Choi MAT 267 ONLINE A Fall 2015
Assignment Test 1 due 09/03/2015 at 11:15am MST
Problem 4. 4. (1 pt) Find the following expressions using
the graph below of vectors u, v, and w.
Problem 1. 1. (1 pt)
Find the equation of the sphere if one of its diameters
MAT 267 Written Homework #1
10.1, 10.2, 10.3, 10.4
SOLUTIONS
Due: September 1
Solve the following problems, showing any necessary work.
1. [1 point] Find an equation for the sphere which contains the points (3, 1, 4) and (1, 3, 2), and whose
center is the
C. HECKMAN
SOLUTIONS
(1) Let f (x, y ) =
267
TEST 2A
x + 2y
.
5 + xy
(a) [10 points] Find the linear approximation of f (x, y ) at the point (4, 1).
Solution: First, calculate the partial derivatives at (4, 1):
fx (4, 1) =
fy (4, 1) =
(5 + xy )(1) (x + 2y
Jennifer Wong
Shotwell MAT 267 Fall 2008 79670
WeBWorK assignment number Section 10.1 is due : 08/31/2008 at 11:59pm MST. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making so
Jennifer Wong
Shotwell MAT 267 Fall 2008 79670
WeBWorK assignment number 10.7a is due : 09/14/2008 at 11:59pm MST. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind
Jennifer Wong
Shotwell MAT 267 Fall 2008 79670
WeBWorK assignment number 11.1a is due : 10/03/2008 at 10:00am MST. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind
Double Integrals over Rectangles
Double Integrals over Rectangles
Consider a closed rectangular region R cfw_( x, y ) | a x b, c y d
and (, ) defined on R. Assume (, ) 0
Let S be the solid bounded below by the region R
and above by the graph of f.
S cfw_
Triple Integral in Cylindrical
Coordinates
Triple Integrals in Cylindrical coordinates
Cylindrical coordinates of the point P: (, , )
and are the polar coordinates of the projection of
the point P onto the -plane.
is the signed vertical distance between
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due October 24th 2016
12.5
This is a cover page and NOT to be written on. It is to be stapled to your actual work. Your work needs to
be neat, organized and in order. T
Triple Integrals
Triple Integrals
Triple integrals over a box: E ( x, y, z ) a x b, c y d , r z s
b d
s
f ( x, y, z)dV a c r
E
f ( x, y, z )dzdydz
By Fubinis Theorem we can
change the order of integration in
six possible ways.
Applications:
1. 1dV is the
Double Integrals
over General Regions
Double Integrals over General Regions Type I
Double integrals over general regions are evaluated as iterated integrals.
Most double integrals fall into two categories determined by the region of
integration D.
Type I
Tangent Planes and
Linear Approximations
Tangent Planes
Suppose a surface S has equation = (, ), where has continuous first
partial derivatives, and let (0 , 0 , 0 ) be a point on S.
Let 1 and 2 be the curves of intersection of S
with the planes = 0 and =
The Chain Rule
The Chain Rule Case I
The function z = (, ) is differentiable at , if the partial derivatives
and exist near , and are continuous at , .
Chain Rule Case I (One Independent Variable)
Let z = (, ) be a differentiable function in its domain w
Gradient and Directional Derivative
Gradient and Directional Derivative - Functions of two variables
Consider the point 0 0 , 0 . Let 0 = (0 , 0 ), then the point
0 , 0 , 0 lies on the surface = , .
Let = 1 , 2 be a unit vector in the xy plane.
The rate o
Maximum and Minimum Values
Maximum and Minimum Values
Problem: Find extrema (maximum or minimum) of (, ) on some domain D.
Consider a function = , .
We say that (, )is a
Local maximum if (, ) (, ) for
(, ) in D around (, )
Absolute maximum if (, ) (, )
fo
Equations of Lines and Planes
Part I - Planes
Equations of Lines and Planes - Planes
To specify a plane we need three points or one point and a direction.
The best way to describe a direction or how a plane is tilted is to give a normal
vector, , that is,
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 12th 2016
Sections 10.8, 10.9
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be neat, organize
NAME_
Calculus III for Engineers
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Arizona State University
Due December 2nd 2016
Sections 13.7
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due August 29th 2016
Sections 10.1 - 10.4
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 26th 2016
Sections 11.3 11.4
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NAME_
Calculus III for Engineers
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Arizona State University
Due September 5th 2016
Sections 10.5 - 10.7
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Calculus III for Engineers
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Arizona State University
Due October 17th 2016
Sections 12.3
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due November 14th 2016
Sections 13.4, 13.5
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due August 29th 2016
Sections 10.1 - 10.4
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 19th 2016
Sections 11.1
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be neat, organized and
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due October 31st 2015
Sections 12.6, 12.7, 13.1
To receive partial credit you MUST show your work. All work must have a solution that is boxed AND
highlighted with a hi
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due October 3rd 2016
Sections 11.5, 11.6
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be neat, organized a
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due November 7th 2016
Sections 13.2, 13.3
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due November 28th 2016
Sections 13.6
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be neat, organized and i
NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due October 10th 2016
Sections 11.7,12.1, 12.2
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 7th 2016
Sections 10.5 - 10.7
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due October 10th 2016
Sections 11.7,12.1, 12.2
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 12th 2016
Sections 10.8, 10.9
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NAME_
Calculus III for Engineers
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Arizona State University
Due October 17th 2016
Sections 12.3
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NAME_
Calculus III for Engineers
Physical Homework Cover Page
Arizona State University
Due September 19th 2016
Sections 11.1
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