Fall Term, 2010
Name:
SOLUTIONS
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 201 Multivariable Calculus
Midterm Test November 4, 2010
Instructor: Dr. Y. Chen
Time Allowed: 80 minutes
Total Value: 75 marks
Number of Pages: 7 plus cover page
Ins
MA201 Lab Report 4 Multiple Integrals
Name:
Student Number:
1. [7 marks] Consider the region D =
Lab
Winter 2015
(x, y) | y x
,0 y
.
2
2
(a) Sketch D on the axes provided.
ZZ
(b) Integrating with respect to x first, express
cos(x2 ) dA as an iterated
MA201 Lab Report 6 Notes
Topic Summary (16.6-16.9)
Parametric Surfaces (16.6):
We can describe a surface by a vector function r(u, v) of two parameters u and v.
We can think of dierent values for u and v as corresponding to latitude and longitude of a p
MA201 Lab Report 6 Surface Integrals; Stokes Theorem; The Divergence Theorem
Name:
Student Number:
Lab
Fall 2014
1. [5 marks] Determine the surface area of the surface S given by the vector equation r(u, v) = u sin v, u cos v, u ,
for u [0, 1] and v [0, ]
WILFRID LAURIER UNIVERSITY
WATERLOO, ONTARIO M
Fall Term, 2014 Name:
Course ID: MA201 Student ID:
Course Title: Multivariable Calculus A 73_ 31, 96
WW! 9
Instructor: Dr. McCluskey 5
Number of Pages: 9 plus cover page
Length of Examination: 80 minutes
Exa
MA201 Lab Report 1 Notes
Topic Summary: 12.1-12.4
Vectors:
Consider the vectors a =< x, y, z >= xi + y j + z k and b =< m, n, p >= mi + nj + pk where i =< 1, 0, 0 >,
j =< 0, 1, 0 >, and k =< 0, 0, 1 > . Then:
1.
a + b =< x + m, y + n, z + p >
6.
k(a + b)
MA201 Lab Report 1 Vectors and the Geometry of Space
Name:
Student Number:
Lab
Spring 2015
x1 + x2 y1 + y2 z1 + z2
,
,
, which all lie on
2
2
2
1. [6 marks] Consider the points P (x1 , y1 , z1 ), Q(x2 , y2 , z2 ), and M
the same straight line.
(a) Prove t
MA201 Lab Report 2 Notes
Topic Summary: 12.5, 12.6, 13.1-13.3
Equations of Lines:
Given v =< a, b, c > (direction vector of line) and r0 =< x0 , y0 , z0 > (position vector of point on line):
The parametric equations of the line are given by:
The vector e
MA201 Lab Report 5 Notes
Topic Summary (16.1-16.5)
Line Integrals:
1. With respect to Arc Length:
- Used to evaluate an integral over a curve, say C, instead of over an interval [a, b] (as we are used to).
If f is a continuous function and C is a smooth c
MA201 Lab Report 5 Vector Calculus
Name:
Student Number:
Lab
(x + y 2 )ds where C is the line from
1. [6 marks] Evaluate
C
Parametrizing C:
7
9
, 6,
2
2
Direction vector of line is
x=
7
5
, 1,
2
2
5
7
, 1,
2
2
to
9
7
, 6,
2
2
Fall 2014
.
= 2, 7, 0
5
7
+ 2
MA201 Lab Report 4 Multiple Integrals
Name:
Student Number:
1. [12 marks] Suppose E is the region under the hemisphere z =
and inside the cylinder x2 + y 2 = 1.
Lab
Fall 2014
25 x2 y 2 , above the paraboloid z = x2 +y 2 ,
(a) Open the le Lab4plot.mw (on M
MA201 Lab Report 6 Vector Calculus
Name:
Student Number:
Lab
Winter 2015
3
9
5
9
1. [5 marks] Evaluate
xy z ds where C is the line from
, 1,
to
, 6,
.
