Math 117 Fall 2014 Lecture 42
(Nov. 20, 2014)
Read:
Innite Series
Proving convergence: Denition; Cauchy; Monotone + bound;
P
If 1 an converges then limn!1 an = 0;
n=1
Comparison:
jan j 6 bn ;
P1
P1
P1
n=1 bn converges =) n=1 an converges;
P1
P1
n=1 bn div
Math 117 Fall 2014 Final Review Problems
Final exam coverage:
Lectures 1 48 and the exercises therein.
Homeworks 1 9.
The exercises below only cover materials after Midterm 3. You should also study the
Review Problems for Midterms 1 3.
Required sections i
Math 117 Fall 2014 Lecture 41
(Nov. 19, 2014)
Read: 314 Dierentiation 4.1.
Higher Order Derivatives.
Definition 1. Let f be dierentiable on (a; b). If f 0(x) is dierentiable at some c 2 (a; b),
we say f is twice dierentiable at c and called (f 0) 0(c) its
Math 117 Fall 2014 Lecture 48
(Dec. 3, 2014)
Higher derivatives.
Leibniz formula:
(f g)(n) =
n
X n
k=0
k
f (k) g (nk):
(1)
Example 1. Calculate (x sin x)(100):
Solution. We notice that x(k) = 0 for all k > 2. Thus
100
100 (1)
(x sin x)(100) =
x (sin x)(10
Math 117 Fall 2014 Lecture 43
(Nov. 24, 2014)
Read: Bowman 4.G; 314 Dierentiation 4.2.
Continuity and Dierentiability as Approximations.
Example 1. Let f (x) be continuous at x0. Then there is exactly one number s0 2 R such
that limx!x0 [f (x) s0] = 0.
Ex
Math 117 Fall 2014 Homework 9 Solutions
Due Thursday Nov. 27 3pm in Assignment Box
Question 1. (5 pts) Prove by denition that f (x) =
0 x=0
/
is integrable on [0; 1].
1 x=0
Proof. Let P = fx0; x1; :; xn g be an arbitrary partition of [0; 1] with 0 = x0 <
Math 117 Fall 2014 Lecture 47
(Dec. 1, 2014)
Proving integrability.
Denition: infPU (f ; P ) = supPL(f ; P ).
If there is fPn g such that limn!1 [U (f ; Pn) L(f ; Pn)] = 0.
If there is fPn g such that limn!1 U (f ; Pn) = limn!1 L(f ; Pn).
Example 1. Prove
Math 117 Fall 2014 Lecture 44
(Nov. 26, 2014)
Read: Bowman 4.G; 314 Dierentiation 4.2.
Taylor Expansion.
Let n 2 N [ f0g, x0 2 R, f : R 7! R. Dene
Taylor polynomial of f at x0 of degree n:
Tn(x) := f (x0) + f 0(x0) (x x0) + +
f (n)(x0)
(x x0)n:
n!
(1)
The
Math 117 Fall 2014 Lecture 40
(Nov. 17, 2014)
Read: Bowman 4.F, 314 Dierentiation 3.2, 3.3.
L'Hospital's Rule.
Theorem 1. Let a 2 R. If there is > 0 such that
i. f ; g dierentiable on (a ; a + ) fag;
ii. limx!af (x) = limx!ag(x) = 0;
f 0(x)
iii. limx!a g
Math 117 Fall 2014 Lecture 46
(Nov. 28, 2014)
Power series
Definition 1. The formal sum
P1
n=0
an (x x0)n is called a power series.
P
Let c 2 R. If the series 1 an (c x0)n converges to L 2 R, then the power series
n=0
P1
n
n=0 an (x x0) becomes a rule of
Math 117 Fall 2014 Lecture 45
(Nov. 27, 2014)
Read: Bowman
Example 1. Prove that e is irrational.
p
Proof. Assume the contrary: e = q . Take n > max fq; eg. We apply Taylor expansion with Lagrange
form of remainder to ex with x0 = 0 and x = 1:
p
1 1
1
ec
Math 117 Fall 2014 Midterm Exam 3 Solutions
Nov. 21, 2014 10am - 10:50am. Total 20+2 Pts
NAME:
ID#:
There are ve questions.
Please write clearly and show enough work.
1
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3
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5
Total
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Math 117 Fall 2014 Midterm Exam 3 Solutions
Question 1. (5 pts) Prov