ASSIGNMENT 6 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 3.1, complete questions: 68, 72
From section 3.2, complete questions: 2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 42, 44, 48
Part B
1. Before proving the Product Rule as we know it, Gottfried Leibniz reportedly assumed that
d
d
d
(f (x)g(x)) =
f (x)
g(x) .
dx
dx
dx
Find an example of functions f and g such that this equation holds.
2. Let f1 , f2 , f3 and f4 be differentiable functions.
d
(f1 (x)f2 (x)f3 (x)).
dx
d
2. (b) Find
(f1 (x)f2 (x)f3 (x)f4 (x)).
dx
2. (a) Find
3. Let f (x) = enx , where n is a positive integer. Find f 0 (x). (Hint: use the Product Rule.)
4. Let f (x) =
x2
. Solve f 00 (x) = 0.
e2x
5. It may be shown that the infinite sum
1+
1
1
1
+ 2 + 3 + ···
2 2
2
is equal to 2; and that, in general, for c ∈ (−1, 1),
1 + c + c2 + c3 + · · · =
1
.
1−c
Using this fact, prove that for c ∈ (−1, 1),
c + 2c2 + 3c3 + 4c4 + · · · =
c
(1 − c)
2.