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
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
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, incl
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.
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
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
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