# Calculus

Publicat: 19 Apr 2011 00:00 By Salman Khan - Khan Academy
Course Description:
Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists).
Lectures:

Lecture 1 - Limit Examples Part 4

More limit examples.

Lecture 2 - Epsilon Delta Limit Definition Part 1

Introduction to the Epsilon Delta Definition of a Limit.

Lecture 3 - Epsilon Delta Limit Definition Part 2

Using the epsilon delta definition to prove a limit.

Lecture 4 - Calculus: Derivatives 1

Calculus-Derivative: Understanding that the derivative is just the slope of a curve at a point (or the slope of the tangent line).

Lecture 5 - Calculus: Derivatives 2

Calculus-Derivative: Finding the slope (or derivative) of a curve at a particular point.

Lecture 6 - Calculus: Derivatives 2.5

Calculus-Derivative: Finding the derivative of y=x^2

Lecture 7 - Derivatives Part 1

Finding the slope of a tangent line to a curve (the derivative). Introduction to Calculus.

Lecture 8 - Derivatives Part 2

More intuition of what a derivative is. Using the derivative to find the slope at any point along f(x)=x^2.

Lecture 9 - Derivatives Part 3

Determining the derivatives of simple polynomials.

Lecture 10 - Derivatives Part 4

Part 4 of derivatives. Introduction to the chain rule.

Lecture 11 - Derivatives Part 5

Examples using the Chain Rule.

Lecture 12 - Derivatives Part 6

Even more examples using the chain rule.

Lecture 13 - Derivatives Part 7

The product rule. Examples using the Product and Chain rules.

Lecture 14 - Derivatives Part 8

Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x.

Lecture 15 - Derivatives Part 9

More examples of taking derivatives.

Lecture 16 - Proof: d/dx(x^n)

Proof that d/dx(x^n) = n*x^(n-1).

Lecture 17 - Proof: d/dx(sqrt(x))

Proof that d/dx (x^.5) = .5x^(-.5).

Lecture 18 - Proof: d/dx(ln x) = 1/x

Taking the derivative of ln x.

Lecture 19 - Proof: d/dx(e^x) = e^x

Proof that the derivative of e^x is e^x.

Lecture 20 - Proofs of Derivatives of Ln(x) and e^x

Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".

Lecture 21 - Extreme Derivative Word Problem

A difficult but interesting derivative word problem.

Lecture 22 - Implicit Differentiation Part 1

Taking the derivative when y is defined implicitly.

Lecture 23 - Implicit Differentiation Part 2

A hairier implicit differentiation problem.

Lecture 24 - More Implicit Differentiation

2 more implicit differentiation examples.

Lecture 25 - More Chain Rule and Implicit Differentiation Intuition

More intuition behind the chain rule and how it applies to implicit differentiation.

Lecture 26 - Trig Implicit Differentiation Example

Implicit differentiation example that involves the tangent function.

Lecture 27 - Derivative of x^(x^x)

Calculus: Derivative of x^(x^x).

Lecture 28 - Maxima Minima Slope Intuition

Intuition on what happens to the slope/derivative and second derivatives at local maxima and minima.

Lecture 29 - Inflection Points and Concavity Intuition

Understanding concave upwards and downwards portions of graphs and the relation to the derivative. Inflection point intuition.

Lecture 30 - Monotonicity Theorem

Using the monotonicity theorem to determine when a function is increasing or decreasing.

Lecture 31 - Maximum and Minimum Values on an Interval

2 examples of finding the maximum and minimum points on an interval.

Lecture 32 - Graphing Using Derivatives

Graphing functions using derivatives.

Lecture 33 - Graphing with Derivatives Example

Using the first and second derivatives to identify critical points and inflection points and to graph the function.

Lecture 34 - Graphing with Calculus

More graphing with calculus.

Lecture 35 - Optimization with Calculus Part 1

Find two numbers whose products is -16 and the sum of whose squares is a minimum.

Lecture 36 - Optimization with Calculus Part 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.

Lecture 37 - Optimization with Calculus Part 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 areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.).

Lecture 38 - Optimization with Calculus Part 4

Minimizing the cost of material for an open rectangular box.

Lecture 39 - Introduction to Rate-of-change

Using derivatives to solve rate-of-change problems.

Lecture 40 - Equation of a Tangent Line

Finding the equation of the line tangent to f(x)=xe^x when x=1.

Lecture 41 - Rates-of-change Part 2

Another (simpler) example of using the chain rule to determine rates-of-change.

Lecture 42 - Ladder rate-of-change Problem

The classic falling ladder problem.

Lecture 43 - Mean Value Theorem

Intuition behind the Mean Value Theorem.

