Theorems and Function Analysis

This page covers important theorems in calculus and techniques for analyzing functions.

Key Theorems

The page presents several fundamental theorems:

• Rolle's Theorem
• Mean Value Theorem (Lagrange's Theorem)
• Cauchy's Mean Value Theorem

Definition: Rolle's Theorem states that for a continuous function f(x) on [a,b] with f(a) = f(b), there exists c in (a,b) where f'(c) = 0.

Function Analysis Techniques

Methods for analyzing functions are provided:

• Finding domain and intervals
• Determining increasing/decreasing intervals using derivatives
• Identifying asymptotes (vertical, horizontal, oblique)

Example: To find a horizontal asymptote, calculate lim(x→∞) f(x)

L'Hôpital's Rule

The page explains L'Hôpital's Rule for evaluating limits of indeterminate forms.

Highlight: L'Hôpital's Rule states that for limits of the form 0/0 or ∞/∞, the limit of the quotient equals the limit of the quotient of derivatives.

Concavity and Inflection Points

Techniques are provided for:

• Finding stationary points
• Determining concavity using the second derivative
• Identifying inflection points

Vocabulary: An inflection point occurs where the concavity of a function changes.

Indefinite Integrals

This page covers techniques for solving indefinite integrals.

Basic Integration Formulas

The page provides formulas for integrating:

• Power functions
• Exponential and logarithmic functions
• Trigonometric functions

Example: ∫ sin x dx = -cos x + C

Integration Methods

Key integration techniques are explained:

• Substitution method
• Integration by parts

Definition: Integration by parts uses the formula: ∫ f'(x)g(x)dx = f(x)g(x) - ∫ f(x)g'(x)dx

Integrating Rational Functions

The page covers techniques for integrating rational functions, including:

• Partial fraction decomposition
• Long division of polynomials

Highlight: Partial fraction decomposition is used when the degree of the numerator is less than the degree of the denominator.

Trigonometric Substitutions

Special substitutions are provided for integrals involving:

• √(1-x²)
• √(1+x²)
• √(x²-1)

Vocabulary: Trigonometric substitutions transform an integral into one involving trigonometric functions.