MAT232H
Assignment 2 - Summer 2022
Due: July 10, 2022 at 11:59pm via CrowdMark.
Unless otherwise stated, you must show your work.
Question 1. Quadric surfaces:
(a) Sketch the cross-sections of the surface x2 + 3y2 = 1 + z2 parallel to the xy-plane, the xz-plane, and the
yz-plane. Identify the shapes of these cross-sections.
(b) The parametric curve r(t) = ⟨1 + cos(t), sin(t), 2 sin( 12 t)⟩ parametrizes the intersection of two quadric surfaces.
Identify the two quadric surfaces by equation (and by name) and explain how you know that your answer is
correct.
Question 2. Computing partial derivatives:
(a) Let f(x, y) be a function of two variables such that fx(5, 3) = 4 and fy(5, 3) = −1. Let g(x, y) = f(2x+3, 3y2).
Find gx(1, 1) and gy(1, 1).
(b) Consider the function
f(x, y) =
∫ sin y
1+x2
0
e5x
2t2 dt.
Evaluate fx(1, π) and fy(1, π).
Question 3. Finding tangent planes through certain “anchors” and certain directions:
(a) Find all planes which (i) are tangent to the elliptic paraboloid z = x2 + y2, and (ii) pass through both points
P = (0, 0,−1) and Q = (2, 0, 3). How many such planes are there?
(b) Find all planes which (i) are tangent to the surface z = x+xy2− y3, (ii) are parallel to the vector v⃗ = ⟨3, 1, 1⟩,
and (iii) pass through the point P = (−1,−2, 3). How many such planes are there?
(c) Find all planes which (i) are tangent to the surface z = x2 + sin y, (ii) are parallel to the x-axis, and (iii) pass
through the point P = (0, 0,−5). How many such planes are there?
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