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MATH20290 Multivariable Calculus for Engineers
Homework 5: Tangent Planes, Directional Derivatives and Gradient Vectors
1. Write the partial derivatives
G
x
and
G
y
of
G(x, y) = (7 + 2x2 )1/2 + (6 + 2y 3 )1/3 .
2. Write an equation of the plane tangent to
MATH20290 Multivariable Calculus for Engineers
Homework 1: Plane Geometry.
1. Draw vectors v1 = 4, 3 and v2 = 1, 7 on the plane and denote the
angle between them by . Compute |v1 |, |v2 |, v1 v2 and cos . Write
down the size of in both degrees and radians
MATH20290 Multivariable Calculus for Engineers
Homework 7: The Chain Rule, Implicit Functions and Jacobians
1. Use the Implicit Function Theorem to show that the identity
z 2 + xy 3 =
xz
y
denes z as a function of the other variables x and y near the poin
MATH20290 Multivariable Calculus for Engineers
Homework 8: Partial Derivatives of Higher Order and Change of Coordinates
in Dierential Operators
1. Find all rst and second order partial derivatives of
a) f (x, y) = (3x2 + y 2 )1/5
and check explicitly tha
MATH20290 Multivariable Calculus for Engineers
Homework 9: Taylor Expension, Critical Points and Extremum Problems
1. Compute the terms of degree two or less in the Taylor expansion
1
= a0 + b1 (x 2) + b2 (y 1) + c1 (x 2)2
1+xy
+c2 (x 2)(y 1) + c3 (y 1)2
MATH20290 Multivariable Calculus for Engineers
Homework 3: Coordinates Systems in 3-space, Quadric Surfaces and Their
Plane Sections.
1. Compute rectangular coordinates (x, y, z) of points whose cylindrical coordinates (, , z) are
a) (2, 5 , 3);
4
b) ( 1
MATH20290 Multivariable Calculus for Engineers
Homework 4: Limits, Continuity and Partial Derivatives.
1. Evaluate the limit
lim
xy 2 sin(xy)
(x,y)( 1 ,)
6
using the continuity of the respective function of two variables. Explain
why this function is cont
MATH20290 Multivariable Calculus for Engineers
Homework 2: Lines and Planes in 3-space.
1. Write a vector of unit length which is parallel to 4, 12, 3 . How many
vectors satisfy this condition?
2. Compute the angle between the vectors v1 = 1, 0, 0 and v2
MATH20290 Multivariable Calculus for Engineers
Homework 10: Ordinary Dierential Equations and Laplace Transform
1. Write generic solutions to the following linear homogeneous dierential
equations:
d2 y
dy
d2 y
dy
d3 y
2y = 0 b) 3 2
+y =0
2
dx
dx
dx
dx
d
MATH20290 Multivariable Calculus for Engineers
Homework 6: Gradient Lines and Chain Rules for Partial Derivatives
1. The height of a hill over (x, y) is given by
H(x, y) = 100 x2 5y 2 .
Describe the projection to the xy-plane of the path of steepest ascen
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FIRST SEMESTER EXAMINATION 20132014
MATH 20290
Multivariable Calculus for Engineers
Dr. Patrick Murphy
Dr. Masha Vlasenko
Time Allowed: 2 hours
Instructions for Candidates
Full marks will be awarded for complete answers to seven out of eight questions.
Sh