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.

Definite Integrals

This final page covers definite integrals and their applications.

Fundamental Theorem of Calculus

The page presents the Fundamental Theorem of Calculus, relating definite integrals to antiderivatives:

∫[a to b] f$x$dx = F$b$ - F$a$

Where F$x$ is an antiderivative of f$x$.

Highlight: The Fundamental Theorem of Calculus provides a powerful method for evaluating definite integrals.

Area Calculation

Techniques are provided for calculating areas using definite integrals:

• Area between a curve and the x-axis
• Area between two curves

Example: The area between f$x$ and g$x$ from a to b is given by: ∫[a to b] $f(x) - g(x)$dx

Integration Techniques for Definite Integrals

The page reviews integration techniques in the context of definite integrals:

• Substitution method
• Integration by parts

Vocabulary: When using substitution in a definite integral, the limits of integration must be adjusted accordingly.

Trigonometric Integrals

Special techniques are provided for integrating products of sine and cosine functions.

Definition: A trigonometric integral involves products of sine and cosine functions, often solved using half-angle formulas or substitutions.

This comprehensive guide covers essential topics in calculus, providing a valuable resource for students preparing for exams or seeking to master key concepts in mathematical analysis.

Limits and Derivatives

This page covers fundamental concepts in calculus related to limits and derivatives.

Key Limit Formulas

The page presents several important limit formulas, including:

• Limits of trigonometric functions as x approaches 0
• Limits involving exponential and logarithmic functions
• Limits of indeterminate forms

Example: lim(x→0) (sin x)/x = 1

Derivative Rules

Basic derivative rules are provided for:

• Power functions
• Exponential and logarithmic functions
• Trigonometric functions

Definition: The derivative f'$x$ represents the slope of the tangent line to f$x$ at a point.

Derivative Techniques

The page covers key techniques for finding derivatives:

• Product rule
• Quotient rule
• Chain rule

Highlight: The chain rule allows differentiating composite functions: D[f(g$x$)] = f'(g$x$) · g'$x$

Points of Non-Differentiability

Types of non-differentiable points are explained:

• Vertical tangent
• Cusp
• Corner point

Vocabulary: A cusp occurs when left and right derivatives are infinite with opposite signs.