Z
Example. Approximate the integral
1
2
ex dx to 2 significant figures.
0
Solution: We will approximate the integrand by a Taylor polynomial so that we can
integrate it! and use Taylors Theorem with remainder to justify the answer.
Substituting x2 for x i
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 4 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. (a) Find the equilibrium solutions of the dierential equation
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 2 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Evaluate
p
R4
(a) 3 (x2 dx8)3=2 = 38 18 2 + C
p
R
2
2
1
(b) p4
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 7 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Determine if the sequence converges or diverges. If it converg
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 8 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas. You may use
calculators, or computers, for numerical work. Expres
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 11 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find the Taylor series for each of the following (you may use
1. Find and classify the equilibrium solutions of dP
= P 2 P 4 as stable, unstable, or semidt
stable.
Solution: P 2 P 4 = P 2 (1 P ) (1 + P ) : Equilibrium solutions: P = 1 (stable), P = 0
(semistable), and P = 1 (unstable).
2. Use Eulers Method with ini
Integration, Differential Equations and Approximation
CALCULUS 21122

Spring 2014
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 1 Solutions
Instructions: Please show all your work, following the instructions posted on Blackboard.
1. Evaluate
R
2
cos 3x + 29 x sin 3x 31 x2 cos 3x + C
(a) x2
Integration, Differential Equations and Approximation
CALCULUS 21122

Spring 2014
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 1
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Evaluate
R
(a) x2 sin (3x) dx
R
(b) x ln (x + 1) dx
R
x
dx
(c) p1+x
2
R
Integration, Differential Equations and Approximation
CALCULUS 21122

Spring 2014
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 3
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Improper Integrals: Evaluate the integral, or show it diverges.
R1
(a) 0
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 10 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Find the radius of convergence and the interval of convergenc
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 9 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Determine if the series converges absolutely, converges condit
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 3 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. Improper Integrals: Evaluate the integral, or show it diverges
Root Test.
X
(i) If lim
p
n
(ii) If lim
X
p
p
n
an  = L > 1 or lim n an  = , then the series
an is divergent.
(iii) If lim
p
n
an  = L < 1, then the series
n
n
n
an is absolutely convergent.
n
an  = 1, the Root Test is inconclusive.
Alternating Se
Eulers Method:
Eulers method is an elementary numerical approach to determining an approximate solution
to initial value problems of the following form:
dy
= f (x, y), y(a) = y0 , a x b
dx
The method rests on the observation that the slope of the tangent
Sequences:
A sequence is a function with domain a set of integers usually the positive integers or
the nonnegative integers.
The domain element is indicated by a subscript as in an =
2
.
2n + 1
Examples.
A sequence of the form
a, ar, ar2 , ar3 , , arn ,
Series Summary:
Bounded sequence theorem
Every bounded monotonic sequence con
verges.
00
A series 2 an converges if the sequence of
11:1 '
partial sums dened by
3n2G1+02+'+an
converges.
Most of the tests on this summary in some
way nd an upper bound0r sho
Theorem. For a given power series
X
cn (x a)n there are only three possibilities:
n=0
(i) The series converges only when x = a.
(ii) The series converges for all x.
(iii) There is a positive number R such that the series converges if
x a < R and diverge
new
Name _1_m_1_i_i_1_i_1_
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21122 Integration, Different
Test 3: April 22, 2014
ial Equations, and Approximation
E TuTh 8:30 PH DH 1211 Chu,
F TuTh 10:30 PH 125C Cheung
G TuTh 12:30 WEH 5302 C
Z
b
Theorem. If f is continuous on [a, b], then
f (x) dx exists.
a
Improper integrals violate this theorem in one of two ways:
Type 1: Integration over an unbounded interval.
Type II: Integration where the integrand becomes unbounded at
one or both endpoi
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 5 Solutions
Instructions: Please show all your work, following the instructions posted on Canvas.
1. (a) Solve the dierential equation
p
y = 4 x2
dy
dx
=
(b) Solve
Integration, Differential Equations and Approximation
CALCULUS 21122

Spring 2014
Carnegie Mellon University
Department of Mathematical Sciences
21122, Spring 2017
Homework 4
Instructions: Please show all your work, following the instructions posted on Canvas.
1. (a) Find and the equilibrium solutions of the dierential equation
dP
= 0