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Transcript
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When I first learned about Taylor series, I definitely
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didn't appreciate just how important they are.
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But time and time again they come up in math, physics,
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and many fields of engineering because they're one of the most
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powerful tools that math has to offer for approximating functions.
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I think one of the first times this clicked for me as a
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student was not in a calculus class but a physics class.
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We were studying a certain problem that had to do with the potential energy of a
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pendulum, and for that you need an expression for how high the weight of the
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pendulum is above its lowest point, and when you work that out it comes out to be
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proportional to 1 minus the cosine of the angle between the pendulum and the vertical.
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The specifics of the problem we were trying to solve are beyond the point here,
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but what I'll say is that this cosine function made the problem awkward and unwieldy,
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and made it less clear how pendulums relate to other oscillating phenomena.
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But if you approximate cosine of theta as 1 minus theta squared over 2,
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everything just fell into place much more easily.
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If you've never seen anything like this before,
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an approximation like that might seem completely out of left field.
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If you graph cosine of theta along with this function, 1 minus theta squared over 2,
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they do seem rather close to each other, at least for small angles near 0,
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but how would you even think to make this approximation,
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and how would you find that particular quadratic?
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The study of Taylor series is largely about taking non-polynomial
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functions and finding polynomials that approximate them near some input.
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The motive here is that polynomials tend to be much easier to deal
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with than other functions, they're easier to compute,
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easier to take derivatives, easier to integrate, just all around more friendly.
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So let's take a look at that function, cosine of x,
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and really take a moment to think about how you might construct a quadratic
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approximation near x equals 0.
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That is, among all of the possible polynomials that look like c0 plus c1
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times x plus c2 times x squared, for some choice of these constants, c0,
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c1, and c2, find the one that most resembles cosine of x near x equals 0,
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whose graph kind of spoons with the graph of cosine x at that point.
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Well, first of all, at the input 0, the value of cosine of x is 1,
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so if our approximation is going to be any good at all,
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it should also equal 1 at the input x equals 0.
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Plugging in 0 just results in whatever c0 is, so we can set that equal to 1.
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This leaves us free to choose constants c1 and c2 to make this
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approximation as good as we can, but nothing we do with them is
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going to change the fact that the polynomial equals 1 at x equals 0.
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It would also be good if our approximation had the same
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tangent slope as cosine x at this point of interest.
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Otherwise the approximation drifts away from the
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cosine graph much faster than it needs to.
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The derivative of cosine is negative sine, and at x equals 0,
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that equals 0, meaning the tangent line is perfectly flat.
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On the other hand, when you work out the derivative of our quadratic,
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you get c1 plus 2 times c2 times x.
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At x equals 0, this just equals whatever we choose for c1.
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So this constant c1 has complete control over the
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derivative of our approximation around x equals 0.
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Setting it equal to 0 ensures that our approximation
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also has a flat tangent line at this point.
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This leaves us free to change c2, but the value and the slope of our
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polynomial at x equals 0 are locked in place to match that of cosine.
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The final thing to take advantage of is the fact that the cosine graph
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curves downward above x equals 0, it has a negative second derivative.
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Or in other words, even though the rate of change is 0 at that point,
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the rate of change itself is decreasing around that point.
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Specifically, since its derivative is negative sine of x,
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its second derivative is negative cosine of x, and at x equals 0, that equals negative 1.
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Now in the same way that we wanted the derivative of our approximation to
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match that of the cosine so that their values wouldn't drift apart needlessly quickly,
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making sure that their second derivatives match will ensure that they
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curve at the same rate, that the slope of our polynomial doesn't drift
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away from the slope of cosine x any more quickly than it needs to.
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Pulling up the same derivative we had before, and then taking its derivative,
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we see that the second derivative of this polynomial is exactly 2 times c2.
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So to make sure that this second derivative also equals negative 1 at x equals 0,
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2 times c2 has to be negative 1, meaning c2 itself should be negative 1 half.
