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.