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.

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.