Section 12.1
Three-Dimensional Coordinate Systems
Section 12.1
Three-Dimensional Coordinate Systems
Goals:
Plot points in a three-dimensional coordinate system.
Use the distance formula.
Study the graph of an implicit function using the following strategi
Math 221: Linear Algebra
Homework assignment 5
Dr. M. H. Mertens
Exercise 1.
For the following matrices, compute all dened products of two matrices.
1 1 1
2 0 0
A := 2 0 1 R33 , B := 1 1 0 R33
0 0 1
3 0 2
2
0 2 0 0
C := 1 1 1 0 R34 , D := 1 R31
3
0 3 0
Math 221: Linear Algebra
Homework assignment 4
Dr. M. H. Mertens
Exercise 1.
Show that R2 cannot be spanned by one vector.
v1
v2
Solution. We look at the span of an arbitrary vector v =
R2 . There are two cases to
distinguish:
1
R2 is certainly not in v
Section 14.1
Functions of Several Variables
Section 14.1
Functions of Several Variables
Goals:
For functions of several variables be able to:
Convert an implicit function to an explicit function.
Calculate the domain.
Calculate level curves and cross sect
Section 12.1
Three-Dimensional Coordinate Systems
Section 12.1
Three-Dimensional Coordinate Systems
Goals:
Plot points in a three-dimensional coordinate system.
Use the distance formula.
Study the graph of an implicit function using the following strategi
Section 15.1
Double Integrals
Section 15.1
Double Integrals
Goals:
Calculate the volume under a surface over a rectangular domain using
a double integral.
Multi-variable Calculus
1 / 61
Section 15.1
Double Integrals
The Single Variable Integral
We approxi
Exam 1 Practice
Solutions
Question 1
Graph the surface x = z 2 + 1. Find its intersection(s) with the line
r(t) = 5i 7j + (1 + t)k.
Multi-variable Calculus
1 / 14
Exam 1 Practice
Solutions
Question 2
Suppose the vector h4, b, ci is parallel to h10, 6, 2i.
Exam 1 Practice
Solutions
Question 1ab
Solve the following dierential equations
(a) ye
(b)
t dy
dt
+t =0
dy
+ 2y = t 2 e
dt
2t
Dierential Equations
1/9
Exam 1 Practice
Solutions
Question 1cd
Solve the following dierential equations
(c) y 00 + 8y 0 + 16y =
Section 13.1
Name:
Quiz 6 (Selfquiz)
Q1 Sketch the space curve defined by r(t) = h7, 2 cos t, 2 sin ti.
Q2 Sketch the space curve defined by r(t) = ti + cos tj t2 k.
Multivariable Calculus
September 15, 2016
1/ 1
Sections 13.1-13.4
Name:
Quiz 8 Practice
Q1 Show how to sketch the graph of the space curve r(t) = 2 sin ti + cos tj + cos2 tk by
projecting a plane curve onto a surface. Indicate the curve and surface you use.
Q2 Write an equation of the tangent line to
Section 12.5
Name:
Quiz 4 (Selfquiz)
Q1 Give the vector equation of a line through (0, 1, 4) and (3, 1, 8).
Q2 Find the equation of a plane that passes through (1, 2, 5), (2, 6, 2) and (4, 2, 3).
Q3 Where does the line r(t) = h3 + 3t, 2 t, 5i intersect th
Exam 1 Practice
Solutions
Question 1
Graph the surface x = z 2 + 1. Find its intersection(s) with the line
r(t) = 5i 7j + (1 + t)k.
Multi-variable Calculus
1 / 14
Exam 1 Practice
Solutions
Question 2
Suppose the vector h4, b, ci is parallel to h10, 6, 2i.
Syllabus - Math 211 Multivariable Calculus
1 Instructor
Name Mike Carr
Office MSC 431
Email mpcarr@emory.edu (expect a response within one business day)
Office Hours
Tues
Weds Thurs
10-11 12:30-2 10-11
and by appointment
2 Catalog Information
Semester Fal
Chapters 12 and 13
Name:
Exam 1 Practice
General Information
The exam covers Chapters 12 and 13.
It will be held in class, October 6.
You will have 70 minutes to complete the exam.
The honor code is in effect for the exam.
You may not access electron
Sections 12.1-12.3
Chapter 12 Practice Problems, Part 1
The following problems are good practice for the method of sections 12.1 to 12.3.
Turn in solutions (with steps shown) to the boldface problems only for full credit. A random
sample of them will be g
Section 13.2
Name:
Quiz 7 (Selfquiz)
Use r(t) = cos ti + t2 j + sin tk for these questions
Q1 Find an equation of the tangent line to r(t) at t = 0.
Z
Q2 Evaluate
r(t)dt.
0
Q3 Find the unit tangent vector T(t) at t = 3.
Multivariable Calculus
September 20
Section 12.2
Name:
Quiz 1S (Selfquiz)
Q1 If AB = 2i 4k and B = (1, 5, 3) what are the coordinates of A?
Q2 If u = h2, 1, 3i and v = h1, 0, 2i, then what is 2v u?
Q3 Let v = h3, 4, 2i.
a
Calculate |v|
b
Give the coordinates of a vector parallel to |v| whos
Section 12.5
Name:
Quiz 4 (Selfquiz)
Q1 Give the vector equation of a line through (0, 1, 4) and (3, 1, 8).
Q2 Find the equation of a plane that passes through (1, 2, 5), (2, 6, 2) and (4, 2, 3).
Q3 Where does the line r(t) = h3 + 3t, 2 t, 5i intersect th
Sections 13.1-13.4
Name:
Quiz 8 Practice
Q1 Show how to sketch the graph of the space curve r(t) = 2 sin ti + cos tj + cos2 tk by
projecting a plane curve onto a surface. Indicate the curve and surface you use.
Q2 Write an equation of the tangent line to
Section 13.2
Name:
Quiz 7 (Selfquiz)
Use r(t) = cos ti + t2 j + sin tk for these questions
Q1 Find an equation of the tangent line to r(t) at t = 0.
Z
Q2 Evaluate
r(t)dt.
0
Q3 Find the unit tangent vector T(t) at t = 3.
Multivariable Calculus
September 20
Sections 12.4-12.5
Chapter 12 Practice Problems, Part 2
The following problems are good practice for the method of sections 12.4 to 12.5.
Turn in solutions (with steps shown) to the boldface problems only for full credit. A random
sample of them will be g
Sections 12.1-12.3
Name:
Quiz 2 Practice
Q1 If a = h2, 3, 0i and 2a b = h3, 6, 2i then what is b?
Q2 Given the stanard basis representation of the vector that has magnitude 4 and points in
direction of the negative y-axis.
Q3 Sketch the graph of the equat
Section 12.5
Name:
Quiz 5 Practice
Q1 Find the vector equation of a line through (2, 3, 5) and parallel to the y-axis.
Q2 Find the equation the line that is the intersection of the planes 3x + 2y z = 0 and
5x 3y = 0
Q3 Find the number b such that the poin