An intuitive overview of Taylor series Sunday, Nov 17 2013 

This is a note written for my fall 2013 Math 100 class, but it was not written “for the exam,” nor does anything on here subtly hint at anything on any exam. But I hope that this will be helpful for anyone who wants to get a basic understanding of Taylor series. What I want to do is try to get some sort of intuitive grasp on Taylor series as approximations of functions. By intuitive, I mean intuitive to those with a good grasp of functions, the basics of a first semester of calculus (derivatives, integrals, the mean value theorem, and the fundamental theorem of calculus) – so it’s a mathematical intuition. In this way, this post is a sort of follow-up of my earlier note, An Intuitive Introduction to Calculus.

PLEASE NOTE that this note was written and typeset for my (now) main site: davidlowryduda.com. You can find it here. Because of this, some of the math shown here will display better there, where I have more control. This will also serve as the announcement that davidlowryduda.com is now on an improved webhost and should be fast and fully operational. Now, back to the math.

We care about Taylor series because they allow us to approximate other functions in predictable ways. Sometimes, these approximations can be made to be very, very, very accurate without requiring too much computing power. You might have heard that computers/calculators routinely use Taylor series to calculate things like {e^x} (which is more or less often true). But up to this point in most students’ mathematical development, most mathematics has been clean and perfect; everything has been exact algorithms yielding exact answers for years and years. This is simply not the way of the world.

Here’s a fundamental fact to both mathematics and life: almost anything worth doing is probably pretty hard and pretty messy.

For a very recognizable example, let’s think about finding zeroes of polynomials. Finding roots of linear polynomials is very easy. If we see {5 + x = 0}, we see that {-5} is the zero. Similarly, finding roots of quadratic polynomials is very easy, and many of us have memorized the quadratic formula to this end. Thus {ax^2 + bx + c = 0} has solutions {x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}. These are both nice, algorithmic, and exact. But I will guess that the vast majority of those who read this have never seen a “cubic polynomial formula” for finding roots of cubic polynomials (although it does exist, it is horrendously messy – look up Cardano’s formula). There is even an algorithmic way of finding the roots of quartic polynomials. But here’s something amazing: there is no general method for finding the exact roots of 5th degree polynomials (or higher degree).

I don’t mean We haven’t found it yet, but there may be one, or even You’ll have to use one of these myriad ways – I mean it has been shown that there is no general method of finding exact roots of degree 5 or higher polynomials. But we certainly can approximate them arbitrarily well. So even something as simple as finding roots of polynomials, which we’ve been doing since we were in middle school, gets incredibly and unbelievably complicated.

So before we hop into Taylor series directly, I want to get into the mindset of approximating functions with other functions.

1. Approximating functions with other functions

We like working with polynomials because they’re so easy to calculate and manipulate. So sometimes we try to approximate complicated functions with polynomials, a problem sometimes called “polynomial interpolation”.

Suppose we wanted to approximate {\sin(x)}. The most naive approximation that we might do is see that {\sin(0) = 0}, so we might approximate {\sin(x)} by {p_0(x) = 0}. We know that it’s right at least once, and since {\sin(x)} is periodic, it’s going to be right many times. I write {p_0} to indicate that this is a degree {0} polynomial, that is, a constant polynomial. Clearly though, this is a terrible approximation, and we can do better.

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Math 100: Before second midterm Thursday, Nov 7 2013 

You have a midterm next week, and it’s not going to be a cakewalk.

As requested, I’m uploading the last five weeks’ worth of worksheets, with (my) solutions. A comment on the solutions: not everything is presented in full detail, but most things are presented with most detail (except for the occasional one that is far far beyond what we actually expect you to be able to do). If you have any questions about anything, let me know. Even better, ask it here – maybe others have the same questions too.

Without further ado –

And since we were unable to go over the quiz in my afternoon recitation today, I’m attaching a worked solution to the quiz as well.

Again, let me know if you have any questions. I will still have my office hours on Tuesday from 2:30-4:30pm in my office (I’m aware that this happens to be immediately before the exam – status not by design). And I’ll be more or less responsive by email.

Study study study!

Math 100: Week 4 Saturday, Sep 28 2013 

This is a post for my math 100 calculus class of fall 2013. In this post, I give the 4th week’s recitation worksheet (no solutions yet – I’m still writing them up). More pertinently, we will also go over the most recent quiz and common mistakes. Trig substitution, it turns out, is not so easy.

