Calculus Study Guide

Calculus Study Guide


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Includes both Calculus I and II
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Table of Contents

Introduction: Functions

Limits and Continuity: Limit of a Sequence | Limit of a Function | Limit of a function at infinity | Continuity | Classification of Discontinuities

Derivative: Computing the derivative | Quotient Rules | The Chain Rule | Implicit Function | Related Rates | Product Rule
Table of derivatives: General differentiation rules | Derivatives of simple functions | Derivatives of exponential and logarithmic functions | Derivatives of trigonometric functions | Derivatives of hyperbolic functions | Derivatives of Inverse Trigonometric Functions

Integration (Antiderivative): Integral | Arbitrary Constant of Integration | The Fundamental Theorem of Calculus
Table of Integrals: Rules for integration of general functions | Integrals of simple functions | Rational functions | Irrational functions | Logarithms | Exponential functions | Trigonometric functions | Inverse Trigonometric Functions | Hyperbolic functions | Inverse hyperbolic functions | Definite integrals lacking closed-form antiderivatives | The "sophomore's dream" | Integral Curve | Euler-Maclaurin Formula | Trapezium rule

Logarithms and Exponentials: E - base of natural logarithm | Ln(x) | Hiperbolic functions

Applications of the Definite Integral in Geometry: Area of a Surface of Revolution | Solid of Revolution

Techniques of Integration: Integration by Parts | The ILATE rule | Integration by Substitution | Trigonometric Substitution | Partial Fractions in Integration of Rational Function | Numeric Integration | Simpson Rule

Principles of Integral Evaluation: Methods of Contour Integration | Cauchy's Integral Formula | Improper Integrals | L'Hopital's Rule

Differential Equations: First-Order Differential Equation | Linear Differential Equation
Examples: A separable first order linear ordinary differential equation | Non-separable first order linear ordinary differential equations | A simple mathematical model | Harmonic Oscillator | Stiff Equation

Numerical Integration Methods: Numerical Ordinary Differential Equations | Euler's Method | Runge-Kutta Methods | Multistep Method

Series: Taylor Polynomials | Taylor Series | List of Taylor series | Lagrange Polynomial

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