Calculus

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What is Calculus? Calculus is concerned with change and motion; it deals with the quantities that approach other quantities.

Sir Isaac Newton invented his version of calculus in order to explain the motion of planets around the sun. Today, calculus is used in calculating the orbits of satellites and spacecrafts, in predicting population sizes, in estimating how fast prices rise, in forecasting weather, in calculating life insurance premiums, and in many other areas.

The topics covered are: Differential Calculus, Integral Calculus, Sequences and Series, Parametric Curves and Polar Coordinates, Multivariable Calculus, and Differential Equations.

AP Calculus AB and BC Past Papers and Solutions

Highlights of Calculus

Highlights of Calculus (Part 1) Big Picture of Calculus
Big Picture: Derivatives
Max and Min and Second Derivative
The Exponential Function
Big Picture: Integrals
Derivative of sin x and cos x
Product Rule and Quotient Rule
Chains f(g(x)) and the Chain Rule

Highlights of Calculus (Part 2) Limits and Continuous Functions
Inverse Functions f^-1(y) & x = ln y
Derivatives of ln y & sin^-1(y)
Growth Rates & Log Graphs
Linear Approximation/Newton's Method
Power Series/Euler's Great Formula
Differential Equations of Motion
Differential Equations of Growth

Differential Calculus

The study of differential calculus is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative.

Introduction to Calculus	The Limits of a Function Definition and Techniques to find Limits
Derivatives Definition and Slope of Tangent	Power Rule Constant Multiple Rule, Sum Rule, Difference Rule
Product Rule Find the derivative of the product of two functions	Quotient Rule Find the derivative of the quotient of two functions
Chain Rule Find the derivative of composite functions	More Chain Rule Examples
Derivative Rules Product Rule, Quotient Rule, Chain Rule, Power Rule, Exponential and Logarithmic Functions, Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions and Inverse Hyperbolic Functions	Examples using the Derivative Rules
More Examples using the Derivative Rules	Trigonometric Derivatives Derivatives of sin, cos, tan, scs, sec, cot
More Derivatives Involving Trigonometric Functions	Inverse Trigonometric Derivatives Derivatives of the inverse of sin, cos, tan, scs, sec, cot
Trigonometric Substitution Videos	Deriving the Derivative Formulas for Tan, Cot, Sec, Cosecant, Arctan
Derivatives of Exponential Functions Derivatives of e^x, a^x, a^g(x)	Derivative of the Natural Log Derivatives of ln(x), ln[g(x)]
Examples of Logarithmic Differentiation	Implicit Differentiation Find the derivatives of non-functions
Examples using Implicit Differentiation	Second Derivative and Higher Derivatives Find the derivatives of derivatives and their applications
Maxima and Minima Maximum amd Minimum Values, Fermat's Theorem, Critical Number, Extreme Value Theorem, Closed Interval Method	Maximum and Minimum Videos
Finding Critical Numbers	Derivative Test First Derivative Test, Second Derivative Test, Minima, Maxima, Increasing/Decreasing Test
Concavity Problems I Analyze functions through increasing, decreasing, concavity, inflection points, critical points, and extremum	Concavity Problems II Analyze functions through increasing, decreasing, concavity, inflection points, critical points, and extremum
Curve Sketching using Calculus	Mean Value Theorem What is Mean Value Theorem?
Asymptotes Vertical Asymptote, Horizontal Asymptote, Oblique Asymptote	Hyperbolic Functions Hyperbolic Identities, Derivatives of Hyperbolic Functions and Derivatives of Inverse Hyperbolic Functions
L'Hopitals Rule Indeterminate Quotients, Product, Differences, and Power	Optimization Problems using Derivatives
More Optimization Problems using Derivatives	Related Rates using Derivatives
More Related Rates using Derivatives	Newton's Method Used to approximate a root

Integral Calculus

Antiderivative Formulas for powers of x, trigonometric functions, exponential functions	Definite Integral Area, Riemann Sum, Properties of Definite Integral
Approximating Integrals using Rectangles, Trapezoid Rule, Simpsons Rule	Calculating a Definite Integral Using Riemann Sums
The Fundamental Theorem of Calculus What is the Fundamental Theorem of Calculus?	Indefinite Integrals Integral Notation, Integral Formulas
Improper Integrals	Integration by Parts Formula and Examples
Integration by Parts Examples	Integration using U-Substitution
Integration using Partial Fractions	Long Partial Fractions Problem Repeated Irreducible Quadratic Factors
Integrating Exponential Functions	Integrating Hyperbolic Functions
Trigonometric Integrals Integration of the Powers of Sine, Cosine, Tangent and Secant	Integration using Inverse Trigonometric Functions
Integral Test To determine whether a series is convergent or divergent	Area under a Curve Calculate area under a curve and between two curves.
Centroids/Center of Mass Calculate the Centroid of a Region	Volume of Revolution Cylindrical Shells
Volume of Revolution Disk/Washers	Work Done using Calculus Tank Problem
Work Done using Calculus Cable/Rope Problem, Spring Problem	Laplace Transform Basic Idea and Properties
Laplace Transform Proof and Tables	Laplace Transform
Inverse Laplace Transform I	Inverse Laplace Transform II