Before we hop into the details, I’d like to encourage you all to avail of each other, your professor, your ta, and the MRC in preparation for the first midterm (next week!).

1. The quiz

There were two versions of the quiz this week, but they were very similar. Both asked about a particular trig substitution

\displaystyle \int_3^6 \sqrt{36 - x^2} \mathrm{d} x

And the other was

\displaystyle \int_{2\sqrt 2}^4 \sqrt{16 - x^2} \mathrm{d}x.

They are very similar, so I’m only going to go over one of them. I’ll go over the first one. We know we are to use trig substitution. I see two ways to proceed: either draw a reference triangle (which I recommend), or think through the Pythagorean trig identities until you find the one that works here (which I don’t recommend).

We see a {\sqrt{36 - x^2}}, and this is hard to deal with. Let’s draw a right triangle that has {\sqrt{36 - x^2}} as a side. I’ve drawn one below. (Not fancy, but I need a better light).

In this picture, note that {\sin \theta = \frac{x}{6}}, or that {x = 6 \sin \theta}, and that {\sqrt{36 - x^2} = 6 \cos \theta}. If we substitute {x = 6 \sin \theta} in our integral, this means that we can replace our {\sqrt{36 - x^2}} with {6 \cos \theta}. But this is a substitution, so we need to think about {\mathrm{d} x} too. Here, {x = 6 \sin \theta} means that {\mathrm{d}x = 6 \cos \theta}.

Some people used the wrong trig substitution, meaning they used {x = \tan \theta} or {x = \sec \theta}, and got stuck. It’s okay to get stuck, but if you notice that something isn’t working, it’s better to try something else than to stare at the paper for 10 minutes. Other people use {x = 6 \cos \theta}, which is perfectly doable and parallel to what I write below.

Another common error was people forgetting about the {\mathrm{d}x} term entirely. But it’s important!.

Substituting these into our integral gives

\displaystyle \int_{?}^{??} 36 \cos^2 (\theta) \mathrm{d}\theta,

where I have included question marks for the limits because, as after most substitutions, they are different. You have a choice: you might go on and put everything back in terms of {x} before you give your numerical answer; or you might find the new limits now.

It’s not correct to continue writing down the old limits. The variable has changed, and we really don’t want {\theta} to go from {3} to {6}.

If you were to find the new limits, then you need to consider: if {x=3} and {\frac{x}{6} = \sin \theta}, then we want a {\theta} such that {\sin \theta = \frac{3}{6}= \frac{1}{2}}, so we might use {\theta = \pi/6}. Similarly, when {x = 6}, we want {\theta} such that {\sin \theta = 1}, like {\theta = \pi/2}. Note that these were two arcsine calculations, which we would have to do even if we waited until after we put everything back in terms of {x} to evaluate.

Some people left their answers in terms of these arcsines. As far as mistakes go, this isn’t a very serious one. But this is the sort of simplification that is expected of you on exams, quizzes, and homeworks. In particular, if something can be written in a much simpler way through the unit circle, then you should do it if you have the time.

So we could rewrite our integral as

\displaystyle \int_{\pi/6}^{\pi/2} 36 \cos^2 (\theta) \mathrm{d}\theta.

How do we integrate {\cos^2 \theta}? We need to make use of the identity {\cos^2 \theta = \dfrac{1 + \cos 2\theta}{2}}. You should know this identity for this midterm. Now we have

\displaystyle 36 \int_{\pi/6}^{\pi/2}\left(\frac{1}{2} + \frac{\cos 2 \theta}{2}\right) \mathrm{d}\theta = 18 \int_{\pi/6}^{\pi/2}\mathrm{d}\theta + 18 \int_{\pi/6}^{\pi/2}\cos 2\theta \mathrm{d}\theta.

The first integral is extremely simple and yields {6\pi} The second integral has antiderivative {\dfrac{\sin 2 \theta}{2}} (Don’t forget the {2} on bottom!), and we have to evaluate {\big[9 \sin 2 \theta \big]_{\pi/6}^{\pi/2}}, which gives {-\dfrac{9 \sqrt 3}{2}}. You should know the unit circle sufficiently well to evaluate this for your midterm.

And so the final answer is {6 \pi - \dfrac{9 \sqrt 2}{2} \approx 11.0553}. (You don’t need to be able to do that approximation).

Let’s go back a moment and suppose you didn’t re\”{e}valuate the limits once you substituted in {\theta}. Then, following the same steps as above, you’d be left with

\displaystyle 18 \int_{?}^{??}\mathrm{d}\theta + 18 \int_{?}^{??}\cos 2\theta \mathrm{d}\theta = \left[ 18 \theta \right]_?^{??} + \left[ 9 \sin 2 \theta \right]_?^{??}.

