ASSIGNMENT 5 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 2.6, complete questions: 4, 6, 8, 14, 16, 18, 20, 22, 24, 26, 30, 32, 40, 42, 44, 52, 56
From section 2.7, complete questions: 4(a) and 4(b), 6, 8, 10(a) and 10(b), 14, 20, 32, 36
From section 2.8, complete questions: 2, 4, 6, 8, 10, 22, 24, 30, 42, 52, 54
From section 3.1, complete questions: 2, 4, 6, 8, 10, 12, 13, 16, 18, 20, 22, 24, 26, 34, 36, 52, 54
From section 1.5, complete questions: 2, 8, 18
From the Review Exercises (on page 74) for Chapter 1, complete questions: 25, 26
Part B
√
x2 + 2
.
x−1
1. (b) A curve y = f (x) has a slant asymptote at y = mx + b if the graph of the curve gets arbitrarily close
to the graph of the line; that is, if
1. (a) Find the vertical and horizontal asymptotes of the curve y =
lim (f (x) − (mx + b)) = 0 or
x→∞
Find the slant asymptote of the curve y =
lim (f (x) − (mx + b)) = 0.
x→−∞
x2 + 2
.
x
2. Consider the point P = (1, 1) on the curve y = x2 .
2. (a) Find the equation of the tangent line at P .
2. (b) Find another point Q such that the tangent line at P is perpendicular to the tangent line at Q.
3. Let f (x) be the slope of the line from (0, −1) to (x, 0), as on question 5(b) of your midterm test. Find
lim f 0 (x),
x→0
or explain why it does not exist.
4. Let
f (x) = cx3 + 2x2 + 3x + 4.
Find c such that the curve y = f (x) has exactly one horizontal tangent line.
5. A differential equation is an equation relating the value of a function and its derivatives. (For example,
population growth is governed by an equation relating population and population growth: if there are too
many deer in an area, they deplete their resources and reproduce more slowly.) Prove that the differential
equation
f 0 (x) = f (x)
has an infinite number of solutions. (Hint: consider exponential functions.)