# 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.

### 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 ex, ax, ag(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

### 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 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." 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?
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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