Thursday, 25 April 2013

Calculus




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Limits




Limit introduction, squeeze theorem, and epsilon-delta definition of limits
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Limits

Limits are the core tool that we build upon for calculus. Many times, a function can be undefined at a point, but we can think about what the function "approaches" as it gets closer and closer to that point (this is the "limit"). Other times, the function may be defined at a point, but it may approach a different limit. There are many, many times where the function value is the same as the limit at a point. Either way, this is a powerful tool as we start thinking about slope of a tangent line to a curve. If you have a decent background in algebra (graphing and functions in particular), you'll hopefully enjoy this tutorial!

Old limits tutorial

This tutorial covers much of the same material as the "Limits" tutorial, but does it with Sal's original "old school" videos. The sound, resolution or handwriting isn't as good, but some people find them more charming.

Limits and infinity

You have a basic understanding of what a limit is. Now, in this tutorial, we can explore situation where we take the limit as x approaches negative or positive infinity (and situations where the limit itself could be unbounded).

Squeeze theorem

If a function is always smaller than one function and always greater than another (i.e. it is always between them), then if the upper and lower function converge to a limit at a point, then so does the one in between. Not only is this useful for proving certain tricky limits (we use it to prove lim (x → 0) of (sin x)/x, but it is a useful metaphor to use in life (seriously). :) This tutorial is useful but optional. It is covered in most calculus courses, but it is not necessary to progress on to the "Introduction to derivatives" tutorial.

Epsilon delta definition of limits

This tutorial introduces a "formal" definition of limits. So put on your ball gown and/or tuxedo to party with Mr. Epsilon Delta (no, this is not referring to a fraternity). This tends to be covered early in a traditional calculus class (right after basic limits), but we have mixed feelings about that. It is cool and rigorous, but also very "mathy" (as most rigorous things are). Don't fret if you have trouble with it the first time. If you have a basic conceptual understanding of what limits are (from the "Limits" tutorial), you're ready to start thinking about taking derivatives.


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Taking derivatives
Calculating derivatives. Power rule. Product and quotient rules. Chain Rule. Implicit differentiation. Derivatives of common functions.
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Introduction to differential calculus

The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.

Introduction to derivatives

Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.

Visualizing derivatives

You understand that a derivative can be viewed as the slope of the tangent line at a point or the instantaneous rate of change of a function with respect to x. This tutorial will deepen your ability to visualize and conceptualize derivatives through videos and exercises. We think you'll find this tutorial incredibly fun and satisfying (seriously).

Power rule

Calculus is about to seem strangely straight forward. You've spent some time using the definition of a derivative to find the slope at a point. In this tutorial, we'll derive and apply the derivative for any term in a polynomial. By the end of this tutorial, you'll have the power to take the derivative of any polynomial like it's second nature!

Chain rule

You can take the derivatives of f(x) and g(x), but what about f(g(x)) or g(f(x))? The chain rule gives us this ability. Because most complex and hairy functions can be thought of the composition of several simpler ones (ones that you can find derivatives of), you'll be able to take the derivative of almost any function after this tutorial. Just imagine.

Product and quotient rules

You can figure out the derivative of f(x). You're also good for g(x). But what about f(x) times g(x)? This is what the product rule is all about. This tutorial is all about the product rule. It also covers the quotient rule (which really is just a special case of the product rule).

Implicit differentiation

Like people, mathematical relations are not always explicit about their intentions. In this tutorial, we'll be able to take the derivative of one variable with respect to another even when they are implicitly defined (like "x^2 + y^2 = 1").

Proofs of derivatives of common functions

We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!


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Derivative applications



Minima, maxima, and critical points. Rates of change. Optimization. Rates of change. L'Hopital's rule. Mean value theorem.
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Minima, maxima, inflection points and critical points

Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.

Optimization with calculus

Using calculus to solve optimization problems

Rates of change

Solving rate-of-change problems using calculus

Mean value theorem

If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).

L'Hôpital's Rule

Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.



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Indefinite and definite integrals



Indefinite integral as anti-derivative. Definite integral as area under a curve. Integration by parts. U-substitution. Trig substitution.
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Indefinite integral as anti-derivative

You are very familiar with taking the derivative of a function. Now we are going to go the other way around--if I give you a derivative of a function, can you come up with a possible original function. In other words, we'll be taking the anti-derivative!

Riemann sums and definite integration

In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)--generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.

Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.

U-substitution

U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverise chain rule. U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

Definite integrals

Until now, we have been viewing integrals as anti-derivatives. Now we explore them as the area under a curve between two boundaries (we will now construct definite integrals by defining the boundaries). This is the real meat of integral calculus!

Trigonometric substitution

We will now do another substitution technique (the other was u-substitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the anti-derivative of.

Fundamental Theorem of Calculus

You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. We'll explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.

Improper integrals

Not everything (or everyone) should or could be proper all the time. Same is true for definite integrals. In this tutorial, we'll look at improper integrals--ones where one or both bounds are at infinity! Mind blowing!




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Solid of revolution
Using definite integrals with the shell and disc methods to find volumes of solids of revolution.
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Disc method

You know how to use definite integrals to find areas under curves. We now take that idea for "spin" by thinking about the volumes of things created when you rotate functions around various lines. This tutorial focuses on the "disc method" and the "washer method" for these types of problems.