Lecture 44 - Indefinite Integrals Part 1

An introduction to indefinite integration of polynomials.

Lecture 45 - Indefinite Integrals Part 2

Examples of taking the indefinite integral (or anti-derivative) of polynomials.

Lecture 46 - Indefinite Integrals Part 3

Integration by doing the chain rule in reverse.

Lecture 47 - Indefinite Integrals Part 4

Integration by substitution (or the reverse-chain-rule).

Lecture 48 - Indefinite Integrals Part 5

Introduction to Integration by Parts (kind of the reverse-product rule).

Lecture 49 - Indefinite Integrals Part 6

Example using Integration by Parts.

Lecture 50 - Indefinite Integrals Part 7

Another example using integration by parts.

Lecture 51 - Another U-substitution Example

Finding the antiderivative using u-substitution.

Lecture 52 - Definite Integrals Part 1

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.

Lecture 53 - Definite Integrals Part 2

More on why the antiderivative and the area under a curve are essentially the same thing.

Lecture 54 - Definite Integrals Part 3

More on why the antiderivative and the area under a curve are essentially the same thing.

Lecture 55 - Definite Integrals Part 4

Examples of using definite integrals to find the area under a curve.

Lecture 56 - Definite Integrals Part 5

More examples of using definite integrals to calculate the area between curves.

Lecture 57 - Definite Integral with Substitution

Solving a definite integral with substitution (or the reverse chain rule).

Lecture 58 - Integrals: Trig Substitution Part 1

Example of using trig substitution to solve an indefinite integral.

Lecture 59 - Integrals: Trig Substitution Part 2

Another example of finding an anti-derivative using trigonometric substitution.

Lecture 60 - Integrals: Trig Substitution Part 3

Example using trig substitution (and trig identities) to solve an integral.

Lecture 61 - Introduction to Differential Equations

3 basic differential equations that can be solved by taking the antiderivatives of both sides.

Lecture 62 - Solid of Revolution Part 1

Figuring out the volume of a function rotated about the x-axis.

Lecture 63 - Solid of Revolution Part 2

The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis.

Lecture 64 - Solid of Revolution Part 3

Figuring out the equation for the volume of a sphere.

Lecture 65 - Solid of Revolution Part 4

More volumes around the x-axis.

Lecture 66 - Solid of Revolution Part 5

Use the "shell method" to rotate about the y-axis.

Lecture 67 - Solid of Revolution Part 6

Using the disk method around the y-axis.

Lecture 68 - Solid of Revolution Part 7

Taking the revolution around something other than one of the axes.

Lecture 69 - Solid of Revolution Part 8

The last part of the problem in part 7.

Lecture 70 - Polynomial Approximation of Functions Part 1

Using a polynomial to approximate a function at f(0).

Lecture 71 - Polynomial Approximation of Functions Part 2

Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series).

Lecture 72 - Polynomial Approximation of Functions Part 3

A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.

Lecture 73 - Polynomial Approximation of Functions Part 4

Approximating cos x with a Maclaurin series.

Lecture 74 - Polynomial Approximation of Functions Part 5

MacLaurin representation of sin x.

Lecture 75 - Polynomial Approximation of Functions Part 6

A pattern emerges!

Lecture 76 - Polynomial Approximation of Functions Part 7

The most amazing conclusion in mathematics!

Lecture 77 - Taylor Polynomials

Approximating a function with a Taylor Polynomial.

Lecture 78 - AP Calculus BC Exams: 2008 1 A

Part 1a of the 2008 BC free response.

Lecture 79 - AP Calculus BC Exams: 2008 1 B & C

Parts b and c of problem 1 (free response).

Lecture 80 - AP Calculus BC Exams: 2008 1 C & D

Parts c&d of problem 1 in the 2008 AP Calculus BC free response.

Lecture 81 - AP Calculus BC Exams: 2008 1 D

Part 1d of the 2008 AP Calculus BC exam (free response).

Lecture 82 - Calculus BC 2008 2 A

2a of 2008 Calculus BC exam (free response).

Lecture 83 - Calculus BC 2008 2 B & C

Parts 2b and 2c of the 2008 BC exam (free response).

Lecture 84 - Calculus BC 2008 2 D

Part 2d of the 2008 Calculus BC exam free-response section.

Lecture 85 - Partial Derivatives Part 1

Introduction to partial derivatives.

Lecture 86 - Partial Derivatives Part 2

More on partial derivatives.

Lecture 87 - Gradient

Introduction to the gradient.

Lecture 88 - Gradient of a Scalar Field

Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.

Lecture 89 - Divergence Part 1

Introduction to the divergence of a vector field.

Lecture 90 - Divergence Part 2

The intuition of what the divergence of a vector field is.

