MATH 207 Spring 2011
Total possible is 50
Dr. Smith
Assignment 1
1. a = < 2, 1, 1> and b = <3, 6, 3>
a. Find a b
< 2, 1, 1> <3, 6, 3> = 6 + 6 3 = 9
b. Find the projection of b onto a
proj a b = [a b / |a|2] a = (9 / [22 + 12 + (1)2]) < 2, 1, 1> = (3/2) <
MATH 207 Spring 2011 Total possible is 50
Dr. Smith
Assignment 1
1. a = < 2, 1, 1> and b = <3, 6, 3> a. Find a b < 2, 1, 1> <3, 6, 3> = 6 + 6 3 = 9 b. Find the scalar and vector projections of b onto a compab = a b / |a| = 9 / [22 + 12 + (1)2] = 9 / 6 pro
MATH 207
Total is 50
Fall 2015
Dr. Smith
Assignment 3
1. Find and sketch the domain of the function
a) f(x,y) = y +
f(x,y) is defined only when y 0 and when 0 x2 + y2 25. Hence the domain is cfw_(x,y) | 0 x2
+ y2 25, and its sketch is the upper half of a
MATH 207
Winter 2015 Dr. Smith
Assignment 3
Total possible is 50
1. Find and sketch the domain of the function
a) f(x,y) = y + 25 x 2 y 2
f(x,y) is defined only when y 0 and when 0 x2 + y2 25. Hence the domain is the upper half
of a circle of radius 5 cen
MATH 207
Spring 2015
Dr. Smith
Assignment 5
1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to
the given constraint
a. f(x,y) = exy, x3 + y3 = 16
b. f(x,y,z) = x2y2z2, x2 + y2 + z2 = 1
2. Find the extreme values
MATH 207
Spring 2015
Dr. Smith
Assignment 6
1a) Electric charge is distributed over the disk x2 + y2 4 so that the charge density at (x,y) is
(x,y) = x + y + x2 + y2 (measured in coulombs per square meter). Find the total charge on the
disk.
b) Find the t
MATH 207 Spring 2015
50 points total
Dr. Smith
Assignment 4
1. Find the linear approximation of the function f(x,y) = ln(x 3y) at (7,2) and use it to
approximate f(6.9, 2.06).
3
3
1
1
|x , y = 7 , 2 =
= 3
f x (7,2) =
= 1 and f y (7,2) =
|x , y = 7, 2 =
x
MATH 207 Spring 2015
Total possible is 50
Dr. Smith
Assignment 8
1a) Use Greens Theorem to evaluate the line integral
along the ellipse
x2 + xy + y2 = 1
b) Use Greens Theorem to evaluate C F dr, where F(x,y) = <y ln(x2 + y2), 2 arctan(y/x)>
and C is the c
MATH 207 Spring 2015 Dr. Smith
Assignment 9
Total possible is 55
Note: if you havent done so already, please pick up from me the handout that covers the Fourier
Series information.
1. (Handout 10.2) Find the Fourier series of the function f(x), which is a
MATH 207 Spring 2015
Total possible is 50
Dr. Smith
Assignment 7
1. Sketch the vector field F.
a) F(x,y) = i + xj
b) F(x,y) =
Six points for a: 3 for directions, 3 for magnitudes of vectors.
2. Evaluate the line integral
a) C sin x dx + cos y dy, C is the
MATH 207
Spring 2015
Dr. Smith
Assignment 8
1.
a) Use Greens Theorem to evaluate the line integral sin y dx + x cos y dy along the ellipse
C
x2 + xy + y2 = 1
b) Use Greens Theorem to evaluate C F dr, where F(x,y) = <y ln(x2 + y2), 2 arctan(y/x)>
and C is
University of Waterloo
Final Examination
Winter 2015
t fr 1'
i/ . /{ , a.»
Name - Student ID #
Math 207
Multivariate Calculus (Non—Specialist)
Section 001
Dr. Paula T. Smith
April 24, 2015
4:00 pm —‘6:30 pm
2.5 hours Closed book
This test contains 10 pag
(ff —~ pm
Name W / C a Student ID#
Math 207 Multivariate Calculus (Non—Specialist)
Midterm Test Winter 2015
February 11, 2015
9:30 pm w. 10:20 pm
This test contains 7 pages, including this cover page and a page at the end for rough work.
Write your name a
University of Waterloo
Final Examination
Winter 2015
Name _
Student ID # _
Math 207
Multivariate Calculus (Non-Specialist)
Section 001
Dr. Paula T. Smith
April 24, 2015
4:00 pm 6:30 pm
2.5 hours Closed book
This test contains 10 pages, including this cove
MATH 207 - Problem Set 9
NOT TO BE SUBMITTED
1. Classify each of the following quadric surfaces by nding the eigenvalues of the corresponding matrix.
(a) z = 2x2 xy + 2y 2
(b) z = 4xy + 3y 2
(c) z = x2 + 2xy + y 2
2. Find and classify the critical points
MATH 207 - Problem Set 1
NOT TO BE SUBMITTED
1. (a) Find the equation of the line through the point x0 = (1, 0, 1) in the direction of
the vector v = (3, 5, 4).
