ASSIGNMENT 9 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 3.5, complete questions: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32(a) and 32(b),
34, 36, 40, 60, 62
Part B
1. Consider the curve x2 + y 2
3
= 8x2 y 2 . Find the equation of the tangent line at the point (−1, 1).
2. Let P be a point on a curve, and l be the tangent line at P . The normal line at P is the line perpendicular
to l and passing through P . Consider the curve x2 + (y − x)3 = 9. Find the equation of the normal line
at x = 1.
y
3. The curve
x2 − xy + y 2 = 3
describes a tilted ellipse, pictured to the right. Find the area of the
smallest box (with sides parallel to the axes) containing this ellipse.
x
4. A (non-tilted) ellipse centred on the origin is described by the equation
x2
y2
+ 2 = 1.
2
a
b
(2a and 2b are the “width” and “height” of the ellipse.) Prove that if the normal line at every point on
the ellipse passes through the origin, then the ellipse is a circle.
5. Describe, in the form of a haiku, how calculus has changed your life.