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Course

# Calculus

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).

Home > Mathematics > Calculus Lectures:
• ### Epsilon Delta Limit Definition Part 1

00:12:48Salman Khan

Introduction to the Epsilon Delta Definition of a Limit.

• ### Epsilon Delta Limit Definition Part 2

00:10:56Salman Khan

Using the epsilon delta definition to prove a limit.

• ### Calculus: Derivatives 1

00:15:43Salman Khan

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

• ### Calculus: Derivatives 2

00:08:28Salman Khan

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

• ### Calculus: Derivatives 2.5

00:11:05Salman Khan

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

• ### Derivatives Part 1

00:09:24Salman Khan

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

• ### Derivatives Part 2

00:09:30Salman Khan

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

• ### Derivatives Part 3

00:09:50Salman Khan

Determining the derivatives of simple polynomials.

• ### Derivatives Part 4

00:09:11Salman Khan

Part 4 of derivatives. Introduction to the chain rule.

• ### Derivatives Part 5

00:09:23Salman Khan

Examples using the Chain Rule.

• ### Derivatives Part 6

00:09:49Salman Khan

Even more examples using the chain rule.

• ### Derivatives Part 7

00:08:48Salman Khan

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

• ### Derivatives Part 8

00:09:31Salman Khan

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.

• ### Derivatives Part 9

00:09:55Salman Khan

More examples of taking derivatives.

• ### Proof: d/dx(x^n)

00:07:03Salman Khan

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

• ### Proof: d/dx(sqrt(x))

00:05:07Salman Khan

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

• ### Proof: d/dx(ln x) = 1/x

00:09:52Salman Khan

Taking the derivative of ln x.

• ### Proof: d/dx(e^x) = e^x

00:04:40Salman Khan

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

• ### Proofs of Derivatives of Ln(x) and e^x

00:12:27Salman Khan

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

• ### Extreme Derivative Word Problem

00:23:13Salman Khan

A difficult but interesting derivative word problem.

• ### Implicit Differentiation Part 1

00:15:57Salman Khan

Taking the derivative when y is defined implicitly.

• ### Implicit Differentiation Part 2

00:10:44Salman Khan

A hairier implicit differentiation problem.

• ### More Implicit Differentiation

00:11:30Salman Khan

2 more implicit differentiation examples.

• ### More Chain Rule and Implicit Differentiation Intuition

00:10:10Salman Khan

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

• ### Trig Implicit Differentiation Example

00:12:08Salman Khan

Implicit differentiation example that involves the tangent function.

• ### Derivative of x^(x^x)

00:09:02Salman Khan

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

• ### Maxima Minima Slope Intuition

00:09:37Salman Khan

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

• ### Inflection Points and Concavity Intuition

00:13:43Salman Khan

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

• ### Monotonicity Theorem

00:09:14Salman Khan

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

• ### Maximum and Minimum Values on an Interval

00:11:42Salman Khan

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

• ### Graphing Using Derivatives

00:20:31Salman Khan

Graphing functions using derivatives.

• ### Graphing with Derivatives Example

00:25:08Salman Khan

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

• ### Graphing with Calculus

00:09:44Salman Khan

More graphing with calculus.

• ### Optimization with Calculus Part 1

00:09:50Salman Khan

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

• ### Optimization with Calculus Part 2

00:11:56Salman Khan

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.

• ### Optimization with Calculus Part 3

00:17:29Salman Khan

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.).

• ### Optimization with Calculus Part 4

00:09:48Salman Khan

Minimizing the cost of material for an open rectangular box.

• ### Introduction to Rate-of-change

00:09:42Salman Khan

Using derivatives to solve rate-of-change problems.

• ### Equation of a Tangent Line

00:08:07Salman Khan

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

• ### Rates-of-change Part 2

00:05:37Salman Khan

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

• ### Ladder rate-of-change Problem

00:10:59Salman Khan

The classic falling ladder problem.

• ### Mean Value Theorem

00:16:48Salman Khan

Intuition behind the Mean Value Theorem.

• ### Indefinite Integrals Part 1

00:09:27Salman Khan

An introduction to indefinite integration of polynomials.

• ### Indefinite Integrals Part 2

00:09:28Salman Khan

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

• ### Indefinite Integrals Part 3

00:09:50Salman Khan

Integration by doing the chain rule in reverse.