(b) Find the equation of the line through the points x1 = (7, 3, 5) and x2 = (2, 4, 1).
(c) Fi
MATH 207 - Problem Set 11
NOT TO BE SUBMITTED
1. Evaluate the following integrals by making an appropriate change of variables. Sketch
the region of integration (in cartesian coordinates).
(a)
dV
x2 +y 2 +z 2
3
, is the region between the spheres of radiu
MATH 207 - Problem Set 10
NOT TO BE SUBMITTED
1. Evaluate the following double integrals. Sketch the region of integration.
x cos y dA , is bounded by x 0, y = x2 , y = x
(a)
(x2 + 2y) dA, is bounded by y = x, y = x3 , x 0
(b)
(c)
2xy dA, is the triangula
Math 207 Dr. Smith Lecture 0
I. Introduction
A. Course Outline
1. Contact ptsmith@uwaterloo.ca
2. TA is Lorena Cid,
lcidmont@uwaterloo.ca
B. Learn
1. Still contact ptsmith@uwaterloo.ca
C. Tutorials 3:30-4:20 Wednesdays in MC
4058; no tutorial this week
D.
MATH 207 Multivariate Calculus (Non-Specialist)
Spring 2015
Lectures: 8:30-9:20 MWF in MC 4042
Tutorials: 3:30-4:20 Wednesdays in MC 4058
Instructor: Dr. Paula Smith
Email: ptsmith@uwaterloo.ca
Phone: 519-888-4567 Ext: 35536
Office: MC 4014
Office hours:
Math 207 final exam syllabus Spring 2015 Paula Smith
1. Vectors, vector operations, equations of planes, lines, cylinders, and quadric surfaces
2. Vector functions, derivatives, integrals
3. Multivariate functions, domain, limits, continuity.
4. Partial d
Math 207 Midterm Syllabus
Dot product
Cross product
Equations of lines in 3-space
Equations of planes
Vector functions
Domain and range
Limits
Derivatives
Integrals
Graphs
Functions of several variables:
Domain and range
Limits
Continuity
Partial
MATH 207 Spring 2015 Dr. Smith
Assignment 9
Note: if you havent done so already, please pick up from me the handout that covers the Fourier
Series information.
1. (Handout 10.2) Find the Fourier series of the function f(x), which is assumed to have period
MATH 207
Spring 2015
Dr. Smith
Assignment 7
1. Sketch the vector field F.
a) F(x,y) = i + xj
j
yi x
b) F(x,y) =
x2 + y 2
2. Evaluate the line integral
a) C sin x dx + cos y dy, C is the top half of the circle x2 + y2 = 1 from (1,0) to ( 1,0) and then
the
MATH 207 Fall 2015
Total possible is 50
Dr. Smith
Assignment 6
1. a. Estimate the volume of the solid that lies below the surface z = x + 2y2 and above the
rectangle R = [0,2] [0,4]. Use a Riemann sum with m = n = 2 and choose the sample points to
be lowe
MATH 207 Fall 2015
Total possible is 50.
Dr. Smith
Assignment 7
1. Evaluate the triple integral
a.
b. E yz cos (x5) dV, where E = cfw_(x,y,z)| 0 x 1, 0 y x, x z 2x
c. E xyz dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (1,1,0), and
MATH 207
Fall 2015
Dr. Smith
Assignment 8
1. Sketch the vector field F(x,y) = (x y)i + xj
Two points.
2. See the exercises at the end of section 13.1.
a. Match the vector fields with the plots labeled I IV, and give reasons; 11 14
11 IV (zero at origin, v
MATH 207
Total is 50
1. Evaluate
Fall 2015
Dr. Smith
Assignment 9
where C is the rectangle with vertices (0,0), (3,0), (3,1), and (0,1)
both directly and also by using Greens Theorem.
Directly: Let C 1 be the line segment from (0,0) to (3,0), C 2 the line
MATH 207 Fall 2015
Dr. Smith
Assignment 10
Note: if you havent done so already, please pick up from me the handout that covers the Fourier
Series information.
Total is 50 points (plus 5 points extra credit).
1. (Handout 10.2) Find the Fourier series of th
Page 1 of8
Name Student ID #
M
Math 207 Multivariate Calculus (NonSpecialist)
Midterm Test Spring 2015
June 10, 201 5
3:30 pm —— 4:20 pm
This test contains 6 pages, including this cover page and a page at the end for rough work.
Write your name and stud