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This gives us the approximation 1 plus 0x minus 1 half x squared.
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To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial,
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you'd estimate it to be 0.995, and this is the true value of cosine of 0.1.
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It's a really good approximation!
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Take a moment to reflect on what just happened.
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You had 3 degrees of freedom with this quadratic approximation,
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the constants c0, c1, and c2.
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c0 was responsible for making sure that the output of the approximation matches that of
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cosine x at x equals 0, c1 was in charge of making sure that the derivatives match at
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that point, and c2 was responsible for making sure that the second derivatives match up.
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This ensures that the way your approximation changes as you move away from x equals 0,
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and the way that the rate of change itself changes,
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is as similar as possible to the behaviour of cosine x,
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given the amount of control you have.
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You could give yourself more control by allowing more terms
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in your polynomial and matching higher order derivatives.
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For example, let's say you added on the term c3 times x cubed for some constant c3.
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In that case, if you take the third derivative of a cubic polynomial,
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anything quadratic or smaller goes to 0.
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As for that last term, after 3 iterations of the power rule,
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it looks like 1 times 2 times 3 times c3.
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On the other hand, the third derivative of cosine x comes out to sine x,
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which equals 0 at x equals 0.
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So to make sure that the third derivatives match, the constant c3 should be 0.
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Or in other words, not only is 1 minus ½ x2 the best possible quadratic
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approximation of cosine, it's also the best possible cubic approximation.
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You can make an improvement by adding on a fourth order term, c4 times x to the fourth.
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The fourth derivative of cosine is itself, which equals 1 at x equals 0.
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And what's the fourth derivative of our polynomial with this new term?
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Well, when you keep applying the power rule over and over,
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with those exponents all hopping down in front,
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you end up with 1 times 2 times 3 times 4 times c4, which is 24 times c4.
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So if we want this to match the fourth derivative of cosine x,
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which is 1, c4 has to be 1 over 24.
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And indeed, the polynomial 1 minus ½ x2 plus 1 24 times x to the fourth,
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which looks like this, is a very close approximation for cosine x around x equals 0.
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In any physics problem involving the cosine of a small angle, for example,
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predictions would be almost unnoticeably different if you substituted this polynomial
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for cosine of x.
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Take a step back and notice a few things happening with this process.
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First of all, factorial terms come up very naturally in this process.
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When you take n successive derivatives of the function x to the n,
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letting the power rule keep cascading on down,
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what you'll be left with is 1 times 2 times 3 on and on up to whatever n is.
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So you don't simply set the coefficients of the polynomial equal to whatever derivative
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you want, you have to divide by the appropriate factorial to cancel out this effect.
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For example, that x to the fourth coefficient was the fourth derivative of cosine,
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1, but divided by 4 factorial, 24.
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The second thing to notice is that adding on new terms,
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like this c4 times x to the fourth, doesn't mess up what the old terms should be,
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and that's really important.
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For example, the second derivative of this polynomial at x equals 0 is still equal
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to 2 times the second coefficient, even after you introduce higher order terms.
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And it's because we're plugging in x equals 0,
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so the second derivative of any higher order term, which all include an x,
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will just wash away.
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And the same goes for any other derivative, which is why each derivative of a
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polynomial at x equals 0 is controlled by one and only one of the coefficients.
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If instead you were approximating near an input other than 0, like x equals pi,
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in order to get the same effect you would have to write your polynomial in
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terms of powers of x minus pi, or whatever input you're looking at.
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This makes it look noticeably more complicated,
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but all we're doing is making sure that the point pi looks and behaves like 0,
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so that plugging in x equals pi will result in a lot of nice cancellation that
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leaves only one constant.
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And finally, on a more philosophical level, notice how what we're doing here is basically
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taking information about higher order derivatives of a function at a single point,
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and translating that into information about the value of the function near that point.
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You can take as many derivatives of cosine as you want.
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It follows this nice cyclic pattern, cosine of x,
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negative sine of x, negative cosine, sine, and then repeat.
