Math 1100
Chapters 12–13 Review
2 December, 2013
Name:
The questions below are intended as a partial review. You should also refer to
your written homework and quizzes (and, to a lesser extent, your WebAssign
work) as you study for the exam.
1. (a) What is an indefinite integral?
(b) What is a definite integral?
(c) How are definite integrals and indefinite integrals related?
2. Evaluate each integral if it exists. For indefinite integrals, remember the constant of
integration.
(a)
(c)
Z
√
1
4x − x + 3 dx
x
Z
5
1
dx
x−1
(d)
(b)
Z
√
3
x 2x2 + 4 dx
Z
5
x4 e(x ) dx
(g)
(e)
Z
2
Z
2
2x − 4 dx
1
−1
(f)
Z
1
3
1
dx
x
∞
1
dx
2x2
3. Find each area shown or described below.
(a) The area enclosed by the curve y = x2 + 1, the x-axis, and the lines x = −1 and
x = 2.
5
4
3
2
1
-4
-3
-2
-1
0
1
2
3
4
5
-1
(b) The area enclosed by y = 4x3 −4x and the x-axis. (Your answer should be a positive
number.)
1
-2
-1
0
1
2
-1
(c) The area enclosed between y = x2 − 2 and y = −x2 − 2x + 2. (Not pictured)
4. Solve each differential equation. If there is more than one solution, give all solutions.
(a)
dy
dx
= 6x + 4
dy
(b) 2y dx
= 4x + 2, passes through the point (4,1).
(c) 2xy +
dy
dx
=y
5. Suppose the
√ marginal revenue for a product is M R = 900 and the marginal cost is
M C = 30 x + 4 with a fixed cost of $1000 (that is, it costs $1000 to produce 0 units).
(a) How many units should be produced to maximize profit?
(b) What is the profit or loss from producing 5 units?