MATH 2132 Tutorial 3
1. (a) Find the rst ve Taylor polynomials for the function cos 2x about x = 0.
(b) Show that the Maclaurin series for cos 2x converges to cos 2x for all x.
2. Find the Taylor series about x = 1 for the function f (x) = 1/(x 2)2 . Expr
THE UNIVERSITY OF MANITOBA
DATE: June 16, 2012
FINAL EXAMINATION
DEPARTMENT & COURSE NO: MATH2132
TIME: 3 hours
EXAMINATION: Engineering Mathematical Analysis 2 EXAMINER: D. Trim
PAGE NO: 1 of 12
10
1. Find the interval of convergence for the power series
MATH2132 Test1
February 4, 2014
70 minutes
1. (a) Determine whether the sequence of functions
cfw_fn (x) =
3n2 x2 + 1
n2 x2 + 2nx + 4
has a limit on the interval 1 x 1. Show your reasoning and all calculations.
(b) Would the series
fn (x) have a sum? Expl
MATH 2132 Test 2
March 4, 2014
Student Name -
75 minutes
Student Number -
Values
12 1. (a) Find the Taylor series about x = 3 for the function 2 + x. Express your answer in sigma
notation simplied as much as possible. You must use a technique that ensures
MATH2132 Test1
8
May 2014
60 minutes
1. Determine whether the sequence of functions
cfw_fn (x) =
n
n+1
x+
n+1
n
n
xn
has a limit on the interval 0 x 1. Show your reasoning and all calculations.
Since lim xn = 0 for 0 < x < 1, it follows that
n
lim fn (x)
THE UNIVERSITY OF MANITOBA
DATE: December 16, 2013 (Afternoon)
FINAL EXAMINATION
DEPARTMENT & COURSE NO: MATH2132
TIME: 3 hours
EXAMINATION: Engineering Mathematical Analysis 2 EXAMINER: D. Trim
6
1. Find the open interval of convergence for the series
(n
THE UNIVERSITY OF MANITOBA
DATE: December 17, 2011
FINAL EXAMINATION
DEPARTMENT & NO: MATH2132
TIME: 3 hours
EXAMINATION: Engineering Mathematical Analysis 2 EXAMINER: M. Despic, D. Trim
PAGE NO: 1 of 10
10
1. For what value of the constant a will the rad
THE UNIVERSITY OF MANITOBA
DATE: June 18, 2011
FINAL EXAMINATION
DEPARTMENT & COURSE NO: MATH2132
TIME: 3 hours
EXAMINATION: Engineering Mathematical Analysis 2 EXAMINER: D. Trim
10
1. Find the interval of convergence for the power series
(1)n
(x 1)n .
n2
MATH2132 Test 2
June, 2014
Student Name -
60 minutes
Student Number -
Values
14 1. Find the Taylor series about x = 2 for the function (x + 2)2 ln (x + 5). Use a method that
guarantees that the series converges to the function. Express your answer in sigm
MATH 2132 Tutorial 7
In Problems 13, nd a one-parameter family of solutions of the dierential equation.
Find any singular solutions.
y 1 dy
dy
dy
= yx2
2.
= x2
3. x2
= y2 1
1. (y 1)
dx
y dx
dx
4. Find an explicit solution of the initial-value problem
dy
x
MATH 2132 Tutorial 9
Find a general solution of the dierential equation in questions 14.
1.
2.
3.
4.
5.
y + 2y 3y = 4e5x x
y + 3y 4y = 2ex + cos 4x
y + 6y 2y = 3x + sin x
4y 3y + 7y + 2y = ex/4
The roots of the auxiliary equation (m) = 0 associated with t
MATH 2132 Tutorial 5
1. Find the Maclaurin series for the function Tan1 (2x2 ). Express your answer in sigma notation,
simplied as much as possible. What is the open interval of convergence of the series?
2. Find the Taylor series for 1/ 10 3x about x = 2
187
EXERCISES 5.1
CHAPTER 5
EXERCISES 5.1
1. With the coordinate system of Figure 5.5, the initial-value problem describing the position x(t) of
the mass is
(1)
d2 x
+ 16x = 0,
dt2
x(0) = 1/10,
x (0) = 0.
The auxiliary equation is m2 +16 = 0 with solution