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And the value of each one of these is easy to compute at x equals 0.
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It gives this cyclic pattern 1, 0, negative 1, 0, and then repeat.
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And knowing the values of all those higher order derivatives is a lot of information
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about cosine of x, even though it only involves plugging in a single number, x equals 0.
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So what we're doing is leveraging that information to get an approximation around this
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input, and you do it by creating a polynomial whose higher order derivatives are designed
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to match up with those of cosine, following this same 1, 0, negative 1, 0, cyclic pattern.
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And to do that, you just make each coefficient of the polynomial follow that
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same pattern, but you have to divide each one by the appropriate factorial.
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Like I mentioned before, this is what cancels out
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the cascading effect of many power rule applications.
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The polynomials you get by stopping this process at
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any point are called Taylor polynomials for cosine of x.
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More generally, and hence more abstractly, if we were dealing with some other function
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other than cosine, you would compute its derivative, its second derivative, and so on,
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getting as many terms as you'd like, and you would evaluate each one of them at x equals
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0.
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Then for the polynomial approximation, the coefficient of each x to the n term should be
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the value of the nth derivative of the function evaluated at 0,
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but divided by n factorial.
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This whole rather abstract formula is something you'll likely
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see in any text or course that touches on Taylor polynomials.
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And when you see it, think to yourself that the constant term ensures that
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the value of the polynomial matches with the value of f,
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the next term ensures that the slope of the polynomial matches the slope
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of the function at x equals 0, the next term ensures that the rate at which
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the slope changes is the same at that point, and so on,
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depending on how many terms you want.
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And the more terms you choose, the closer the approximation,
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but the tradeoff is that the polynomial you'd get would be more complicated.
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And to make things even more general, if you wanted to approximate near some input
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other than 0, which we'll call a, you would write this polynomial in terms of powers
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of x minus a, and you would evaluate all the derivatives of f at that input, a.
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This is what Taylor polynomials look like in their fullest generality.
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Changing the value of a changes where this approximation is hugging the original
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function, where its higher order derivatives will be equal to those of the original
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function.
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One of the simplest meaningful examples of this is
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the function e to the x around the input x equals 0.
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Computing the derivatives is super nice, as nice as it gets,
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because the derivative of e to the x is itself,
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so the second derivative is also e to the x, as is its third, and so on.
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So at the point x equals 0, all of these are equal to 1.
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And what that means is our polynomial approximation should look like
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1 plus 1 times x plus 1 over 2 times x squared plus 1 over 3 factorial times x cubed,
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and so on, depending on how many terms you want.
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These are the Taylor polynomials for e to the x.
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Ok, so with that as a foundation, in the spirit of showing you just how connected all
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the topics of calculus are, let me turn to something kind of fun,
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a completely different way to understand this second order term of the Taylor
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polynomials, but geometrically.
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It's related to the fundamental theorem of calculus,
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which I talked about in chapters 1 and 8 if you need a quick refresher.
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Like we did in those videos, consider a function that gives the area
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under some graph between a fixed left point and a variable right point.
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What we're going to do here is think about how to approximate this area function,
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not the function for the graph itself, like we've been doing before.
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Focusing on that area is what's going to make the second order term pop out.
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Remember, the fundamental theorem of calculus is that this graph itself represents the
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derivative of the area function, and it's because a slight nudge dx to the right bound
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of the area gives a new bit of area approximately equal to the height of the graph times
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dx.
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And that approximation is increasingly accurate for smaller and smaller choices of dx.
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But if you wanted to be more accurate about this change in area,
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given some change in x that isn't meant to approach 0,
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you would have to take into account this portion right here,
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which is approximately a triangle.
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Let's name the starting input a, and the nudged input above it x, so that change is x-a.
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The base of that little triangle is that change, x-a,
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and its height is the slope of the graph times x-a.
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Since this graph is the derivative of the area function,
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its slope is the second derivative of the area function, evaluated at the input a.