• ### Indefinite Integrals Part 4

00:09:48Salman Khan

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

• ### Indefinite Integrals Part 5

00:09:10Salman Khan

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

• ### Indefinite Integrals Part 6

00:09:33Salman Khan

Example using Integration by Parts.

• ### Indefinite Integrals Part 7

00:09:38Salman Khan

Another example using integration by parts.

• ### Another U-substitution Example

00:05:57Salman Khan

Finding the antiderivative using u-substitution.

• ### Definite Integrals Part 1

00:09:18Salman Khan

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.

• ### Definite Integrals Part 2

00:09:15Salman Khan

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

• ### Definite Integrals Part 3

00:09:45Salman Khan

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

• ### Definite Integrals Part 4

00:09:51Salman Khan

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

• ### Definite Integrals Part 5

00:09:57Salman Khan

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

• ### Definite Integral with Substitution

00:08:54Salman Khan

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

• ### Integrals: Trig Substitution Part 1

00:07:38Salman Khan

Example of using trig substitution to solve an indefinite integral.

• ### Integrals: Trig Substitution Part 2

00:08:11Salman Khan

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

• ### Integrals: Trig Substitution Part 3

00:17:52Salman Khan

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

• ### Introduction to Differential Equations

00:14:25Salman Khan

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

• ### Solid of Revolution Part 1

00:10:05Salman Khan

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

• ### Solid of Revolution Part 2

00:07:31Salman Khan

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

• ### Solid of Revolution Part 3

00:08:59Salman Khan

Figuring out the equation for the volume of a sphere.

• ### Solid of Revolution Part 4

00:08:22Salman Khan

More volumes around the x-axis.

• ### Solid of Revolution Part 5

00:09:29Salman Khan

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

• ### Solid of Revolution Part 6

00:09:19Salman Khan

Using the disk method around the y-axis.

• ### Solid of Revolution Part 7

00:10:06Salman Khan

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

• ### Solid of Revolution Part 8

00:04:15Salman Khan

The last part of the problem in part 7.

• ### Polynomial Approximation of Functions Part 1

00:09:45Salman Khan

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

• ### Polynomial Approximation of Functions Part 2

00:10:03Salman Khan

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

• ### Polynomial Approximation of Functions Part 3

00:09:59Salman Khan

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

• ### Polynomial Approximation of Functions Part 4

00:09:59Salman Khan

Approximating cos x with a Maclaurin series.

• ### Polynomial Approximation of Functions Part 5

00:09:24Salman Khan

MacLaurin representation of sin x.

• ### Polynomial Approximation of Functions Part 6

00:09:59Salman Khan

A pattern emerges!

• ### Polynomial Approximation of Functions Part 7

00:10:18Salman Khan

The most amazing conclusion in mathematics!

• ### Taylor Polynomials

00:18:05Salman Khan

Approximating a function with a Taylor Polynomial.

• ### AP Calculus BC Exams: 2008 1 A

00:08:54Salman Khan

Part 1a of the 2008 BC free response.

• ### AP Calculus BC Exams: 2008 1 B & C

00:09:34Salman Khan

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

• ### AP Calculus BC Exams: 2008 1 C & D

00:09:20Salman Khan

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

• ### AP Calculus BC Exams: 2008 1 D

00:05:22Salman Khan

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

• ### Calculus BC 2008 2 A

00:08:45Salman Khan

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

• ### Calculus BC 2008 2 B & C

00:10:42Salman Khan

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

• ### Calculus BC 2008 2 D

00:04:28Salman Khan

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

• ### Partial Derivatives Part 1

00:11:11Salman Khan

Introduction to partial derivatives.

• ### Partial Derivatives Part 2

00:10:26Salman Khan

More on partial derivatives.

00:09:45Salman Khan

Introduction to the gradient.

• ### Gradient of a Scalar Field

00:10:54Salman Khan

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

• ### Divergence Part 1

00:10:20Salman Khan

Introduction to the divergence of a vector field.

• ### Divergence Part 2

00:10:52Salman Khan

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

• ### Divergence Part 3

00:10:48Salman Khan

Analyzing a vector field using its divergence.

• ### Curl Part 1

00:09:32Salman Khan

Introduction to the curl of a vector field.

• ### Curl Part 2

00:10:17Salman Khan

The mechanics of calculating curl.

