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. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists).

• Epsilon Delta Limit Definition Part 1
Salman Khan

Introduction to the Epsilon Delta Definition of a Limit.

• Epsilon Delta Limit Definition Part 2
Salman Khan

Using the epsilon delta definition to prove a limit.

• Calculus: Derivatives 1
Salman 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
Salman Khan

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

• Calculus: Derivatives 2.5
Salman Khan

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

• Derivatives Part 1
Salman Khan

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

• Derivatives Part 2
Salman 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
Salman Khan

Determining the derivatives of simple polynomials.

• Derivatives Part 4
Salman Khan

Part 4 of derivatives. Introduction to the chain rule.

• Derivatives Part 5
Salman Khan

Examples using the Chain Rule.

• Derivatives Part 6
Salman Khan

Even more examples using the chain rule.

• Derivatives Part 7
Salman Khan

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

• Derivatives Part 8
Salman 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
Salman Khan

More examples of taking derivatives.

• Proof: d/dx(x^n)
Salman Khan

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

• Proof: d/dx(sqrt(x))
Salman Khan

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

• Proof: d/dx(ln x) = 1/x
Salman Khan

Taking the derivative of ln x.

• Proof: d/dx(e^x) = e^x
Salman Khan

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

• Proofs of Derivatives of Ln(x) and e^x
Salman Khan

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

• Extreme Derivative Word Problem
Salman Khan

A difficult but interesting derivative word problem.

• Implicit Differentiation Part 1
Salman Khan

Taking the derivative when y is defined implicitly.

• Implicit Differentiation Part 2
Salman Khan

A hairier implicit differentiation problem.

• More Implicit Differentiation
Salman Khan

2 more implicit differentiation examples.

• More Chain Rule and Implicit Differentiation Intuition
Salman Khan

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

• Trig Implicit Differentiation Example
Salman Khan

Implicit differentiation example that involves the tangent function.

• Derivative of x^(x^x)
Salman Khan

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

• Maxima Minima Slope Intuition
Salman Khan

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

• Inflection Points and Concavity Intuition
Salman Khan

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

• Monotonicity Theorem
Salman Khan

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

• Maximum and Minimum Values on an Interval
Salman Khan

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

• Graphing Using Derivatives
Salman Khan

Graphing functions using derivatives.

• Graphing with Derivatives Example
Salman Khan

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

• Graphing with Calculus
Salman Khan

More graphing with calculus.

• Optimization with Calculus Part 1
Salman Khan

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

• Optimization with Calculus Part 2
Salman 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
Salman 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
Salman Khan

Minimizing the cost of material for an open rectangular box.

• Introduction to Rate-of-change
Salman Khan

Using derivatives to solve rate-of-change problems.

• Equation of a Tangent Line
Salman Khan

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

• Rates-of-change Part 2
Salman Khan

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

• Ladder rate-of-change Problem
Salman Khan

The classic falling ladder problem.

• Mean Value Theorem
Salman Khan

Intuition behind the Mean Value Theorem.

• Indefinite Integrals Part 1
Salman Khan

An introduction to indefinite integration of polynomials.

• Indefinite Integrals Part 2
Salman Khan

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

• Indefinite Integrals Part 3
Salman Khan

Integration by doing the chain rule in reverse.

• Indefinite Integrals Part 4
Salman Khan

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

• Indefinite Integrals Part 5
Salman Khan

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

• Indefinite Integrals Part 6
Salman Khan

Example using Integration by Parts.

• Indefinite Integrals Part 7
Salman Khan

Another example using integration by parts.

• Another U-substitution Example
Salman Khan

Finding the antiderivative using u-substitution.

• Definite Integrals Part 1
Salman 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
Salman Khan

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

• Definite Integrals Part 3
Salman Khan

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

• Definite Integrals Part 4
Salman Khan

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

• Definite Integrals Part 5
Salman Khan

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

• Definite Integral with Substitution
Salman Khan

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

• Integrals: Trig Substitution Part 1
Salman Khan

Example of using trig substitution to solve an indefinite integral.

• Integrals: Trig Substitution Part 2
Salman Khan

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

• Integrals: Trig Substitution Part 3
Salman Khan

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

• Introduction to Differential Equations
Salman Khan

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

• Solid of Revolution Part 1
Salman Khan

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

• Solid of Revolution Part 2
Salman Khan

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

• Solid of Revolution Part 3
Salman Khan

Figuring out the equation for the volume of a sphere.

• Solid of Revolution Part 4
Salman Khan

More volumes around the x-axis.

• Solid of Revolution Part 5
Salman Khan

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

• Solid of Revolution Part 6
Salman Khan

Using the disk method around the y-axis.

• Solid of Revolution Part 7
Salman Khan

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

• Solid of Revolution Part 8
Salman Khan

The last part of the problem in part 7.

• Polynomial Approximation of Functions Part 1
Salman Khan

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

• Polynomial Approximation of Functions Part 2
Salman Khan

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

• Polynomial Approximation of Functions Part 3
Salman Khan

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

• Polynomial Approximation of Functions Part 4
Salman Khan

Approximating cos x with a Maclaurin series.

