# Calculus - Khan Academy

Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

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## Lessons hide description |
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## 1) Newton Leibniz and Usain Bolt## Why we study differential calculus More free lessons at: http://www.khanacademy.org/video?v=EKvHQ |

## 2) Introduction to limits | Limits | Differential Calculus## Introduction to limits |

## 3) Introduction to limits 2 | Limits | Precalculus## Introduction to the intuition behind limits |

## 4) Limit examples (part 1) | Limits | Differential Calculus## Some limit exercises |

## 5) Limit examples (part 2) | Limits | Differential Calculus## More limit examples |

## 6) Limit examples (part 3) | Limits | Differential Calculus## More limit examples |

## 7) Limit examples w/ brain malfunction on first prob (part 4) | Differential Calculus## 3 interesting limit examples (correct answer for problem 1 is 3/16 (6/(4*8) NOT 6/(4+8)) |

## 9) Proof: lim (sin x)/x | Limits | Differential Calculus## Using the squeeze theorem to prove that the limit as x approaches 0 of (sin x)/x =1 |

## 10) More limits | Limits | Differential Calculus## More limit examples |

## 11) Epsilon-delta limit definition 1 | Limits | Differential Calculus## Introduction to the Epsilon Delta Definition of a Limit. |

## 12) Epsilon-delta limit definition 2 | Limits | Differential Calculus## Using the epsilon delta definition to prove a limit |

## 13) Derivative as slope of a tangent line | Taking derivatives | Differential Calculus## Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line) |

## 14) Calculating slope of tangent line using derivative definition | Differential Calculus## Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 15) The derivative of f(x)=x^2 for any x | Taking derivatives | Differential Calculus## Watch the next lesson: https://www.khanacademy.org/math/differe ... |

## 16) Derivative intuition module | Taking derivatives | Differential Calculus## Sal walking through the derivative intuition module made by Ben Eater Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 17) Calculus: Derivatives 1 | Taking derivatives | Differential Calculus## Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus. |

## 18) Calculus: Derivatives 2 | Taking derivatives | Differential Calculus## More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2 |

## 19) Power rule introduction (old) | Taking derivatives | Differential Calculus## Determining the derivatives of simple polynomials. |

## 20) The Chain Rule## Part 4 of derivatives. Introduction to the chain rule. |

## 21) Chain Rule Examples## Examples using the Chain Rule |

## 22) Even More Chain Rule## Even more examples using the chain rule. |

## 25) Derivatives (part 9)## More examples of taking derivatives |

## 28) Proof: d/dx(ln x) = 1/x | Taking derivatives | Differential Calculus## Taking the derivative of ln x |

## 29) Proof: d/dx(e^x) = e^x | Taking derivatives | Differential Calculus## Proof that the derivative of e^x is e^x. |

## 30) Proofs of derivatives of ln(x) and e^x | Taking derivatives | Differential Calculus## Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular". |

## 32) Implicit Differentiation## Taking the derivative when y is defined implicitly. |

## 33) Implicit Differentiation (part 2)## A hairier implicit differentiation problem. |

## 34) More implicit differentiation## 2 more implicit differentiation examples |

## 35) More chain rule and implicit differentiation intuition## More intuition behind the chain rule and how it applies to implicit differentiation. |

## 36) Trig Implicit Differentiation Example## Implicit differentiation example that involves the tangent function |

## 37) Derivative of x^(x^x) | Taking derivatives | Differential Calculus## Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 38) Introduction to l'Hôpital's rule | Derivative applications | Differential Calculus## Introduction to L'Hôpital's Rule Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 39) L'Hôpital's rule example 1 | Derivative applications | Differential Calculus## L'Hôpital's Rule Example 1 Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 40) L'Hôpital's rule example 2 | Derivative applications | Differential Calculus## L'Hôpital's Rule Example 2 Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 41) L'Hôpital's rule example 3 | Derivative applications | Differential Calculus## L'Hôpital's Rule Example 3 Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 42) Maxima Minima Slope Intuition## Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima. |

## 43) Inflection Points and Concavity Intuition## Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition |

## 44) Monotonicity Theorem## Using the monotonicity theorem to determine when a function is increasing or decreasing. |

## 45) Calculus: Maximum and minimum values on an interval## 2 examples of finding the maximum and minimum points on an interval. |

## 46) Graphing using derivatives | Derivative applications | Differential Calculus## Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/diffe ... |

## 47) Another example graphing with derivatives | Differential Calculus |

## 48) Graphing with Calculus## More graphing with calculus. |

## 49) Optimization with Calculus 1## Find two numbers whose products is -16 and the sum of whose squares is a minimum. |

## 50) Optimization with Calculus 2## Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides. |

## 51) Optimization with Calculus 3## A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two are ... |

## 52) Optimization Example 4## Minimizing the cost of material for an open rectangular box. |

## 53) Introduction to rate-of-change problems## Using derivatives to solve rate-of-change problems |

## 55) Rates-of-change (part 2)## Another (simpler) example of using the chain rule to determine rates-of-change |

## 56) Ladder rate-of-change problem## The classic falling ladder problem |

## 57) Mean value theorem | Derivative applications | Differential Calculus## Intuition behind the Mean Value Theorem |

## 58) The Indefinite Integral or Anti-derivative## An introduction to indefinite integration of polynomials. |

## 59) Indefinite integrals (part II)## Examples of taking the indefinite integral (or anti-derivative) of polynomials. |