Shell method

You want to rotate a function around a vertical line, but do all your integrating in terms of x and f(x), then the shell method is your new friend. It is similarly fantastic when you want to rotate around a horizontal line but integrate in terms of y.

Solid of revolution volume

Using definite integration, we know how to find the area under a curve. But what about the volume of the 3-D shape generated by rotating a section of the curve about one of the axes (or any horizontal or vertical line for that matter). This in an older tutorial that is now covered in other tutorials. This tutorial will give you a powerful tool and stretch your powers of 3-D visualization!



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Sequences, series and function approximation
Sequences, series and approximating functions. Maclaurin and Taylor series.
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Sequences and series review

You want to learn about Maclaurin and Taylor series but are a little rough on your sequences and series. This tutorial will get you brushed up on the concepts, vocabulary and ideas behind sequences and series.

Maclaurin and Taylor series

In this tutorial, we will learn to approximate differentiable functions with polynomials. Beyond just being super cool, this can be useful for approximating functions so that they are easier to calculate, differentiate or integrate. So whether you will have to write simulations or become a bond trader (bond traders use polynomial approximation to estimate changes in bond prices given interest rate changes and vice versa), this tutorial could be fun. If that isn't motivation enough, we also come up with one of the most epic and powerful conclusions in all of mathematics in this tutorial: Euler's identity.

Sal's old Maclaurin and Taylor series tutorial

Everything in this tutorial is covered (with better resolution and handwriting) in the "other" Maclaurin and Taylor series tutorial, but this one has a bit of old-school charm so we are keeping it here for historical reasons.



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AP Calculus practice questions
Sample questions from the A.P. Calculus AB and BC exams (both multiple choice and free answer).
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Calculus AB example questions

Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).

Calculus BC sample questions

The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle!





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Double and triple integrals
Volume under a surface with double integrals. Triple integrals as well.
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Double integrals

A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface!

Triple integrals

This is about as many integrals we can use before our brains explode. Now we can sum variable quantities in three-dimensions (what is the mass of a 3-D wacky object that has variable mass)!




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Partial derivatives, gradient, divergence, curl
Thinking about forms of derivatives in multi-dimensions and for vector-valued functions: partial derivatives, gradient, divergence and curl.
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Partial derivatives

Let's jump out of that boring (okay, it wasn't THAT boring) 2-D world into the exciting 3-D world that we all live and breath in. Instead of functions of x that can be visualized as lines, we can have functions of x and y that can be visualized as surfaces. But does the idea of a derivative still make sense? Of course it does! As long as you specify what direction you're going in. Welcome to the world of partial derivatives!

Gradient

Ever walk on hill (or any wacky surface) and wonder which way would be the fastest way up (or down). Now you can figure this out exactly with the gradient.

Divergence

Is a vector field "coming together" or "drawing apart" at a given point in space. The divergence is a vector operator that gives us a scalar value at any point in a vector field. If it is positive, then we are diverging. Otherwise, we are converging!

Curl

Curl measures how much a vector field is "spinning". A bit of a pain to calculate, but could come in handy when we work with Stokes' and Greens' theorems later on.


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Line integrals and Green's theorem
Line integral of scalar and vector-valued functions. Green's theorem and 2-D divergence theorem.
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Line integrals for scalar functions

With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).

Position vector functions and derivatives

In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).

Line integrals in vector fields

You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

Green's theorem

It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.

2-D Divergence theorem

Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.


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Surface integrals and Stokes' theorem
Parameterizing a surface. Surface integrals. Stokes' theorem.
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Parameterizing a surface

You can parameterize a line with a position vector valued function and understand what a differential means in that context already. This tutorial will take things further by parametrizing surfaces (2 parameters baby!) and have us thinking about partial differentials.

Surface integrals

Finding line integrals to be a bit boring? Well, this tutorial will add new dimension to your life by explore what surface integrals are and how we can calculate them.

Flux in 3-D and constructing unit normal vectors to surface

Flux can be view as the rate at which "stuff" passes through a surface. Imagine a next placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.

Stokes' theorem intuition and application

Stokes' theorem relates the line integral around a surface to the curl on the surface. This tutorial explores the intuition behind Stokes' theorem, how it is an extension of Green's theorem to surfaces (as opposed to just regions) and gives some examples using it. We prove Stokes' theorem in another tutorial. Good to come to this tutorial having experienced the tutorial on "flux in 3D".

Proof of Stokes' theorem

You know what Stokes' theorem is and how to apply it, but are craving for some real proof that it is true. Well, you've found the right tutorial!



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Divergence theorem
Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D.
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Divergence theorem (3D)

An earlier tutorial used Green's theorem to prove the divergence theorem in 2-D, this tutorial gives us the 3-D version (what most people are talking about when they refer to the "divergence theorem"). We will get an intuition for it (that the flux through a close surface--like a balloon--should be equal to the divergence across it's volume). We will use it in examples. We will prove it in another tutorial.

Types of regions in three dimensions

This tutorial classifies regions in three dimensions. Comes in useful for some types of double integrals and we use these ideas to prove the divergence theorem.

Divergence theorem proof

You know what the divergence theorem is, you can apply it and you conceptually understand it. This tutorial will actually prove it to you (references types of regions which are covered in the "types of regions in 3d" tutorial.







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