Solutions to Sample Questions

The following are solutions to sample questions of the CollegeBoard AP/AB and AP/BC Calculus examination

AP / AB Calculus Free Response Question	AP / AB Calculus Test - Sample Questions 1 to 8
AP / AB Calculus Test - Sample Questions 9 to 16	AP / AB Calculus Test - Sample Questions 17 to 24
AP / AB Calculus Test - Sample Questions 25 to 28	AP / BC Calculus Test - Sample Questions 1 to 8
AP / BC Calculus Test - Sample Questions 9 to 16

Sequences and Series Videos

Arithmetic Sequences and Series Review	Geomertic Sequences and Series Review
Absolute Convergence, Conditional Convergence and Divergence Sequences and Series Convergence and Divergence	Alternating Series
Binomial Series	Geometric Series and the Test for Divergence
Integral Test	Limit Comparison Test and Direct Comparison Test
Partial Sums: Showing a Series Diverges	Power Series
Ratio Test Root Test	Taylor and Maclaurin Series
More Taylor and Maclaurin Series	Telescoping Series Strategies for Testing Series

Parametric Curves and Polar Coordinates

Parametric Curves	Calculus with Parametric Curves
Polar Coordinates	Calculus and Area in Polar Coordinates
Graphing Polar Curves

Multivariable Calculus Videos

Vectors and the Geometry of Space

Three-Dimensional Coordinate System	Vectors - Domains and Limits
The Dot Product	The Cross Product
Vector Equation of a Line	Equation of a Plane
Intersection of a Line and Plane	Cylinders and Quadric Surfaces
Cylindrical and Spherical Coordinates

Vector Functions

Vector Functions	Derivatives of Vector Functions
Arc Length of Vector Functions

Partial Derivatives

Partial Derivatives	Implicit Differentiation
Tangent Planes and Linear Approximations	The Chain Rule
Directional Derivatives and the Gradient Vector	Maximum and Minimum Values
Absolute Maximum/Minimum Values of Multivariable Functions	Lagrange Multipliers

Multiple Integrals

Double Integrals	Iterated Integrals
Double Integrals over General Regions	Double Integrals in Polar Coordinates
Applications of Double Integrals	Triple Integrals
Triple Integrals in Spherical Coordinates	Change of Variables in Multiple Integrals

Vector Calculus

Vector Fields	Line Integrals
Applications of Line Integrals	The Fundamental Theorem for Line Integrals
Green's Theorem	Curl of a Vector Field
Use of Curl to Show that a Vector Field is Conservative	Divergence of a Vector Field
Surface Integrals

Differential Equations

Introduction to differential equations Separable Differential Equations	Exact Equations Intuition Exact Equations Examples
Integrating factors	First Order Linear Differential Equations
First Order Homogenous Differential Equations	2nd Order Linear Homogeneous Differential Equations
Homogeneous Second-Order Differential Equations	Non-homogeneous Second-Order Differential Equations
Homogeneous Differential Equations Change of Variables	Complex roots of the characteristic equations
Repeated roots of the characteristic equation	Undetermined Coefficients
Laplace Transform 1	Laplace Transform 2
Laplace Transform to solve an equation More Laplace Transform tools Using the Laplace Transform	Laplace Transform of : L{t} Laplace Transform of tⁿ: L{tⁿ}
Laplace Transform of the Unit Step Function Inverse Laplace Examples Laplace/Step Function Differential Equation	Dirac Delta Function Function Laplace Transform of the Dirac Delta Function
Introduction to the Convolution The Convolution and the Laplace Transform Using the Convolution Theorem

Videos

Big Picture of Calculus
Calculus is about change. One function tells how quickly another function is changing. Professor Strang shows how calculus applies to ordinary life situations, such as: * driving a car * climbing a mountain * growing to full adult height

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Big Picture: Derivatives
Calculus finds the relationship between the distance traveled and the speed - easy for constant speed, not so easy for changing speed. Professor Strang is finding the "rate of change" and the "slope of a curve" and the "derivative of a function."

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Big Picture: Integrals
The second half of calculus looks for the distance traveled even when the speed is changing. Finding this "integral" is the opposite of finding the derivative. Professor Strang explains how the integral adds up little pieces to recover the total distance. I know the speed at each moment of my trip, so how far did I go?

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Highlights of Calculus Part 1 (More lectures by Prof. Strang)

Highlights of Calculus Part 2

Calculus Calculator with step by step solutions
Functions, Operations on Functions,
Polynomial and Rational Functions,
Exponential and Logarithmic Functions,
Sequences and Series
Evaluating Limits, Derivatives,
Applications of Differentiation, Integrals,
Applications of Integration,
Techniques of Integration,
Parametric Equations and Polar Coordinates

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