• Polynomial Approximation of Functions Part 5
Salman Khan

MacLaurin representation of sin x.

• Polynomial Approximation of Functions Part 6
Salman Khan

A pattern emerges!

• Polynomial Approximation of Functions Part 7
Salman Khan

The most amazing conclusion in mathematics!

• Taylor Polynomials
Salman Khan

Approximating a function with a Taylor Polynomial.

• AP Calculus BC Exams: 2008 1 A
Salman Khan

Part 1a of the 2008 BC free response.

• AP Calculus BC Exams: 2008 1 B & C
Salman Khan

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

• AP Calculus BC Exams: 2008 1 C & D
Salman Khan

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

• AP Calculus BC Exams: 2008 1 D
Salman Khan

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

• Calculus BC 2008 2 A
Salman Khan

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

• Calculus BC 2008 2 B & C
Salman Khan

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

• Calculus BC 2008 2 D
Salman Khan

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

• Partial Derivatives Part 1
Salman Khan

Introduction to partial derivatives.

• Partial Derivatives Part 2
Salman Khan

More on partial derivatives.

• Gradient
Salman Khan

Introduction to the gradient.

• Gradient of a Scalar Field
Salman Khan

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

• Divergence Part 1
Salman Khan

Introduction to the divergence of a vector field.

• Divergence Part 2
Salman Khan

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

• Divergence Part 3
Salman Khan

Analyzing a vector field using its divergence.

• Curl Part 1
Salman Khan

Introduction to the curl of a vector field.

• Curl Part 2
Salman Khan

The mechanics of calculating curl.

• Curl Part 3
Salman Khan

More on curl.

• Double Integrals Part 1
Salman Khan

Introduction to the double integral.

• Double Integrals Part 2
Salman Khan

Figuring out the volume under z=xy^2.

• Double Integrals Part 3
Salman Khan

Let's integrate dy first!

• Double Integrals Part 4
Salman Khan

Another way to conceptualize the double integral.

• Double Integrals Part 5
Salman Khan

Finding the volume when we have variable boundaries.

• Double Integrals Part 6
Salman Khan

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

• Triple Integrals Part 1
Salman Khan

Introduction to the triple integral.

• Triple Integrals Part 2
Salman Khan

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

• Triple Integrals Part 3
Salman Khan

Figuring out the boundaries of integration.

• (2^ln x)/x Antiderivative Example
Salman Khan

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

• Line Integrals
Salman Khan

Introduction to the Line Integral.

• Line Integral Example 1
Salman Khan

Concrete example using a line integral.

• Line Integral Example 2 Part 1
Salman Khan

Line integral over a closed path (part 1).

• Line Integral Example 2 Part 2
Salman Khan

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

• Position Vector Valued Functions
Salman Khan

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

• Derivative of a Position Vector Valued Function
Salman Khan

Visualizing the derivative of a position vector valued function.

• Differential of a Vector Valued Function
Salman Khan

Understanding the differential of a vector valued function.

• Differential of a Vector Valued Function Example
Salman Khan

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

• Line Integrals and Vector Fields
Salman 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
Salman Khan

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

• Parametrization of a Reverse Path
Salman Khan

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

• Scalar Field Line Integral Independent of Path Direction
Salman Khan

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

• Vector Field Line Integral Dependent of Path Direction
Salman Khan

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

• Path Independence for Line Integrals
Salman 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
Salman Khan

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

• Example of Closed Line Integral of Conservative Field
Salman Khan

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

• Second Example of Line Integral of Conservative Vector Field
Salman Khan

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

• Green's Theorem Proof Part 1
Salman Khan

Part 1 of the proof of Green's Theorem.

• Green's Theorem Proof Part 2
Salman Khan

Part 2 of the proof of Green's Theorem.

• Green's Theorem Example Part 1
Salman Khan

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

• Green's Theorem Example Part 2
Salman Khan

Another example applying Green's Theorem.

• Introduction to Parametrizing a Surface with Two Parameters
Salman Khan

Introduction to Parametrizing a Surface with Two Parameters.

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

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

• Partial Derivatives of Vector-Valued Functions
Salman Khan

Partial Derivatives of Vector-Valued Functions.

• Introduction to the Surface Integral
Salman Khan

Introduction to the surface integral.

• Calculating a Surface Integral Example Part 1
Salman Khan

Example of calculating a surface integral part 1.

• Calculating a Surface Integral Example Part 2
Salman Khan

Example of calculating a surface integral part 2.

• Calculating a Surface Integral Example Part 3
Salman Khan

Example of calculating a surface integral part 3.

• L'Hopital's Rule
Salman Khan

Introduction to L'Hopital's Rule.

• L'Hopital's Rule Example 1
Salman Khan

L'Hopital's Rule Example 1.

• L'Hopital's Rule Example 2
Salman Khan

L'Hopital's Rule Example 2.

• L'Hopital's Rule Example 3
Salman Khan

L'Hopital's Rule Example 1.