The integral test (Sect. 10.3)
Review: Bounded and monotonic sequences.
Application: The harmonic series.
Testing with an integral.
Error estimation in the integral test.
The integral test (Sect. 10.3)
Review: Bounded and monotonic sequences.
Application:
Improper integrals (Sect. 8.7)
Review: Improper integrals type I and II.
Examples: I =
1
dx
, and I =
xp
1
0
dx
.
xp
Convergence test: Direct comparison test.
Convergence test: Limit comparison test.
Improper integrals (Sect. 8.7)
Review: Improper integra
Innite sequences (Sect. 10.1)
Todays Lecture:
Overview: Sequences, series, and calculus.
Denition and geometrical representations.
The limit of a sequence, convergence, divergence.
Properties of sequence limits.
The Sandwich Theorem for sequences.
Next Le
Improper integrals (Sect. 8.7)
This class:
Integrals on innite domains (Type I).
dx
The case I =
.
xp
1
Integrands with vertical asymptotes (Type II).
1
dx
The case I =
.
p
0 x
Next class:
Convergence tests:
Direct comparison test.
Limit comparison test.
Innite sequences (Sect. 10.1)
Todays Lecture:
Review: Innite sequences.
The Continuous Function Theorem for sequences.
Using LHpitals rule on sequences.
o
Table of useful limits.
Bounded and monotonic sequences.
Previous Lecture:
Overview: Sequences, seri
Innite series (Sect. 10.2)
Series and partial sums.
Geometric series.
The n-term test for a divergent series.
Operations with series.
Adding-deleting terms and re-indexing.
Innite series (Sect. 10.2)
Series and partial sums.
Geometric series.
The n-term t
Integrating using tables (Sect. 8.5)
Remarks on:
Using Integration tables.
Reduction formulas.
Computer Algebra Systems.
Non-elementary integrals.
Limits using LHpitals Rule (Sect. 7.5).
o
Integrating using tables (Sect. 8.5)
Remarks on:
Using Integration
Integration by parts (Sect. 8.1)
Integral form of the product rule.
Exponential and logarithms.
Trigonometric functions.
Denite integrals.
Substitution and integration by parts.
Integral form of the product rule
Remark: The integration by parts formula is
Review for Exam 3.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers: 8.3, 8.4, 7.5, 8.7, 10.1.
Trigonometric substitutions (8.3).
Integration using partial fractions (8.4).
LHpit
Trigonometric substitutions (Sect. 8.3)
Substitutions to cancel the square root
Integrals involving a2 x 2 : Use x = a sin().
Integrals involving a2 + x 2 : Use x = a tan().
Integrals involving x 2 a2 : Use x = a sec().
Substitutions to cancel the square
Trigonometric integrals (Sect. 8.2)
Product of sines and cosines.
Eliminating square roots.
Integrals of tangents and secants.
Products of sines and cosines.
Product of sines and cosines
Remark: There is a procedure to compute integrals of the form
I =
si
Review for Exam 2.
5 or 6 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers: 7.4, 7.6, 7.7, 8-IT, 8.1, 8.2.
Solving dierential equations (7.4).
Inverse trigonometric functions (7.6).
Hyp
Limits using LHpitals Rule (Sect. 7.5)
o
0
Review: LHpitals rule for indeterminate limits .
o
0
Indeterminate limit
.
Indeterminate limits 0 and .
Overview of improper integrals (Sect. 8.7).
0
LHpitals rule for indeterminate limits
o
0
Remarks:
f (x)
in
x