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What is integration?
It is the Inverse O eration of differentiation.
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Remember what 1nverse operations do?
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Math 125
Sec 12.3
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LHpitals Rule and the Indeterminate form 0/0
One-sided Limits and Limits at 00
LH6pitals Rule and the Indeterminate form 00/00
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Lets first look at what things are NOT the indeterminate form.
If:
lim fOC)
lim 106)
lint f(x)
Unit: Sequences and Series
Module: The Ratio and Root Tests
The Ratio Test
To apply the ratio test to a series
n =1
an , let
= lim
n
an +1 . an
If < 1, then the series converges absolutely. If > 1, then the series diverges. If = 1, then the test is inco
Unit: Sequences and Series
Module: Power Series Representations of Functions
Finding Power Series Representations by Differentiation
You can do calculus on a power series inside its interval of convergence. The derivative of the power series
n =0
an (
Unit: Sequences and Series
Module: Power Series Representations of Functions
Differentiation and Integration of Power Series
You can do calculus on a power series inside its interval of convergence. The derivative of the power series
n =0
an ( x c )n
Unit: Sequences and Series
Module: Power Series
Interval and Radius of Convergence
The interval of convergence of a power series is the collection of points for which the series converges. The radius of convergence of a power series is the distance betw
Unit: Sequences and Series
Module: Power Series
Definition of Power Series
The general form of a power series centered at x = c is a sequence of numbers that only depend on n.
n =0
an ( x c )n , where an is
All power series converge for x = c. If a pow
Unit: Sequences and Series
Module: Taylor and Maclaurin Series
Taylor Series
The Taylor series expansion of f ( x ) centered at x = c is
f ( n ) (c ) ( x c )n n! n =0
assuming that f ( x ) is differentiable an infinite number of times. For a given x, if
Unit: Sequences and Series
Module: Taylor and Maclaurin Polynomials
Maclaurin Polynomials
The Maclaurin polynomial of f (x ) is the Taylor polynomial of f (x ) centered at k f n0 n f (2) (0) 2 f ( k ) (0) k x = 0: x = f (0) (0) + f (1) (0)x + x +K+ x. 2 k
Unit: Sequences and Series
Module: Taylor and Maclaurin Polynomials
Taylor Polynomials
Higher-degree polynomial approximations result in more accurate representations. To find a Taylor polynomial, find all the necessary derivatives and evaluate them at
Unit: Sequences and Series
Module: Polynomial Approximations of Elementary Functions
Polynomial Approximation of Elementary Functions
Polynomials can approximate complicated functions. The tangent line approximation of f (x ) at x = c is y = f (c ) + f
The Root Test
To apply the nth root test to a series
n =1
an , let
= lim
n
n
an .
If < 1, then the series converges absolutely. If > 1, then the series diverges. If = 1, then the test is inconclusive. The nth root test is another way to see if the terms
Unit: Sequences and Series
Module: Power Series Representations of Functions
Finding Power Series Representations by Integration
You can do calculus on a power series inside its interval of convergence. The integral of the power series
n =0
an ( x c )n
Unit: Sequences and Series
Module: Infinite Series
[page 1 of 2]
Introduction to Infinite Series
Binary operations combine two values to yield a single result. You can add as many numbers as you want as long as you only have finitely many of them. Since
[page 1 of 2]
Monotonic and Bounded Sequences
A sequence is increasing if each term is larger than the previous one. A sequence is decreasing if each term is smaller than the previous one. A sequence is monotonic if the terms are nonincreasing (a1 a2 an
Unit: Sequences and Series
Module: Sequences
The Limit of a Sequence
A sequence is a pattern of numbers. If a sequence approaches a particular number as the index n gets larger, then the sequence is said to converge to that number. If not, the sequence
Unit: Applications of Integration
Module: Work
Introduction to Work
Work is the energy used when applying a force over a distance. For a constant force F, work is the product of the force and the change in distance. For a changing or variable force F(x)
Unit: Applications of Integral Calculus
Module: Arc Length
Introduction to Arc Length
Arc length is the length of the curve. The arc length of a smooth curve given by the function f (x) between a and b:
b a
1 + [f ( x )]2 dx .
W hen measuring how long a
Unit: Applications of Integration
Module: Disks and Washers
Solids of Revolution
Revolving a plane region about a line forms a solid of revolution. The volume of a solid of revolution using the disk method where R (x) is the radius of the solid of revol
Unit: Applications of Integration
Module: Finding Volumes Using Cross-Sections
Finding Volumes Using Cross-Sectional Slices
The volume of a solid with vertical cross-sections of area A (x) is V where:
V=
b
A( x ) dx .
a
The volume of a solid with hori
Unit: Improper Integrals
Module: Convergence and Divergence of Improper Integrals
The Second Type of Improper Integral
If a function is not continuous on the integration interval, then the standard procedure will not work. Use the discontinuity as an endp
Unit: Improper Integrals
Module: Convergence and Divergence of Improper Integrals
The First Type of Improper Integral
An improper integral is a definite integral with one of the following properties: the integration takes place over an infinite interval
Unit: Techniques of Integration
Module: Trigonometric Substitution Strategy
An Overview of Trig Sub Strategy
Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares. 1. Match the square root e
Unit: Techniques of Integration
Module: Introduction to Trigonometric Substitution
Trigonometric Substitutions on Rational Powers
Use trigonometric substitution to evaluate integrals involving the square root of the sum or difference of two squares. 1. Ma