Lecture 91 - Divergence Part 3

Analyzing a vector field using its divergence.

Lecture 92 - Curl Part 1

Introduction to the curl of a vector field.

Lecture 93 - Curl Part 2

The mechanics of calculating curl.

Lecture 94 - Curl Part 3

More on curl.

Lecture 95 - Double Integrals Part 1

Introduction to the double integral.

Lecture 96 - Double Integrals Part 2

Figuring out the volume under z=xy^2.

Lecture 97 - Double Integrals Part 3

Let's integrate dy first!

Lecture 98 - Double Integrals Part 4

Another way to conceptualize the double integral.

Lecture 99 - Double Integrals Part 5

Finding the volume when we have variable boundaries.

Lecture 100 - Double Integrals Part 6

Let's evaluate the double integrals with y=x^2 as one of the boundaries.

Lecture 101 - Triple Integrals Part 1

Introduction to the triple integral.

Lecture 102 - Triple Integrals Part 2

Using a triple integral to find the mass of a volume of variable density.

Lecture 103 - Triple Integrals Part 3

Figuring out the boundaries of integration.

Lecture 104 - (2^ln x)/x Antiderivative Example

Finding ?(2^ln x)/x dx.

Lecture 105 - Line Integrals

Introduction to the Line Integral.

Lecture 106 - Line Integral Example 1

Concrete example using a line integral.

Lecture 107 - Line Integral Example 2 Part 1

Line integral over a closed path (part 1).

Lecture 108 - Line Integral Example 2 Part 2

Part 2 of an example of taking a line integral over a closed path.

Lecture 109 - Position Vector Valued Functions

Using a position vector valued function to describe a curve or path.

Lecture 110 - Derivative of a Position Vector Valued Function

Visualizing the derivative of a position vector valued function.

Lecture 111 - Differential of a Vector Valued Function

Understanding the differential of a vector valued function.

Lecture 112 - Differential of a Vector Valued Function Example

Concrete example of the derivative of a vector valued function to better understand what it means.

Lecture 113 - Line Integrals and Vector Fields

Using line integrals to find the work done on a particle moving through a vector field.

Lecture 114 - Using a Line Integral to Find a Vector Field Example

Using a line integral to find the work done by a vector field example.

Lecture 115 - Parametrization of a Reverse Path

Understanding how to parametrize a reverse path for the same curve.

Lecture 116 - Scalar Field Line Integral Independent of Path Direction

Showing that the line integral of a scalar field is independent of path direction.

Lecture 117 - Vector Field Line Integral Dependent of Path Direction

Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.

Lecture 118 - Path Independence for Line Integrals

Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.

Lecture 119 - Closed Curve Line Integrals of Conservative Vector Fields

Showing that the line integral along closed curves of conservative vector fields is zero.

Lecture 120 - Example of Closed Line Integral of Conservative Field

Example of taking a closed line integral of a conservative field.

Lecture 121 - Second Example of Line Integral of Conservative Vector Field

Using path independence of a conservative vector field to solve a line integral.

Lecture 122 - Green's Theorem Proof Part 1

Part 1 of the proof of Green's Theorem.

Lecture 123 - Green's Theorem Proof Part 2

Part 2 of the proof of Green's Theorem.

Lecture 124 - Green's Theorem Example Part 1

Using Green's Theorem to solve a line integral of a vector field.

Lecture 125 - Green's Theorem Example Part 2

Another example applying Green's Theorem.

Lecture 126 - Introduction to Parametrizing a Surface with Two Parameters

Introduction to Parametrizing a Surface with Two Parameters.

Lecture 127 - Position Vector-Valued Function for a Parametrization of Two Parameters

Determining a Position Vector-Valued Function for a Parametrization of Two Parameters.

Lecture 128 - Partial Derivatives of Vector-Valued Functions

Partial Derivatives of Vector-Valued Functions.

Lecture 129 - Introduction to the Surface Integral

Introduction to the surface integral.

Lecture 130 - Calculating a Surface Integral Example Part 1

Example of calculating a surface integral part 1.

Lecture 131 - Calculating a Surface Integral Example Part 1

Example of calculating a surface integral part 2.

Lecture 132 - Calculating a Surface Integral Example Part 1

Example of calculating a surface integral part 3.

Lecture 133 - L'Hopital's Rule

Introduction to L'Hopital's Rule.

Lecture 134 - L'Hopital's Rule Example 1

L'Hopital's Rule Example 1.

Lecture 135 - L'Hopital's Rule Example 2

L'Hopital's Rule Example 2.

Lecture 136 - L'Hopital's Rule Example 3

L'Hopital's Rule Example 1.

Source: http://academicearth.org/courses/calculus

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