# 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).

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

• ##### Calculus: Derivatives 2.5
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Calculus-Derivative: Finding the derivative of y=x^2

• ##### Derivatives Part 1
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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
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Part 4 of derivatives. Introduction to the chain rule.

• ##### Derivatives Part 5
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Examples using the Chain Rule.

• ##### Derivatives Part 6
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Even more examples using the chain rule.

• ##### Derivatives Part 7
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The product rule. Examples using the Product and Chain rules.

• ##### Derivatives Part 8
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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
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More examples of taking derivatives.

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

• ##### Proof: d/dx(sqrt(x))
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Proof that d/dx (x^.5) = .5x^(-.5).

• ##### Proof: d/dx(ln x) = 1/x
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Taking the derivative of ln x.

• ##### Proof: d/dx(e^x) = e^x
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Proof that the derivative of e^x is e^x.

• ##### Proofs of Derivatives of Ln(x) and e^x
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Doing both proofs in the same video to clarify any misconceptions that the original proof was "circular".

• ##### Extreme Derivative Word Problem
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A difficult but interesting derivative word problem.

• ##### Implicit Differentiation Part 1
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Taking the derivative when y is defined implicitly.

• ##### Implicit Differentiation Part 2
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A hairier implicit differentiation problem.

• ##### More Implicit Differentiation
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2 more implicit differentiation examples.

• ##### More Chain Rule and Implicit Differentiation Intuition
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More intuition behind the chain rule and how it applies to implicit differentiation.

• ##### Trig Implicit Differentiation Example
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Implicit differentiation example that involves the tangent function.

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

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

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

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

• ##### Maximum and Minimum Values on an Interval
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2 examples of finding the maximum and minimum points on an interval.

• ##### Graphing Using Derivatives
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Graphing functions using derivatives.

• ##### Graphing with Derivatives Example
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Using the first and second derivatives to identify critical points and inflection points and to graph the function.

• ##### Graphing with Calculus
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More graphing with calculus.

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

• ##### Optimization with Calculus Part 2
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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
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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
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Minimizing the cost of material for an open rectangular box.

• ##### Introduction to Rate-of-change
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Using derivatives to solve rate-of-change problems.

• ##### Equation of a Tangent Line
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Finding the equation of the line tangent to f(x)=xe^x when x=1.

• ##### Rates-of-change Part 2
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Another (simpler) example of using the chain rule to determine rates-of-change.

Salman Khan

• ##### Mean Value Theorem
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Intuition behind the Mean Value Theorem.

• ##### Indefinite Integrals Part 1
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An introduction to indefinite integration of polynomials.

• ##### Indefinite Integrals Part 2
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Examples of taking the indefinite integral (or anti-derivative) of polynomials.

• ##### Indefinite Integrals Part 3
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Integration by doing the chain rule in reverse.

• ##### Indefinite Integrals Part 4
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Integration by substitution (or the reverse-chain-rule).

• ##### Indefinite Integrals Part 5
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Introduction to Integration by Parts (kind of the reverse-product rule).

• ##### Indefinite Integrals Part 6
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Example using Integration by Parts.

• ##### Indefinite Integrals Part 7
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Another example using integration by parts.

• ##### Another U-substitution Example
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Finding the antiderivative using u-substitution.

• ##### Definite Integrals Part 1
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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
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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
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Examples of using definite integrals to find the area under a curve.

• ##### Definite Integrals Part 5
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More examples of using definite integrals to calculate the area between curves.

• ##### Definite Integral with Substitution
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Solving a definite integral with substitution (or the reverse chain rule).

• ##### Integrals: Trig Substitution Part 1
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Example of using trig substitution to solve an indefinite integral.

• ##### Integrals: Trig Substitution Part 2
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Another example of finding an anti-derivative using trigonometric substitution.

• ##### Integrals: Trig Substitution Part 3
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Example using trig substitution (and trig identities) to solve an integral.

• ##### Introduction to Differential Equations
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3 basic differential equations that can be solved by taking the antiderivatives of both sides.

• ##### Solid of Revolution Part 1
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Figuring out the volume of a function rotated about the x-axis.

• ##### Solid of Revolution Part 2
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The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis.

• ##### Solid of Revolution Part 3
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Figuring out the equation for the volume of a sphere.

• ##### Solid of Revolution Part 4
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More volumes around the x-axis.

• ##### Solid of Revolution Part 5
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Use the "shell method" to rotate about the y-axis.

• ##### Solid of Revolution Part 6
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Using the disk method around the y-axis.

• ##### Solid of Revolution Part 7
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Taking the revolution around something other than one of the axes.

• ##### Solid of Revolution Part 8
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The last part of the problem in part 7.

• ##### Polynomial Approximation of Functions Part 1
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Using a polynomial to approximate a function at f(0).

• ##### Polynomial Approximation of Functions Part 2
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Approximating a function with a polynomial by making the derivatives equal at f(0) (Maclauren Series).

• ##### Polynomial Approximation of Functions Part 3
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A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.

• ##### Polynomial Approximation of Functions Part 4
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Approximating cos x with a Maclaurin series.

• ##### Polynomial Approximation of Functions Part 5
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MacLaurin representation of sin x.

