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 Free Response Past Papers and Solutions

• AP Calculus AB Free Response Solutions
  AP Calculus AB 2019 Free Response Solutions
  AP Calculus AB 2018 Free Response Solutions
  AP Calculus AB 2017 Free Response Solutions
  AP Calculus AB 2016 Free Response Solutions
  AP Calculus AB 2015 Free Response Solutions
  AP Calculus AB 2014 Free Response Solutions
  AP Calculus AB 2013 Free Response Solutions
  AP Calculus AB 2012 Free Response Solutions
  AP Calculus AB 2011 Free Response Solutions
  AP Calculus AB 2003 Free Response Solutions
• AP Calculus BC Free Response Solutions
  AP Calculus BC 2019 Free Response Solutions
  AP Calculus BC 2018 Free Response Solutions
  AP Calculus BC 2017 Free Response Solutions
  AP Calculus BC 2016 Free Response Solutions
  AP Calculus BC 2015 Free Response Solutions
  AP Calculus BC 2014 Free Response Solutions

AP Calculus AB and BC Multiple Choice Past Papers & Solutions

• AP Calculus AB Multiple Choice Solutions
  AP Calculus AB 2012 Multiple Choice Solutions Part A
  AP Calculus AB 2012 Multiple Choice Solutions Part B
  AP Calculus AB 2008 Multiple Choice Solutions Part A
  AP Calculus AB 2008 Multiple Choice Solutions Part B
  AP Calculus AB 1998 Multiple Choice Solutions Part A
  AP Calculus AB 1998 Multiple Choice Solutions Part B
• AP Calculus BC Multiple Choice Solutions
  AP Calculus BC 2012 Multiple Choice Solutions Part A
  

Solutions to Sample Questions

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

• AP Calculus AB & BC sample questions & solutions
  AP Calculus BC Sample Free Response Solutions 2020
  AP / AB Calculus Free Response Solutions
  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

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.

• Highlights of Calculus
  Highlights of Calculus (Part 1)
  Big Picture of Calculus
  Big Picture: Derivatives
  Max and Min & Second Derivative
  The Exponential Function
  Big Picture: Integrals
  Derivative of sin x & cos x
  Product Rule & Quotient Rule
  Chains f(g(x)) & the Chain Rule
  
  Highlights of Calculus (Part 2)
  Limits & 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
• Limits
  Introduction to Calculus
  The Limits of a Function Definition & Techniques to find Limits
  Mean Value Theorem What is Mean Value Theorem?
  L'Hopitals Rule Indeterminate Quotients, Product, Differences, & Power
  Newton's Method Used to approximate a root
  
• Derivative Rules
  Derivatives Definition & 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 & Logarithmic Functions, Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions & Inverse Hyperbolic Functions
  Examples using the Derivative Rules
  More Examples using the Derivative Rules
• Trigonometric Derivatives
  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
  Deriving the Derivative Formulas for Tan, Cot, Sec, Cosecant, Arctan
• Exponential & Logarithmic Differentiation
  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
  Implicit Differentiation Find the derivatives of non-functions
  Examples using Implicit Differentiation
• Derivatives of Hyperbolic Functions
  Hyperbolic Functions Hyperbolic Identities, Derivatives of Hyperbolic Functions & Derivatives of Inverse Hyperbolic Functions
• Derivative Tests, Maxima & Minima
  Second Derivative & Higher Derivatives Find the derivatives of derivatives & their applications
  Maxima and Minima Maximum amd Minimum Values, Fermat's Theorem, Critical Number, Extreme Value Theorem, Closed Interval Method
  Maximum & Minimum
  Finding Critical Numbers
  Derivative Test First Derivative Test, Second Derivative Test, Minima, Maxima, Increasing/Decreasing Test
  
• Concavity Problems
  Concavity Problems I Analyze functions through increasing, decreasing, concavity, inflection points, critical points, & extremum
  Concavity Problems II Analyze functions through increasing, decreasing, concavity, inflection points, critical points, & extremum
  
• Optimization & Rate Problems
  Optimization Problems using Derivatives
  More Optimization Problems using Derivatives
  Related Rates using Derivatives
  More Related Rates using Derivatives
• Curve Sketching & Asymptotes
  Curve Sketching using Calculus
  Asymptotes Vertical Asymptote, Horizontal Asymptote, Oblique Asymptote

Integral Calculus

• Antiderivatives & Integrals
  The Fundamental Theorem of Calculus What is the Fundamental Theorem of Calculus?
  Approximating Integrals using Rectangles, Trapezoid Rule, Simpsons Rule
  Calculating a Definite Integral Using Riemann Sums
  Definite Integral Area, Riemann Sum, Properties of Definite Integral
  
  Antiderivative Formulas for powers of x, trigonometric functions, exponential functions
  Indefinite Integrals Integral Notation, Integral Formulas
  Improper Integrals
  
• Integration Methods
  Integration by Parts Formula & Examples
  Integration by Parts Examples
  Integration using U-Substitution
  Integration using Partial Fractions
  Long Partial Fractions Problem Repeated Irreducible Quadratic Factors
• Area, Volume & Work Done using Integration
  Area under a Curve Calculate area under a curve & 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
• Integrating Exponential, Hyperbolic & Trigonometric Functions
  Integrating Exponential Functions
  Integrating Hyperbolic Functions
  Trigonometric Integrals Integration of the Powers of Sine, Cosine, Tangent & Secant
  Integration using Inverse Trigonometric Functions

Sequences, Series & Tests

• Sequences & Series
  Arithmetic Sequences & Series Review
  Geometric Sequences & Series Review
  Alternating Series
  Binomial Series Power Series
  Taylor & Maclaurin Series
  More Taylor & Maclaurin Series
  
• Convergence & Divergence Tests
  Absolute Convergence, Conditional Convergence & Divergence Sequences & Series Convergence & Divergence
  Geometric Series & the Test for Divergence
  Integral Test To determine whether a series is convergent or divergent
  Integral Test
  
  Limit Comparison Test & Direct Comparison Test
  Partial Sums: Showing a Series Diverges
  Ratio Test Root Test
  Telescoping Series
  Strategies for Testing Series

Parametric Curves & Polar Coordinates

• Parametric Curves & Polar Coordinates
  Parametric Curves
  Calculus with Parametric Curves
  Polar Coordinates
  Calculus & Area in Polar Coordinates
  Graphing Polar Curves

Multivariable Calculus

• Vectors & the Geometry of Space
  Three-Dimensional Coordinate System
  Vectors: Domains & Limits
  The Dot Product
  The Cross Product
  
  Vector Equation of a Line
  Equation of a Plane
  Intersection of a Line & Plane
  Cylinders & Quadric Surfaces
  Cylindrical & Spherical Coordinates
• Vector Functions
  Vector Functions
  Derivatives of Vector Functions
  Arc Length of Vector Functions
• Partial Derivatives
  Partial Derivatives
  Implicit Differentiation
  Tangent Planes & Linear Approximations
  The Chain Rule
  Directional Derivatives & the Gradient Vector
  
  Maximum & 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 Homogeneous 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
  Laplace Transform Basic Idea & Properties
  Laplace Transform Proof & Tables
  Laplace Transform
  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 Laplace Transform Using the Convolution Theorem Inverse Laplace Transform I
  Inverse Laplace Transform II
  

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

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