First name:   Last name:

1. 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

2. 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.  $\frac{-1}{2}$

3. If $\int_0^5 f(x) dx = 10$, and $\int_0^5 g(x) dx = 4$, find $\int_0^5 f(x)+2g(x) dx$

a.  10

b.  14

c.  18

d.  22

e.  There is no answer

For questions 4 and 5, 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$^2$.

4. 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.  $v(t) = 2t - 1$

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

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

7. 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.