Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
1. Consider the sequence dened by the recursion an+1 =
p
an .
(a) Show that if 0 < a0 < 1, then an is convergent.
Solution.
p
p
Observe that if x 2 (0, 1), then (i) x 2 (0, 1) and (ii) x > x.
Thus if an 2 (0, 1) then 0 < an < an+1 < 1. By induction (forma
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
July 29, 2013 Calculus 1501 30 minutes
Quiz 2
Nme(PRINT)C}SmlA atmd was Student Number _i_
Signature
Instructions
1. No calculators. No Books. No Notes.
2. Show all your work. Justify all your answers.
3. Note that questions are printed on both si
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
v: 20130122
CALC 1501 LECTURE NOTES
RASUL SHAFIKOV
2. Techniques of Integration
The rules of dierentiation give us an explicit algorithm for calculating derivatives of all elementary functions, including trigonometric and exponential functions, as well
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
v: 20130204
CALC 1501 LECTURE NOTES
RASUL SHAFIKOV
1. Mean Value Theorem
1.1. Review: limit, continuity, dierentiability. We denote by R the set of real numbers. A
domain D of R is any subset of R. Typically this will be on open interval (a, b) or a clo
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
v: 20130121
CALC 1501 LECTURE NOTES
RASUL SHAFIKOV
3. Improper Intergrals.
So far we dealt with integration of continuous functions on bounded intervals. In this section
we will discuss integration of continuous functions on unbounded intervals, and als
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
v: 20130131
CALC 1501 LECTURE NOTES
RASUL SHAFIKOV
4. Sequences
Denition 4.1. A sequence s is a function s : N R. It can be thought of as a list of numbers
s1 , s2 , s3 , . . . ,
where sn = s(n) for n N.
Example 4.1.
1
1
(i) cfw_sn = 1, 1 , 3 , . . . .
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
Calculus 1501a
Additional Exercises for Section 11.10
1. Find the Maclaurin series of the following:
Zx
sin(t)
dt
(a)
t
0
Zx t
e
1
dt
(b)
t
0
Z 1+x
ln(t)
(c)
dt
t1
1
Zx
(d)
cos(t2 )dt
0
(e)
Z
x
tan
t2
0
(f)
Z
1 (t2 )
x
e
t2
dt
dt
0
(g)
Z
x
(1 + t3 )1/4 dt
Western University (Ontario)  Also known as University of Western Ontario
Calculus II for the Mathematical and Physical Sciences
MATH 1501

Summer 2013
v: 20130211
CALC 1501 LECTURE NOTES
RASUL SHAFIKOV
5. Series
5.1. Basic Denitions. Given a sequence of real numbers
a1 , a2 , a3 , . . . , an , . . .
a formal expression
(5.1)
a1 + a2 + a3 + + an + =
an
n=1
is called an innite series, or just a series.