16 June 2011
1. (6 points) You do not need to simplify your answer for these problemsjust make sure
you have fully computed all derivatives. For example, leaving your answer in the form
(x2 1)(x2 + 2x) is acceptable, but (x2 1) dx [x
24 May 2011
1. (3 points) Solve the following inequalities. Write your answers in interval notation.
(a) |3x + 4| 3
Solution: We could rewrite the inequality as 3 3x + 4 3, and then
subtract 4 from all parts of the inequality to attain
23 June 2011
= 1, show that lim
1. (2 points) Using the equality lim
Solution: We can rewrite
cos2 (x) 1
cos(x) 1 cos(x) + 1
cos(x) + 1
x(cos(x) + 1)
Homework 1 Solutions
2. Solve the following inequalities. Write your answer in interval notation and draw it on
the number line.
(a) (1/2, )
(c) (0, 1)
3. Draw the following curves in the xy-plane.
(a) This is a sideways parabola, opening up to the right,
9 June 2011
1. (4 points) Solve (1 x)(x2 1) < 0. Give your answer in interval notation.
Solution: The zeros of (1 x)(x2 1) occur at x = 1 and 1. Since f (x) =
(1 x)(x2 1) is a continuous function, the intermediate value theorem tells u
21 July 2011
1. (a) (2 points) Find the sum of the areas of ve rectangles, whose bases are equally
spaced subintervals of [1, 4], and whose heights are the values of f (x) = x3 x + 1
at the left endpoint of each subinterval.
1. (3 points) Let f : R \ cfw_1 RAN GE be given by f (x) =
2 June 2011
(a) State what it means for f to be one-to-one.
Solution: f is one-to-one if whenever f (x) = f (y), we must have x = y.
Equivalently, f is one-to-one if th
1. (2 points) Compute
14 July 2011
3x4 + csc2 (x) dx.
3x4 + csc2 (x) dx = x5 + x2/3 + cot(x) + C.
2. (2 points) Compute
x 1 2x dx.
Solution: Use the u-substitution: u = 12x, so du = 2 dx. Replace x by (1u)
30 June 2011
1. Compute the following. [You may use any standard identities without proof, but anybody caught evaluating a limit by plugging in 0 or will be punished.]
(a) (2 points) lim x csc(2x)
x csc(2x) =
7 July 2011
1. Compute the following integrals using the fundamental theorem of calculus and, if you
need to, u-substitution. [If you use u-substitution, remember to write the correct bounds
for your integrals!]
(a) (2 points)