## 60) Indefinite Integration (part III)## Integration by doing the chain rule in reverse. |

## 61) Indefinite Integration (part IV)## Integration by substitution (or the reverse-chain-rule) |

## 62) Indefinite Integration (part V)## Introduction to Integration by Parts (kind of the reverse-product rule) |

## 63) Integration by Parts (part 6 of Indefinite Integration)## Example using Integration by Parts |

## 64) Indefinite Integration (part 7)## Another example using integration by parts. |

## 65) Another u-subsitution example## Finding the antiderivative using u-substitution. |

## 66) Introduction to definite integrals## Using the definite integral to solve for the area under a curve. Intuition on why the antiderivative is the same thing as the area under a curve. |

## 67) Definite integrals (part II)## More on why the antiderivative and the area under a curve are essentially the same thing. |

## 68) Definite Integrals (area under a curve) (part III)## More on why the antiderivative and the area under a curve are essentially the same thing. |

## 69) Definite Integrals (part 4)## Examples of using definite integrals to find the area under a curve |

## 70) Definite Integrals (part 5)## More examples of using definite integrals to calculate the area between curves |

## 71) Definite integral with substitution## Solving a definite integral with substitution (or the reverse chain rule) |

## 72) Integrals: Trig Substitution 1## Example of using trig substitution to solve an indefinite integral More free lessons at: http://www.khanacademy.org/video?v=n4EK9 |

## 73) Integrals: Trig Substitution 2## Another example of finding an anti-derivative using trigonometric substitution More free lessons at: http://www.khanacademy.org/video?v=fD7Mb |

## 74) Integrals: Trig Substitution 3 (long problem)## Example using trig substitution (and trig identities) to solve an integral. More free lessons at: http://www.khanacademy.org/video?v=sw2p2 |

## 75) Periodic Definite Integral## 2010 IIT JEE Paper 1 Problem 52 Periodic Definite Integral. The second term at about minute 14 should have a positive sign. Luckily, it doesn't effect the final answer! More free lessons at: http:/ ... |

## 76) Simple Differential Equations## 3 basic differential equations that can be solved by taking the antiderivatives of both sides. |

## 77) Solid of Revolution (part 1)## Figuring out the volume of a function rotated about the x-axis. More free lessons at: http://www.khanacademy.org/video?v=R_aqS |

## 78) Solid of Revolution (part 2)## The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis More free lessons at: http://www.khanacademy.org/video?v=iUzfs |

## 79) Solid of Revolution (part 3)## Figuring out the equation for the volume of a sphere. More free lessons at: http://www.khanacademy.org/video?v=tqfU9 |

## 80) Solid of Revolution (part 4)## More volumes around the x-axis. More free lessons at: http://www.khanacademy.org/video?v=OtmjN |

## 81) Solid of Revolution (part 5)## Use the "shell method" to rotate about the y-axis More free lessons at: http://www.khanacademy.org/video?v=NIdqk |

## 82) Solid of Revolution (part 6)## Using the disk method around the y-axis. More free lessons at: http://www.khanacademy.org/video?v=F2psx |

## 83) Solid of Revolution (part 7)## Taking the revolution around something other than one of the axes. More free lessons at: http://www.khanacademy.org/video?v=IZ8W- |

## 84) Solid of Revolution (part 8)## The last part of the problem in part 7 More free lessons at: http://www.khanacademy.org/video?v=4Flj9 |

## 85) Sequences and Series (part 1)## Introduction to the arithmetic and geometric series |

## 86) Sequences and series (part 2)## Finding the sum of an infinite geometric series. |

## 87) Maclauren and Taylor Series Intuition## Approximating a function at 0 using a polynomial More free lessons at: http://www.khanacademy.org/video?v=epgwu |

## 88) Cosine Taylor Series at 0 (Maclaurin)## Approximating f(x)=cos x using a Maclauren Series (special case of a Taylor series at x=0) More free lessons at: http://www.khanacademy.org/video?v=WWe7p |

## 89) Sine Taylor Series at 0 (Maclaurin)## Sine Taylor Series at 0 (Maclaurin) More free lessons at: http://www.khanacademy.org/video?v=LlKvu |

## 90) Taylor Series at 0 (Maclaurin) for e to the x## Taylor Series at 0 (Maclaurin) for e to the x More free lessons at: http://www.khanacademy.org/video?v=JYQqm |

## 91) Euler's Formula and Euler's Identity## Rationale for Euler's Formula and Euler's Identity More free lessons at: http://www.khanacademy.org/video?v=mgNtP |

## 92) Visualizing Taylor Series Approximations## Using Wolfram Alpha to approximate sin(x) More free lessons at: http://www.khanacademy.org/video?v=8dMLK |

## 93) Generalized Taylor Series Approximation## Approximating a function around a non-zero x value More free lessons at: http://www.khanacademy.org/video?v=1LxhX |

## 94) Visualizing Taylor Series for e^x## Visualizing Taylor Series for e^x More free lessons at: http://www.khanacademy.org/video?v=AFMXi |

## 95) Polynomial approximation of functions (part 1)## Using a polynomial to approximate a function at f(0). More free lessons at: http://www.khanacademy.org/video?v=sy132 |

## 96) Polynomial approximation of functions (part 2)## Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series) More free lessons at: http://www.khanacademy.org/video?v=3JG3q |

## 97) Approximating functions with polynomials (part 3)## A glimpse of the mystery of the Universe as we approximate e^x with an infinite series. More free lessons at: http://www.khanacademy.org/video?v=XZDGr |

## 98) Polynomial approximation of functions (part 4)## Approximating cos x with a Maclaurin series. More free lessons at: http://www.khanacademy.org/video?v=gcJeg |

## 99) Polynomial approximations of functions (part 5)## MacLaurin representation of sin x More free lessons at: http://www.khanacademy.org/video?v=9AoDu |

## 100) Polynomial approximation of functions (part 6)## A pattern emerges! More free lessons at: http://www.khanacademy.org/video?v=-gRNR |

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