Since {\frac{x}{6} = \sin \theta}, we know that {\theta = \arcsin (x/6)}. This is how we evaluate the left integral, and we are left with {[18 \arcsin(x/6)]_3^6}. This means we need to know the arcsine of {1} and {\frac 12}. These are exactly the same two arcsine computations that I referenced above! Following them again, we get {6\pi} as the answer.

We could do the same for the second part, since {\sin ( 2 \arcsin (x/6))} when {x = 3} is {\sin (2 \arcsin \frac{1}{2} ) = \sin (2 \cdot \frac{\pi}{6} ) = \frac{\sqrt 3}{2}}; and when {x = 6} we get {\sin (2 \arcsin 1) = \sin (2 \cdot \frac{\pi}{2}) = \sin (\pi) = 0}.

Putting these together, we see that the answer is again {6\pi - \frac{9\sqrt 3}{2}}.

Or, throwing yet another option out there, we could do something else (a little bit wittier, maybe?). We have this {\sin 2\theta} term to deal with. You might recall that {\sin 2 \theta = 2 \sin \theta \cos \theta}, the so-called double-angle identity.

Then {9 \sin 2\theta = 18 \sin \theta \cos \theta}. Going back to our reference triangle, we know that {\cos \theta = \dfrac{\sqrt{36 - x^2}}{6}} and that {\sin \theta = \dfrac{x}{6}}. Putting these together,

\displaystyle 9 \sin 2 \theta = \dfrac{ x\sqrt{36 - x^2} }{2}.

When {x=6}, this is {0}. When {x = 3}, we have {\dfrac{ 3\sqrt {27}}{2} = \dfrac{9\sqrt 3}{2}}.

And fortunately, we get the same answer again at the end of the day. (phew).

2. The worksheet

Finally, here is the worksheet for the day. I’m working on their solutions, and I’ll have that up by late this evening (sorry for the delay).

Ending tidbits – when I was last a TA, I tried to see what were the good predictors of final grade. Some things weren’t very surprising – there is a large correlation between exam scores and final grade. Some things were a bit surprising – low homework scores correlated well with low final grade, but high homework scores didn’t really have a strong correlation with final grade at all; attendance also correlated weakly. But one thing that really stuck with me was the first midterm grade vs final grade in class: it was really strong. For a bit more on that, I refer you to my final post from my Math 90 posts.

Math 100: Week 3 and pre-midterm Tuesday, Sep 24 2013 

This is a post for my Math 100 class of fall 2013. In this post, I give the first three weeks’ worksheets from recitation and the set of solutions to week three’s worksheet, as well as a few administrative details.

Firstly, here is the recitation work from the first three weeks:

  1. (there was no recitation the first week)
  2. A worksheet focusing on review.
  3. A worksheet focusing on integration by parts and u-substitution, with solutions.

In addition, I’d like to remind you that I have office hours from 2-4pm (right now) in Kassar 018. I’ve had multiple people set up appointments with me outside of these hours, which I’m tempted to interpret as suggesting that I change when my office hours are. If you have a preference, let me know, and I’ll try to incorporate it.

Finally, there will be an exam next Tuesday. I’ve been getting a lot of emails about what material will be on the exam. The answer is that everything you have learned up to now and by the end of this week is fair game for exam material. This also means there could be exam questions on material that we have not discussed in recitation. So be prepared. However, I will be setting aside a much larger portion of recitation this Thursday for questions than normal. So come prepared with your questions.

Best of luck, and I’ll see you in class on Thursday.

An intuitive introduction to calculus Saturday, Sep 7 2013 

This is a post written for my fall 2013 Math 100 class but largely intended for anyone with knowledge of what a function is and a desire to know what calculus is all about. Calculus is made out to be the pinnacle of the high school math curriculum, and correspondingly is thought to be very hard. But the difficulty is bloated, blown out of proportion. In fact, the ideas behind calculus are approachable and even intuitive if thought about in the right way.

Many people managed to stumble across the page before I’d finished all the graphics. I’m sorry, but they’re all done now! I was having trouble interpreting how WordPress was going to handle my gif files – it turns out that they automagically resize them if you don’t make them of the correct size, which makes them not display. It took me a bit to realize this. I’d like to mention that this actually started as a 90 minute talk I had with my wife over coffee, so perhaps an alternate title would be “Learning calculus in 2 hours over a cup of coffee.”

