ASSIGNMENT 18 for SECTION 001
This assignment is to be handed in. There are two parts: Part A and Part B.
Part A will be graded for completeness. You will receive full marks only if every question has been completed.
Part B will be graded for correctness. You will receive full marks on a question only if your answer is correct
and your reasoning is clear. In both parts, you must show your work.
Please submit Part A and Part B separately, with your name on each part.
Part A
From Calculus: Early Transcendentals:
From section 4.4, complete questions: 6, 8, 10, 12, 16, 18, 20, 22, 40, 46, 48, 50, 52, 60, 64, 70
From section 4.9, complete questions: 2, 4, 6, 8, 10, 12, 24, 26, 28, 30, 32, 34, 50
From section 5.3, complete questions: 20, 22
Part B
1. Evaluate the following limits:
f (x + h) − 2f (x) + f (x − h)
, where f is twice-differentiable
h2
n
1
(b) lim 1 +
n→∞
n
(a) lim
h→0
2. Let
f (x) = xm a0 + a1 x + a2 x2 + · · ·
and g(x) = xn b0 + b1 x + b2 x2 + · · ·
where m and n are positive integers and neither a0 nor b0 is equal to zero. Prove that
lim f (x)g(x) = 1.
x→0+
(This proves that all smooth, nonzero functions f and g with f (0) = g(0) = 0 satisfy the given limit.)
3. Find an antiderivative of each of the following functions:
(a) f (x) =
1
(2x + 1)2
(b) g(x) = √
(c) h(x) = √
4
x+1
2x
x2 + 1
4. Find the area under the curve y = sin x on the interval [0, π].
5. It is a well-known fact that calculus is better than ice cream (in fact, better than anything else that you’ve
tried). Explain why.