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~ ~ ~ ~ ~ ~ ~ a i n * ~ ~ m ~ ~ o s l b a ~ a r a y
If the objective function is optimized at two adjacent vertices of S, then it is optimized at every point on the line
segmentjoining these vertices. In this case, there are infinitely many solutio
MAT 211 Mathematics for Business Analysis
Fall 2016
NOTE: THIS SYLLABUS MAY BE MODIFIED AT ANY TIME BY ANNOUNCEMENTS MADE IN CLASS.
Instructor: David Fishman
Office: ECA 337
SLN:71541
Office Hours: MWF 2:15-3:15 PM; TTH 1:00-2:00 PM
Place: SCOB 150 @TTH 7
MAT 211 SYLLABUS ACKNOWLEDGEMENT FORM
Instructions: You are expected to read the syllabus completely so that
you are fully aware of everything expected of you in this course.
Complete the form below and sign in the appropriate space indicating
that you ha
MAE 215: Introduction to Programming in MATLAB
Summer B 2016
Homework Assignment #3
Due: Tuesday, July 19th by 11.59 pm
Submission Instructions: Your solutions to this assignment must be typed and neatly
presented. Unorganized solu
TAYLOR SERIES
1. Find the nth degree Taylor polynomial for f ( x)
1
about x 1 . Include sigma notation.
x 1
2. The table below shows how well the Taylor polynomials approximate the value of f (x ) for various
values of x. What do you notice?
3. Create an
Solve the following differential equations algebraically.
Section 11.4
1.
dy
kx
dx
2.
dy
ky
dx
3.
dy
x 2 k 2
dx
4.
dy
y2 k 2
dx
5.
dy
y ky
dx
6.
dy
y k
dx
7.
dy
kx x
dx
8.
dy
ky ( x 1)
dx
Solve the initial value problems:
dy
x ( y 2)
dx
x2 4
y (2) 3
9.
SET UP THE INTEGRALS NEEDED TO SOLVE EACH OF THE PROBLEMS BELOW. INCLUDE
AN ILLUSTRATION OF THE VARIABLE.
1. A tank contains 288 ft3 of water. If the density of water is 62.4 lbs/ft3, how much work is needed to
pump all of the water out of the tank? How w
Find the radius and interval of convergence of the following power series:
1.
1
( x 2) k
k 1 k
2.
n 1
3. f ( x) 1 x 4 x 2 9 x 3 16 x 4 .
4.
1
2
n
xn
1
k 2x
k
k 0
5.
( 1) k
k 0
3k k
x
k 0 2 k
1
x 2 k 1
( 2k 1)!
6.
n
n 0
7.
(1)
8.
(1)
k 0
1
( x 1) n
n!
INTEGRALS AND SERIES
[7.7]
Definition of convergence of improper integrals:
Suppose f(x) is positive for x a .
If
b
lim
f ( x) dx
a
b
is a finite number, we say that
a
a
f ( x ) dx converges and define
b
f ( x ) dx lim f ( x ) dx .
b
a
Otherwise, we say
SET UP THE INTEGRALS NEEDED TO SOLVE EACH OF THE PROBLEMS BELOW. INCLUDE
SKETCHES OF THE REGIONS AND/OR SOLIDS.
Area Problems
1. The area of the region bounded between y x 2 2 x 3 and y 4 x 45 .
2. The area of the region bounded by y ln x , the x and y ax
Series and Improper Integrals
1. Show that the series
1
n
n 1
2. Show that the series
4
converges by comparing it to the improper integral
1
1
n
n 1
1
x
4
dx .
1
diverges by comparing it to the improper integral
n 1 n
3. Show that the series
Section 9.2
Application to geometry:
Section 8.2
1. Consider the region bounded by the curve y e x , the x-axis, and the lines x 1 and x 1 .
Find the volume of the following solids. Include sketches.
A. The solid obtained by rotating the region about the x-axis.
B. T
Specific Example
General
The problem:
The problem:
Find the area of the irregular shaped
region bounded by f ( x ) , the x-axis,
over the interval [a, b].
The quantity we want to find depends on something
that varies. Applications come from geometry (leng
Area/mass
1. Find the area of the region bounded by y x 6 ,
A. Use slices perpendicular to the x-axis.
y 5
x
Section 8.4
and the x-axis.
B. Use slices perpendicular to the y-axis
2 A. Find the area of the region bounded by the first arch of y sin ax and y
Practice with Limits
Name_
Find the following limits. Show all work and use LHopitals Rule whenever possible. Express your
answer in exact form. For example, if the value of the limit is do not write 3.14.