So read on if you would like to understand what calculus is, or if you’re looking for a refresher of the concepts from a first semester in calculus (like for Math 100 students at Brown), or if you’re looking for a bird’s eye view of AP Calc AB subject material.

1. An intuitive and semicomplete introduction to calculus

We will think of a function {f(\cdot)} as something that takes an input {x} and gives out another number, which we’ll denote by {f(x)}. We know functions like {f(x) = x^2 + 1}, which means that if I give in a number {x} then the function returns the number {f(x) = x^2 + 1}. So I put in {1}, I get {1^2 + 1 = 2}, i.e. {f(1) = 2}. Primary and secondary school overly conditions students to think of functions in terms of a formula or equation. The important thing to remember is that a function is really just something that gives an output when given an input, and if the same input is given later then the function spits the same output out. As an aside, I should mention that the most common problem I’ve seen in my teaching and tutoring is a fundamental misunderstanding of functions and their graphs

For a function that takes in and spits out numbers, we can associate a graph. A graph is a two-dimensional representation of our function, where by convention the input is put on the horizontal axis and the output is put on the vertical axis. Each axis is numbered, and in this way we can identify any point in the graph by its coordinates, i.e. its horizontal and vertical position. A graph of a function {f(x)} includes a point {(x,y)} if {y = f(x)}.

The graph of the function x^2 + 1 is in blue. The emphasized point appears on the graph because it is of the form (x, f(x)). In particular, this point is (1, 2).

Thus each point on the graph is really of the form {(x, f(x))}. A large portion of algebra I and II is devoted to being able to draw graphs for a variety of functions. And if you think about it, graphs contain a huge amount of information. Graphing {f(x)= x^2 + 1} involves drawing an upwards-facing parabola, which really represents an infinite number of points. That’s pretty intense, but it’s not what I want to focus on here.

1.1. Generalizing slope – introducing the derivative

You might recall the idea of the ‘slope’ of a line. A line has a constant ratio of how much the {y} value changes for a specific change in {x}, which we call the slope (people always seem to remember rise over run). In particular, if a line passes through the points {(x_1, y_1)} and {(x_2, y_2)}, then its slope will be the vertical change {y_2 - y_1} divided by the horizontal change {x_2 - x_1}, or {\dfrac{y_2 - y_1}{x_2 - x_1}}.

The graph of a line appears in blue. The two points (0,1) and (1,3) are shown on the line. The horizontal red line shows the horizontal change. The vertical red line shows the vertical change. The ‘slope’ of the blue line is the length of the vertical red line divided by the length of the horizontal red line.

So if the line is given by an equation {f(x) = \text{something}}, then the slope from two inputs {x_1} and {x_2} is {\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}}. As an aside, for those that remember things like the ‘standard equation’ {y = mx + b} or ‘point-slope’ {(y - y_0) = m(x - x_0)} but who have never thought or been taught where these come from: the claim that lines are the curves of constant slope is saying that for any choice of {(x_1, y_1)} on the line, we expect {\dfrac{y_2 - y_1}{x_2 - x_1} = m} a constant, which I denote by {m} for no particularly good reason other than the fact that some textbook author long ago did such a thing. Since we’re allowing ourselves to choose any {(x_1, y_1)}, we might drop the subscripts – since they usually mean a constant – and rearrange our equation to give {y_2 - y = m(x_2 - x)}, which is what has been so unkindly drilled into students’ heads as the ‘point-slope form.’ This is why lines have a point-slope form, and a reason that it comes up so much is that it comes so naturally from the defining characteristic of a line, i.e. constant slope.

But one cannot speak of the ‘slope’ of a parabola.

The parabola f(x) = x^2 + 1 is shows in blue. Slope is a measure of how much the function f(x) changes when x is changed. Some tangent lines to the parabola are shown in red. The slope of each line seems like it should be the ‘slope’ of the parabola when the line touches the parabola, but these slopes are different.

