Section 12.2
12.2.1: Because f (x, y ) = 4 3x 2y is dened for all x and y , the domain of f is the entire two-dimensional plane. 12.2.2: Because x2 + 2y 2 x2 + 2y 2 is the entire two-
0 for all x and y , the domain of f (x, y ) =
dimensional plane. 12.2.3
Section 13.1
13.1.1: Part (a): f (1, 2) 1 + f (2, 2) 1 + f (1, 1) 1 + f (2, 1) 1 + f (1, 0) 1 + f (2, 0) 1 = 198. Part (b): f (2, 1) 1 + f (3, 1) 1 + f (2, 0) 1 + f (3, 0) 1 + f (2, 1) 1 + f (3, 1) 1 = 480. The average of the two answers is 335, fairly cl
Math 223
Test 3 - Review 1
cos y
x sin y dx dy
1.
0
0
1x
2
(4 y) dy dx
2.
1
0
(x2 + 4y) dA where R is the region bounded by the graphs of
3. Evaluate the integral
R
y = 2x and y = x2 .
1
2
4.
2x
0
1
1
5.
0
2
ey dy dx.
cos y 2 dy dx.
x
6. Use a double inte
17.1 Greens Theorem
Remember: the circulation of a conservative vector field around every closed path is zero.
Consider a domain whose boundary is a simple closed curve 1. We follow standard usage
and denote the boundary curve C by D
1 A closed curve that
17.3 Divergence Theorem
Recall:
integral
oriented
the
boundary of the domain
The Fundamental Theorem of Calculus
relates the integral of f(x) over an
interval [a, b] to the integral of f(x) over
the boundary of [a, b] consisting of two
integral
over the
p
17.2 Stokess Theorem
Stokess Theorem expands upon Greens Theorem to three dimensions in which circulation is
related to a surface integral in R3 (rather than to a double integral in the plane).
Before stating the theorem, new definitions and terminology i
15.5 Applications of Multiple Integrals
Density. Consider quantities that are distributed with a given density, , in two- or threedimensional space, such as mass, charge, and population. Remember that the total amount is
defined as the integral of density
Exam 2 Notes
09/22/2014
09/22/2014
09/22/2014
09/22/2014
09/22/2014
Chapter 14: Differentiation in Several Variables
14.1 Functions of Two or More Variables. A function of n variables is a function, f(x_1, x_2,
x_n) that assigns a real number to each n-t
MATH 223
(17)
(18)
(19)
(20)
Final Exam Review Problems (16.1)
F= 1,1,1
F= x , 0, z
F= x , y , z
F=e r
(22) Prove that
F= yz , xz , y is not conservative.
(25) Find a potential function for the vector field
F= y z , x z , 2 xyz
2
2
F by inspection.
MA
17.2 Stokess Theorem
Stokess Theorem expands upon Greens Theorem to three dimensions in which circulation is
related to a surface integral in R3 (rather than to a double integral in the plane).
Before stating the theorem, new definitions and terminology i
17.1 Greens Theorem
Remember: the circulation of a conservative vector field around every closed path is
Consider a domain whose boundary is a simple closed curve1. We
follow standard usage and denote the boundary curve C by
D . The counterclockwise orie
Analyzing Multivariable Change:
Optimization
Chapter 8.1
Extreme Points and Saddle Points
8.1- Extreme Points and Saddle Points
The optimization techniques for functions with a
single input variable readily generalize to
multivariable functions.
In the
14
PARTIAL DERIVATIVES
PARTIAL DERIVATIVES
14.6
Directional Derivatives
and the Gradient Vector
In this section, we will learn how to find:
The rate of changes of a function of
two or more variables in any direction.
INTRODUCTION
This weather map shows a
Multivariate Calculus Final Exam Review Checklist
Section: Title:
Content:
Types of Problems:
13.1 Vectors in the Plane Vector equivalence
Parallelogram Law
Vector properties
Linear combinations
Unit vectors
Standard basis vectors
Triangle Ineq
Math 122 Test 1
June 20, 2011
Name
1
2
3
4
5
6
7
8
Total
Directions:
1. No books, notes, or belly-opping before your rst swim meet. You
may use a calculator to do routine arithmetic computations. You may
not use your calculator to store notes or formulas.
