ASSIGNMENT 10 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 1.6, complete questions: 10, 12, 18, 22, 24, 26, 34, 36, 38, 46, 48, 50
From section 3.6, complete questions: 2, 4, 8, 10, 12, 14, 16, 20, 24, 26, 28, 30, 32, 34, 38, 40, 42, 44, 46
Part B
√
d −1
f (x) at x = −2.
dx
√
d −1
1. (b) Let f (x) = x x2 + 3. Find the value of
f (x) at x = −2.
dx
1. (a) Let f (x) =
x2 + 3. Find the value of
2. Let a > 0 be a constant. Solve x
xx
..
.
= a. (The exponent tower is infinite.)
3. Let a > 1 be a constant. Find the equation of the line tangent to the curve y = ax which passes through
the origin.
4. The function f (x) = xx is defined for x > 0. On that domain, find the coordinates of the point where the
graph of y = f (x) has a horizontal tangent line.
5. An alternate definition for Euler’s number e is the limit of the sequence
( )
n ∞
1
2
3
4
1
2
3
4
5
1+
=
,
,
,
, . . . = {2, 2.25, 2.37, 2.44, . . .} .
n
1
2
3
4
n=1
Prove the general version of this identity. That is, prove that, for any x,
x n
= ex .
lim 1 +
n→∞
n
(Hint: take the natural logarithm of both sides; then let h = x/n, and look for the limit definition of a
certain derivative — or adapt the proof given in the last part of section 3.6.)