Intuitively, we look at our parabola {x^2 + 1} and see that the ‘slope,’ or an estimate of how much the function {f(x)} changes with a change in {x}, seems to be changing depending on what {x} values we choose. (This should make sense – if it didn’t change, and had constant slope, then it would be a line). The first major goal of calculus is to come up with an idea of a ‘slope’ for non-linear functions. I should add that we already know a sort of ‘instantaneous rate of change’ of a nonlinear function. When we’re in a car and we’re driving somewhere, we’re usually speeding up or slowing down, and our pace isn’t usually linear. Yet our speedometer still manages to say how fast we’re going, which is an immediate rate of change. So if we had a function {p(t)} that gave us our position at a time {t}, then the slope would give us our velocity (change in position per change in time) at a moment. So without knowing it, we’re familiar with a generalized slope already. Now in our parabola, we don’t expect a constant slope, so we want to associate a ‘slope’ to each input {x}. In other words, we want to be able to understand how rapidly the function {f(x)} is changing at each {x}, analogous to how the slope {m} of a line {g(x) = mx + b} tells us that if we change our input by an amount {h} then our output value will change by {mh}.

How does calculus do that? The idea is to get closer and closer approximations. Suppose we want to find the ‘slope’ of our parabola at the point {x = 1}. Let’s get an approximate answer. The slope of the line coming from inputs {x = 1} and {x = 2} is a (poor) approximation. In particular, since we’re working with {f(x) = x^2 + 1}, we have that {f(2) = 5} and {f(1) = 2}, so that the ‘approximate slope’ from {x = 1} and {x = 2} is {\frac{5 - 2}{2 - 1} = 3}. But looking at the graph,

The parabola x^2 + 1 is shown in blue, and the line going through the points (1,2) and (2,5) is shown. The line immediately goes above and crosses the parabola, so it seems like this line is rising faster (changing faster) than the parabola. It’s too steep, and the slope is too high to reflect the ‘slope’ of the parabola at the indicated point.

we see that it feels like this slope is too large. So let’s get closer. Suppose we use inputs {x = 1} and {x = 1.5}. We get that the approximate slope is {\frac{3.25 - 2}{1.5 - 1} = 2.5}. If we were to graph it, this would also feel too large. So we can keep choosing smaller and smaller changes, like using {x = 1} and {x = 1.1}, or {x = 1} and {x = 1.01}, and so on. This next graphic contains these approximations, with chosen points getting closer and closer to {1}.

The parabola x^2 + 1 is shown in blue. Two points are chosen on the parabola and the line between them is drawn in red. As the points get closer to each other, the red line indicates the rate of growth of the parabola at the point (1,2) better and better. So the slope of the red lines seems to be getting closer to the ‘slope’ of the parabola at (1,2).

Let’s look a little closer at the values we’re getting for our slopes when we use {1} and {2, 1.5, 1.1, 1.01, 1.001} as our inputs. We get

\displaystyle \begin{array}{c|c} \text{second input} & \text{approx. slope} \\ \hline 2 & 3 \\ 1.5 & 2.5 \\ 1.1 & 2.1 \\ 1.01 & 2.01 \\ 1.001 & 2.001 \end{array}

It looks like the approximate slopes are approaching {2}. What if we plot the graph with a line of slope {2} going through the point {(1,2)}?

The parabola x^2 + 1 is shown in blue. The line in red has slope 2 and goes through the point (1,2). We got this line by continuing the successive approximations done above. It looks like it accurately indicates the ‘slope’ of the parabola at (1,2).

It looks great! Let’s zoom in a whole lot.

When we zoom in, the blue parabola looks almost like a line, and the red line looks almost like the parabola! This is why we are measuring the ‘slope’ of the parabola in this fashion – when we zoom in, it looks more and more like a line, and we are getting the slope of that line.

That looks really close! In fact, what I’ve been allowing as the natural feeling slope, or local rate of change, is really the line tangent to the graph of our function at the point {(1, f(1))}. In a calculus class, you’ll spend a bit of time making sense of what it means for the approximate slopes to ‘approach’ {2}. This is called a ‘limit,’ and the details are not important to us right now. The important thing is that this let us get an idea of a ‘slope’ at a point on a parabola. It’s not really a slope, because a parabola isn’t a line. So we’ve given it a different name – we call this ‘the derivative.’ So the derivative of {f(x) = x^2 + 1} at {x = 1} is {2}, i.e. right around {x = 1} we expect a rate of change of {2}, so that we expect {f(1 + h) - f(1) \approx 2h}. If you think about it, we’re saying that we can approximate {f(x) = x^2 + 1} near the point {(1, 2)} by the line shown in the graph above: this line passes through {(1,2)} and it’s slope is {2}, what we’re calling the slope of {f(x) = x^2 + 1} at {x = 1}.

Let’s generalize. We were able to speak of the derivative at one point, but how about other points? The rest of this post is below the ‘more’ tag below.

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