• ### Curl Part 3

00:10:22Salman Khan

More on curl.

• ### Double Integrals Part 1

00:10:29Salman Khan

Introduction to the double integral.

• ### Double Integrals Part 2

00:09:50Salman Khan

Figuring out the volume under z=xy^2.

• ### Double Integrals Part 3

00:08:03Salman Khan

Let's integrate dy first!

• ### Double Integrals Part 4

00:09:24Salman Khan

Another way to conceptualize the double integral.

• ### Double Integrals Part 5

00:09:51Salman Khan

Finding the volume when we have variable boundaries.

• ### Double Integrals Part 6

00:09:58Salman Khan

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

• ### Triple Integrals Part 1

00:10:38Salman Khan

Introduction to the triple integral.

• ### Triple Integrals Part 2

00:07:26Salman Khan

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

• ### Triple Integrals Part 3

00:11:48Salman Khan

Figuring out the boundaries of integration.

• ### (2^ln x)/x Antiderivative Example

00:08:40Salman Khan

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

• ### Line Integrals

00:18:48Salman Khan

Introduction to the Line Integral.

• ### Line Integral Example 1

00:13:02Salman Khan

Concrete example using a line integral.

• ### Line Integral Example 2 Part 1

00:12:53Salman Khan

Line integral over a closed path (part 1).

• ### Line Integral Example 2 Part 2

00:09:58Salman Khan

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

• ### Position Vector Valued Functions

00:07:45Salman Khan

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

• ### Derivative of a Position Vector Valued Function

00:14:45Salman Khan

Visualizing the derivative of a position vector valued function.

• ### Differential of a Vector Valued Function

00:05:04Salman Khan

Understanding the differential of a vector valued function.

• ### Differential of a Vector Valued Function Example

00:12:45Salman Khan

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

• ### Line Integrals and Vector Fields

00:16:46Salman Khan

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

• ### Using a Line Integral to Find a Vector Field Example

00:11:32Salman Khan

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

• ### Parametrization of a Reverse Path

00:07:26Salman Khan

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

• ### Scalar Field Line Integral Independent of Path Direction

00:16:22Salman Khan

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

• ### Vector Field Line Integral Dependent of Path Direction

00:15:51Salman Khan

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

• ### Path Independence for Line Integrals

00:17:14Salman Khan

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

• ### Closed Curve Line Integrals of Conservative Vector Fields

00:08:25Salman Khan

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

• ### Example of Closed Line Integral of Conservative Field

00:11:02Salman Khan

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

• ### Second Example of Line Integral of Conservative Vector Field

00:10:19Salman Khan

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

• ### Green's Theorem Proof Part 1

00:14:26Salman Khan

Part 1 of the proof of Green's Theorem.

• ### Green's Theorem Proof Part 2

00:19:26Salman Khan

Part 2 of the proof of Green's Theorem.

• ### Green's Theorem Example Part 1

00:10:31Salman Khan

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

• ### Green's Theorem Example Part 2

00:07:07Salman Khan

Another example applying Green's Theorem.

• ### Introduction to Parametrizing a Surface with Two Parameters

00:19:02Salman Khan

Introduction to Parametrizing a Surface with Two Parameters.

• ### Position Vector-Valued Function for a Parametrization of Two Parameters

00:16:32Salman Khan

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

• ### Partial Derivatives of Vector-Valued Functions

00:10:07Salman Khan

Partial Derivatives of Vector-Valued Functions.

• ### Introduction to the Surface Integral

00:22:14Salman Khan

Introduction to the surface integral.

• ### Calculating a Surface Integral Example Part 1

00:10:45Salman Khan

Example of calculating a surface integral part 1.

• ### Calculating a Surface Integral Example Part 2

00:10:29Salman Khan

Example of calculating a surface integral part 2.

• ### Calculating a Surface Integral Example Part 3

00:09:54Salman Khan

Example of calculating a surface integral part 3.

• ### L'Hopital's Rule

00:08:51Salman Khan

Introduction to L'Hopital's Rule.

• ### L'Hopital's Rule Example 1

00:07:43Salman Khan

L'Hopital's Rule Example 1.

• ### L'Hopital's Rule Example 2

00:05:15Salman Khan

L'Hopital's Rule Example 2.

• ### L'Hopital's Rule Example 3

00:07:50Salman Khan

L'Hopital's Rule Example 1.