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So the area of this triangle, 1 half base times height,
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is 1 half times the second derivative of this area function, evaluated at a,
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multiplied by x-a2.
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And this is exactly what you would see with a Taylor polynomial.
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If you knew the various derivative information about this area function at the point a,
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how would you approximate the area at the point x?
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Well you have to include all that area up to a, f of a,
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plus the area of this rectangle here, which is the first derivative, times x-a,
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plus the area of that little triangle, which is 1 half times the second derivative,
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times x-a2.
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I really like this, because even though it looks a bit messy all written out,
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each one of the terms has a very clear meaning that you can just point to on the diagram.
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If you wanted, we could call it an end here, and you would have a
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phenomenally useful tool for approximating these Taylor polynomials.
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But if you're thinking like a mathematician, one question you might ask is
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whether or not it makes sense to never stop and just add infinitely many terms.
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In math, an infinite sum is called a series, so even though one of these
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approximations with finitely many terms is called a Taylor polynomial,
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adding all infinitely many terms gives what's called a Taylor series.
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You have to be really careful with the idea of an infinite series,
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because it doesn't actually make sense to add infinitely many things,
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you can only hit the plus button on the calculator so many times.
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But if you have a series where adding more and more of the terms,
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which makes sense at each step, gets you increasingly close to some specific value,
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what you say is that the series converges to that value.
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Or, if you're comfortable extending the definition of equality to
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include this kind of series convergence, you'd say that the series as a whole,
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this infinite sum, equals the value it's converging to.
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For example, look at the Taylor polynomial for e to the x,
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and plug in some input, like x equals 1.
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As you add more and more polynomial terms, the total sum gets closer and
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closer to the value e, so you say that this infinite series converges to the number e,
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or what's saying the same thing, that it equals the number e.
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In fact, it turns out that if you plug in any other value of x, like x equals 2,
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and look at the value of the higher and higher order Taylor polynomials at this value,
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they will converge towards e to the x, which is e squared.
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This is true for any input, no matter how far away from 0 it is,
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even though these Taylor polynomials are constructed only from derivative information
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gathered at the input 0.
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In a case like this, we say that e to the x equals its own Taylor series at all inputs x,
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which is kind of a magical thing to have happen.
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Even though this is also true for a couple other important functions,
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like sine and cosine, sometimes these series only converge within a
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certain range around the input whose derivative information you're using.
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If you work out the Taylor series for the natural log of x around the input x equals 1,
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which is built by evaluating the higher order derivatives of the natural
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log of x at x equals 1, this is what it would look like.
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When you plug in an input between 0 and 2, adding more and more terms of this
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series will indeed get you closer and closer to the natural log of that input.
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But outside of that range, even by just a little bit,
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the series fails to approach anything.
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As you add on more and more terms, the sum bounces back and forth wildly.
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It does not, as you might expect, approach the natural log of that value,
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even though the natural log of x is perfectly well defined for inputs above 2.
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In some sense, the derivative information of ln
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of x at x equals 1 doesn't propagate out that far.
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In a case like this, where adding more terms of the series doesn't approach anything,
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you say that the series diverges.
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And that maximum distance between the input you're approximating
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near and points where the outputs of these polynomials actually
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converge is called the radius of convergence for the Taylor series.
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There remains more to learn about Taylor series.
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There are many use cases, tactics for placing bounds on the error of
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these approximations, tests for understanding when series do and don't converge,
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and for that matter, there remains more to learn about calculus as a whole,
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and the countless topics not touched by this series.
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The goal with these videos is to give you the fundamental intuitions
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that make you feel confident and efficient in learning more on your own,
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and potentially even rediscovering more of the topic for yourself.
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In the case of Taylor series, the fundamental intuition to keep in mind
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as you explore more of what there is, is that they translate derivative
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information at a single point to approximation information around that point.
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Thank you once again to everybody who supported this series.
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The next series like it will be on probability,
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and if you want early access as those videos are made, you know where to go.
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Thank you.
— end of transcript —
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