - 2\)

e.  \(v(t) = 2t - 1\)

7. Find a formula for the height, \(h\) of the balloon in m.

a.  \(h(t) = t^2 + 100t\)

b.  \(h(t) = \frac{1}{2}t^2 + 100t + 100\)

c.  \(h(t) = \frac{1}{2}t^2 + t + 100\)

d.  \(h(t) = \frac{1}{2}t^2 - t + 100\)

e.  \(h(t) = \frac{1}{2}t^2 - 2t + 100\)

8. Which one of the following represents a right hand sum for the function \(f(x) = 2^x\) between \(x = 0\) and \(x = 2\) with \(n = 4\) equal subintervals.

a.  \((\frac{1}{2})(\sqrt{2}) + (\frac{1}{2})(2) + (\frac{1}{2})(2\sqrt{2}) + (\frac{1}{2})(4)\)

b.  \((\frac{1}{4})(\sqrt{2}) + (\frac{1}{4})(2) + (\frac{1}{4})(2\sqrt{2}) + (\frac{1}{4})(4)\)

c.  \((\frac{1}{4})(\sqrt{2}) + (\frac{1}{2})(2) + (\frac{3}{4})(2\sqrt{2}) + (1)(4)\)

d.  \((\frac{1}{2})(\sqrt{2}) + (1)(2) + (\frac{3}{2})(2\sqrt{2}) + (2)(4)\)

e.  \((\frac{1}{2})(\frac{1}{4}) + (\frac{1}{2})(1) + (\frac{1}{2})(\frac{9}{4}) + (\frac{1}{2})(4)\)

9. Consider the area under the curve \(y = f(x) = 1 - x^2\) between \(x = 0\) and \(x = 1\). (It will help you answer this question if you draw the curve first.)

a.  The left-hand-sum with \(n=4\) subintervals is an underestimate of the true area under the curve.

b.  The right-hand-sum with \(n=4\) subintervals is an underestimate of the true area under the curve.

c.  The right-hand-sum with \(n=4\) subintervals is an overestimate of the true area under the curve.

d.  None of the above are true.