Math 223
Test 1 - Review 1
1. For a = 2, 5, 4 and b = 1. 2, 3 nd (a) 2a + b, (b) a b, (c) |a b|, and (d) a
|a|
2. For a = 3, 2, 0 and b = 0, 3, 2 , nd a b.
3. Find two dierent unit vectors both perpendicular to a = 1, 2, 3 and b = 2, 3, 5
4. Find the area
Math 223
Test 2 - Review 1
Describe the domain of:
1. f (x, y) = 4 x2 y 2
2. f (x, y) = arcsin(x + y)
+
3. z = x xy y
Describe the level curves of the function:
c = 1, 0, 2, 4
4. z = x + y,
5. f (x, y) =
25 x2 y 2 ,
c = 0, 1, 2, 3, 4, 5
Find the limit:
x+
Math 223
Final - Review
1. Let F = x3 ln z, xey , (y 2 + 2z) , nd div F at (2, ln 2, 1)
2. Let F = cos x, sin y, z , nd the divergence of F at ( , , 1)
2
3. Let F = cos x, sin y, exy , nd the curl of F at (1, 1, 1)
4. Let F = x3 ln z, xey , (y 2 + 2z) , n
MATH 223 HOMEWORK 2
NAME:
(1) Planar points A = (1, 1), B = (2, 3), C = (1, 4), D = (3, 0) determine displace
ment vectors AB and CD. Are these two vectors parallel? If so, do they point in
the same direction or the opposite direction?
(2) Decide if the t
MATH 223 HOMEWORK 6
NAME:
1. Label each graph A)-F) according to its equation:
A)
x2
9
+
y2
16
+
z2
9
= 1,
B) 15x2 4y 2 + 15z 2 = 4,
D) y 2 = 4x2 + 9z 2 , E) x2 y + z 2 = 0,
C) 4x2 y 2 + 4z 2 = 4
F) 4x2 y 2 + 4z = 0
2. In parts a), and b), sketch the cyli
Is Algebra Necessary? - NYTimes.com
1 of 5
http:/www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.
July 28, 2012
Is Algebra Necessary?
By ANDREW HACKER
A TYPICAL American school day finds some six million high school students and two million
MATH 223 HOMEWORK 8
NAME:
1. Write an equation in rectangular coordinates x, y for the curve r(t) = et i + et j,
< t < . Identify this well-known planar curve and sketch.
2. Let C be the curve of intersection of the quadric surfaces x2 + y 2 = 1 and z =
MATH 223 Fall 2014
14.6 The Chain Rule Problem Types
Exam #2 Review
The Chain Rule of Paths can be extended to general composite functions
o If
x , y ,z are differentiable functions of st such that
then their composite,
x=x ( s , t ) , y= y ( s , t ) ,z
MATH 223 Fall 2014
Exam #2 Review
14.5 The Gradient and Directional Derivatives Problem Types
I. The gradient is the vector whose components are the partial derivatives of a multivariable function, f, at a
point P.
Definition. The gradient of a function,
Math 223 Test 1
June 17, 2013
Name
1
2
3
4
5
6
7
8
Total
Directions:
1. No books, notes, or trying to catch a foul ball while holding a baby.
You may use a calculator to do routine arithmetic computations. You
may not use your calculator to store notes or
Math 223 Test 1
June 18, 2012
Name
1
2
3
4
5
6
7
8
9
Total
Directions:
1. No books, notes, or over 1400 incoming freshmen. You may use a calculator to do routine arithmetic computations. You may not use your
calculator to store notes or formulas. You may
Math 223 Test 1
June 15, 2015
Name
1
2
3
4
5
6
7
8
Total
Directions:
1. No books, notes, or swim meets in thunder storms. You may use a calculator to do routine arithmetic computations. You may not use your
calculator to store notes or formulas. You may n
Math 223 Test 1
June 20, 2016
Name
1
2
3
4
5
6
7
8
Total
Directions:
1. No books, notes, or beating the Cavs. You may use a calculator to do
routine arithmetic computations. You may not use your calculator to
store notes or formulas. You may not share a c