ASSIGNMENT 4
There are two parts to this assignment. The first part consists of 15 questions on WeBWorK — the link is
available on the course webpage. The second part consists of the questions on this page. You are expected
to provide full solutions with complete justifications. You will be graded on the mathematical, logical and
grammatical coherence and elegance of your solutions. Your solutions must be typed, with your name and
student number at the top of the first page. If your solutions are on multiple pages, the pages must be stapled
together.
Your written assignment must be handed in before the start of your recitation on Friday, February
6. The online assignment will close at 9:00 a.m. on Friday, February 6.
Z
1. Evaluate
0
1
t2
2t + 5
dt.
+ 5t + 6
Z
r
Z
2. Let f (x) be continuous everywhere. Prove that
0
0
s
Z r
f (t) dt ds =
(r − u)f (u) du.
0
3. Let f (t) be a function such that f (t), f 0 (t), f 00 (t), . . . , f (n+1) (t) are continuous on an interval containing
a. Then for any x in that interval, Taylor’s Theorem (with integral remainder) states that
Z
f 0 (a)
f 00 (a)
1 x (n+1)
2
f (x) = f (a) +
(x − a) +
(x − a) + · · · +
f
(t)(x − t)n dt.
(1)
1!
2!
n! a
(Recall that we define 0! = 1, 1! = 1, 2! = (2)(1), 3! = (3)(2)(1), and so on.) In this question, you will
prove and apply this theorem.
(a) Explain why equation (1) is true in the case n = 0; that is, explain why
Z
1 x (0+1)
f (x) = f (a) +
f
(t)(x − t)0 dt.
0! a
(b) Suppose equation (1) is true in the case n = k. Prove that it is true in the case n = k + 1. (Hint:
integrate the last term using the technique of parts.)
(c) Explain in one or two sentences why parts (a) and (b) imply that Taylor’s Theorem is true.
(d) Prove that, for any x, sin(x) = x −
x3
x5
x7
|x|9
+
−
+ c for some constant |c| ≤
.
3!
5!
7!
9!