1
sin(q )
sin( x )
1. lim
x
2. qlim
1
4. lim s
TAYLOR SERIES
Name_
USE WELL-KNOWN SERIES TO ANSWER THE FOLLOWING.
1. Find 3
27 243 2187
.
3! 5!
7!
2. Find x 2
x 4 x 6 x8
. .
3! 5! 7!
(1) k 1 x k
.
k
k 1
3. Find
4. Use series to find f (5) (0) and f (6) (0) for f ( x )
x
.
1 x2
5. Use the values i
Homework for 7.8
Use the technique that we discussed in section 7.8 to determine if the integrals converge or not. If they
converge, find an upper bound.
1.
4.
2
1
1
3
x 1
2
dx
sin 2 t 2t
dt
t3 4
2.
5.
1
1
3
dy
cos y 2 y
3.
2
3cos x
dx
x
6.
1
0
1
dx
CREATE A NEW SERIES FROM AN OLD SERIES
1. Find the Taylor series for g ( x) x 2 e x about x 0 . Include the general term.
2. Find the Taylor series for h( x) ln(4 8 x) about x 0 . Include the interval of convergence.
USE A SERIES TO EVALUATE OR APPROXIMAT
1. Find the Taylor series about 0 for the functions below. Include at least four nonzero terms.
A. f ( x ) arcsin x
B. f ( x ) ln(3 x 1)
C. f ( x)
sin( x 2 )
x
D. f ( x ) esin x
1
x
2. Use an appropriate Taylor polynomial about 0 to find an approximation
Power Series
1. Find the radius and interval of convergence of
2k k
x using the steps below:
k
k 0 5
(i) a k
(ii) a k 1 =
(iii) Simplify
(iv) klim
a k 1
a k 1
ak
ak
=
=
(v) Radius
(vi) Interval
2. Repeat the process shown in problem 1 for the following s
Section 8.5
1. A worker on a roof 50 ft above the ground needs to lift a 300 lb bucket of cement from the ground
to a point 20 ft above the ground by pulling on a rope weighing 4 lb/ft. How much work is required?
2. A water tank is in shape of right circu
Geometric Series
1. Find the sum of the series:
4 4
4
L
A. 4
5 25 125
C. e
E.
n 0
e 2 e3 e 4
L
5 25 125
2 n 1 3
4n
1 1 1
L
B. 3
3 27 81
D.
1
4
n 3
n
F.
5n
n
n 0 3
3. A ball is dropped from a height of 9.0 m. On each upward bounce the ball returns to
Section 10.2/10.3
1. The following parts refer to f ( x )
1
.
3 x
A. Write the Taylor series expansion for f ( x) about x = 0.
B. Expand f ( x) in a Taylor series in terms of
x
about x = 0.
3
C. Find the Taylor series expansion for f ( x) about x = 2 wit
MTE 280 Ch. 3 Review
Use the lattice method to add:
31)
673
32)
218
7401
3156
Use the lattice method to multiply:
33)
73
34)
16
527
Power rules:
column:
35)
814
Write answers in answer
2 x 8x
5
13
36)
b b c b
5
3 0
37)
5x
6
y
7
3
35)_
36)_
37)_
38)_
38
Problem Set 4
Directions:
1. Solve the following problems. Hand write your solutions and explanations on your
own paper. Do not write answers on this page.
2. Show your work AND explain your reasoning using complete English sentences.
Explanations must di
MAT 342, Quiz 10, Spring 2015
Answers.
1. (8 pts) Let S be the subspace in R4 that is spanned by the vectors (3, 1, 3, 1)T and
(1, 1, 3, 3)T . Find a basis for the subspace S . If (x1 , x2 , x3 )T is orthogonal to the
two given equations we have that thei
MAT 342, Quiz 11, Spring 2015. Answers.
1. (4 pts) The method of least squares is to be used to fit a quadratic function y = a + bx + cx2
to the data:
x 1 2 3 4 5
y 5 4 3 1 0
What is the system of equations for a, b, and c that needs to be treated with th
MAT 342, Quiz 9 Answers, Spring 2015,
1. (8 pts) Consider the points P = (1, 1, 1), Q = (1, 2, 3) and R = (4, 3, 2).
(a) Find a vector that is perpendicular to both P Q and P R
Answer. P Q= (0, 1, 2)T and
P Q P R=
P R= (3, 2, 1)T so that
i j k
0 1 2 = 3