2
2
2
2
C
5
9
3
9
Parametrizing C: Direction vector of line is
, 6,
, 1,
= h1, 7, 0i
2
2
2
2
Z
x=
Z
xy
MA201 Lab Report 5 Multiple Integrals
Name:
Student Number:
Lab
Winter 2015
1. [10 marks] Suppose E is the solid region in the first octant thats bounded above by the plane 3x + 2y + z = 9.
(a) Determine the line of intersection between 3x + 2y + z = 9 an
MA201 Lab Report 2 Vectors and The Geometry of Space; Vector Functions;
Functions of Several Variables
Name:
Student Number:
Lab
Winter 2015
Equations of Lines and Planes:
1. [4 marks] Determine the parametric and symmetric equations for the line of inter
MA201 Lab Report 3 Functions of Several Variables
Name:
Student Number:
Lab
Winter 2015
1. [12 marks] In this question, we are going to use the Hessian matrix to determine any relative extrema and
saddle points of f (x, y) = x4 + y 4 4xy + 2. The Hessian
MA201 Lab Report 1 Vectors and the Geometry of Space; Vector Functions
Name:
Student Number:
Lab
Fall 2014
Vectors and the Geometry of Space:
1. [7 marks] Consider the vectors r = 4, 5, 1 , s = 7, 6, 3 , and t = 2, 7, 1 .
(a) Dene the vectors r, s, and t
MA201 Lab Report 2 Notes
Topic Summary (14.1-14.6):
Functions of More Than One Variable:
A function z may be defined in terms of two [z = f (x, y)] or more [z = f (x1, x2, ., xn )] independent variables.
Such a function assigns to each ordered pair, or n-
MA201 Lab Report 2 Functions of Several Variables
Name:
Student Number:
Lab
Fall 2014
2x2 xy
, (x, y) = (0, 0)
4x2 2y 2
1. [8 marks] Consider the function g(x, y) =
1
,
(x, y) = (0, 0)
2
(a) Using Maple, dene g(x, y) as a function and complete the chart
MA201 Lab Report 1 Notes
Topic Summary: 12.1-12.6, 13.1-13.3
Vectors:
Consider the vectors a =< x, y, z >= xi + y j + z k and b =< m, n, p >= mi + nj + pk where i =< 1, 0, 0 >,
j =< 0, 1, 0 >, and k =< 0, 0, 1 > . Then:
1.
a + b =< x + m, y + n, z + p >
MA201 Lab Report 3 Notes
Topic Summary (14.7, 14.8, 15.1-15.3, p. 1005 of 15.5):
Method For Finding Relative Extrema:
Consider z = f (x, y). To nd any relative extrema:
Step 1:
Determine all (a, b) such that both fx (a, b) = 0 and fy (a, b) = 0.
Step 2:
F
MA201 Lab Report 4 Notes
Topic Summary (15.4, 15.7-15.9)
Double Integrals in Polar Coordinates:
- In general, given f (x, y) in cartesian coordinates, the corresponding function in polar coordinates is determined
using the substitutions: x = r cos , y = r
MA201 Lab Report 2 Vectors and The Geometry of Space; Vector Functions
Name:
Student Number:
Lab
Spring 2015
1. [4 marks] Determine the angle (in radians) at which each of the following pairs of planes intersect (if at all).
Be sure to justify your answer
MA201 Lab Report 3 Notes
Topic Summary: 14.1 - 14.8
Functions of More Than One Variable:
A function z may be defined in terms of two [z = f (x, y)] or more [z = f (x1, x2, ., xn )] independent variables.
Such a function assigns to each ordered pair, or n-
MATHEMATICS 201 A - Multivariable Calculus
Spring Term, 2015
Instructor: Dr. A. Allison
Oce: BA303E (Bricker Academic)
Oce Hours: to be announced
Email : [email protected]
Include the course number in the subject or it might not be read.
Lectures: Mon, Wed
MA201 Midterm Test Information
Date: Wednesday, June 24
Time: 2:303:50pm
Location: DAWB 2-104 2-101
ONLY the Casio FX-300MS Plus calculator will be allowed during the test.
(This stipulation is stated in the course outline.)
You are responsible for:
All