• ##### Polynomial Approximation of Functions Part 6
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A pattern emerges!

• ##### Polynomial Approximation of Functions Part 7
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The most amazing conclusion in mathematics!

• ##### Taylor Polynomials
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Approximating a function with a Taylor Polynomial.

• ##### AP Calculus BC Exams: 2008 1 A
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Part 1a of the 2008 BC free response.

• ##### AP Calculus BC Exams: 2008 1 B & C
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Parts b and c of problem 1 (free response).

• ##### AP Calculus BC Exams: 2008 1 C & D
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Parts c&d of problem 1 in the 2008 AP Calculus BC free response.

• ##### AP Calculus BC Exams: 2008 1 D
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Part 1d of the 2008 AP Calculus BC exam (free response).

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

• ##### Calculus BC 2008 2 B & C
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Parts 2b and 2c of the 2008 BC exam (free response).

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

• ##### Partial Derivatives Part 1
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Introduction to partial derivatives.

• ##### Partial Derivatives Part 2
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More on partial derivatives.

Salman Khan

• ##### Gradient of a Scalar Field
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Intuition of the gradient of a scalar field (temperature in a room) in 3 dimensions.

• ##### Divergence Part 1
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Introduction to the divergence of a vector field.

• ##### Divergence Part 2
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The intuition of what the divergence of a vector field is.

• ##### Divergence Part 3
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Analyzing a vector field using its divergence.

• ##### Curl Part 1
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Introduction to the curl of a vector field.

• ##### Curl Part 2
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The mechanics of calculating curl.

• ##### Curl Part 3
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More on curl.

• ##### Double Integrals Part 1
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Introduction to the double integral.

• ##### Double Integrals Part 2
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Figuring out the volume under z=xy^2.

• ##### Double Integrals Part 3
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Let's integrate dy first!

• ##### Double Integrals Part 4
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Another way to conceptualize the double integral.

• ##### Double Integrals Part 5
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Finding the volume when we have variable boundaries.

• ##### Double Integrals Part 6
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Let's evaluate the double integrals with y=x^2 as one of the boundaries.

• ##### Triple Integrals Part 1
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Introduction to the triple integral.

• ##### Triple Integrals Part 2
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Using a triple integral to find the mass of a volume of variable density.

• ##### Triple Integrals Part 3
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Figuring out the boundaries of integration.

• ##### (2^ln x)/x Antiderivative Example
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Finding ?(2^ln x)/x dx.

• ##### Line Integrals
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Introduction to the Line Integral.

• ##### Line Integral Example 1
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Concrete example using a line integral.

• ##### Line Integral Example 2 Part 1
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Line integral over a closed path (part 1).

• ##### Line Integral Example 2 Part 2
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Part 2 of an example of taking a line integral over a closed path.

• ##### Position Vector Valued Functions
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Using a position vector valued function to describe a curve or path.

• ##### Derivative of a Position Vector Valued Function
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Visualizing the derivative of a position vector valued function.

• ##### Differential of a Vector Valued Function
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Understanding the differential of a vector valued function.

• ##### Differential of a Vector Valued Function Example
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Concrete example of the derivative of a vector valued function to better understand what it means.

• ##### Line Integrals and Vector Fields
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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
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Using a line integral to find the work done by a vector field example.

• ##### Parametrization of a Reverse Path
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Understanding how to parametrize a reverse path for the same curve.

• ##### Scalar Field Line Integral Independent of Path Direction
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Showing that the line integral of a scalar field is independent of path direction.

• ##### Vector Field Line Integral Dependent of Path Direction
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Showing that, unlike line integrals of scalar fields, line integrals over vector fields are path direction dependent.

• ##### Path Independence for Line Integrals
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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
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Showing that the line integral along closed curves of conservative vector fields is zero.

• ##### Example of Closed Line Integral of Conservative Field
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Example of taking a closed line integral of a conservative field.

• ##### Second Example of Line Integral of Conservative Vector Field
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Using path independence of a conservative vector field to solve a line integral.

• ##### Green's Theorem Proof Part 1
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Part 1 of the proof of Green's Theorem.

• ##### Green's Theorem Proof Part 2
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Part 2 of the proof of Green's Theorem.

• ##### Green's Theorem Example Part 1
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Using Green's Theorem to solve a line integral of a vector field.

• ##### Green's Theorem Example Part 2
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Another example applying Green's Theorem.

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

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

• ##### Partial Derivatives of Vector-Valued Functions
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Partial Derivatives of Vector-Valued Functions.

• ##### Introduction to the Surface Integral
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Introduction to the surface integral.

• ##### Calculating a Surface Integral Example Part 1
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Example of calculating a surface integral part 1.

• ##### Calculating a Surface Integral Example Part 2
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Example of calculating a surface integral part 2.

• ##### Calculating a Surface Integral Example Part 3
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Example of calculating a surface integral part 3.

• ##### L'Hopital's Rule
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Introduction to L'Hopital's Rule.

• ##### L'Hopital's Rule Example 1
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L'Hopital's Rule Example 1.

• ##### L'Hopital's Rule Example 2
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L'Hopital's Rule Example 2.

• ##### L'Hopital's Rule Example 3
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L'Hopital's Rule Example 1.