MTH 201: MULTIVARIABLE CALCULUS
HOMEWORK 1 (90 POINTS)
DUE ON SEPTEMBER 25, 2021 BY 11:59 PM
Instructions:
• There are six questions and a bonus problem. All problems carry 15 points.
• Start solution to each problem on a fresh page. You are required to upload solution to each
question in a separate file.
• Write your roll number on each file.
• Give proper justification to receive full credit.
• You can use class notes/material. No help from any other person or internet sources is allowed.
If you are found to be in violation of these instructions, 0 points will be given on this exam.
Questions:
(1) a) Give parametric equation (point-direction form) of the line which lies on both of the planes:
x + y + z = 1 and −x + 2y + 10z = 2. What is the direction d of this line?
b) Let n1 and n2 be the normal vectors to the two given planes. Without actual computation,
describe the relationship between d and n1 × n2 .
(2) Let P QR be a triangle with coordinates P = (x1 , y1 , z1 ), Q = (x2 , y2 , z2 ) and R = (x3 , y3 , z3 ).
a) Prove that the triangle P QR lies on a plane. Find the equation of this plane.
b) Let (x, y, z) be a point in the plane containing the triangle. Prove that
x − x1 y − y1 z − z1
x − x2 y − y2 z − z2 = 0.
x − x3 y − y3 z − z3
(3) Consider the function
 p

 x x2 + y 2
, x 6= 0
f (x, y) =
|x|

0,
x = 0.
a) Is f continuous at (0, 0)?
b) Find all the directions in which the directional derivative of f exist at the point (0, 0).
c) Is f differentiable at (0, 0)?
(4) Let f (x, y, z) be a differentiable function. At the point (1, 1, 2), the directional derivative is 4,
3, 2 in the direction i + j, j + k and i + k, respectively.
a) Find the directional derivative at the point (1, 1, 2) in the direction 3i + 3j + 3k.
b) Compute ∇f (1, 1, 2).
c) In which direction does the function f increases most rapidly? In which direction does the
function f decreases most rapidly?
(5) a) Find the linear and quadratic approximations of
sin(0.96π) tan(0.26π) + (0.96)2 (0.26)
b) Compare these approximations with the actual value.
(6) Can the surface x5 y 3 + x2 y + z = 3 be expressed as the graph of a function x = f (y, z) near
the point (2, 1/2, −3)? If yes, compute fy (1/2, −3) and fz (1/2, −3).
Date: September 18, 2021.
1
2
HW 1
(Bonus problem)
Let D = {(x, y) ∈ R2 : x > 0, 0 < y < x3 }. Define
(
0, (x, y) ∈
/D
f (x, y) =
1, (x, y) ∈ D.
a) Approaching (0, 0) along the line y = mx for each real number m and the y-axis, prove that
lim(x,y)→(0,0) f (x, y) exists and compute the limit.
b) Argue whether f is continuous at (0, 0).
Hint: Drawing a picture will help!