ASSIGNMENT 8 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 3.4, complete questions: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 26, 28, 30, 32, 34, 36, 38, 40,
42, 46, 48, 50, 52, 54, 62, 66, 68, 74, 76
Part B
1. Find constants c and k such that the function f (x) = ecx + k satisfies the differential equation
3f 00 (x) + 13f 0 (x) = 10f (x).
2. (a) The Cissoid of Diocles is a classical curve given by the equation
y2 =
x3
.
2−x
Find the slope of the tangent line at the point (1, 1).
2. (b) Consider the point (3, −4) on the circle centred at the origin. Find the equation of the tangent line
at this point.
3. Let P and Q be points on the plane with coordinates (0, 5) and (12, 0), respectively. Suppose P is moving
toward the origin with velocity 5, and Q is moving away from the origin with velocity 12. Determine the
rate at which the distance between P and Q is changing.
4. Find a function f such that, where f is differentiable,
f 0 (x)
= 1 and f (0) = 1.
f (x)2
(Hint: express both sides of the first equation as derivatives.)
5. Recall that a function f is even if it satisfies f (x) = f (−x) for all x, and odd if it satisfies −f (x) = f (−x)
for all x. Let f be even and differentiable. Prove that its derivative is odd.