Taylor series | Chapter 11, Essence of calculus | Highlights and Annotations by Gistr.

This video explains Taylor series, a powerful tool for approximating functions using polynomials. It demonstrates how to construct Taylor polynomials by matching derivatives at a point, emphasizing the role of each coefficient and the use of factorials. The video then extends this to Taylor series (infinite sums), discussing convergence and the radius of convergence, using examples like eˣ and ln x. Finally, it connects Taylor series to the fundamental theorem of calculus through a geometric interpretation of the second-order term. well, first of all, at the input 0, the value of cosine of x is 1, so if our approximation is going to be any good at all, it should also equal 1 at the input x equals 0. plugging in 0 just results in whatever C0 is, so we can set that equal to 1. this leaves us free to choose constants C1 and C2 to make this approximation as good as we can, but nothing we do with them is going to change the fact that the polynomial equals 1 at x equals 0. it would also be good if our approximation had the same tangent slope as cosine x at this point of interest,. otherwise the approximation drifts away from the cosine graph much faster than it needs to.! the derivative of cosine is negative, sine, and at x equals 0,, that equals 0,, meaning the tangent line is perfectly flat..! on the other hand,, when you work out the derivative of our quadratic, you get C1 plus 2 times C2 times x. at x equals 0,, this just equals whatever we choose for C1..! so this constant, C1 has complete control over the derivative of our approximation around x equals 0.. setting it equal to 0 ensures that our approximation also has a flat tangent line at this point..! this leaves us free to change C2,; but the value and the slope of our polynomial at x equals 0 are locked in place to match that of cosine..-the final thing to take advantage of is the fact that the cosine graph curves downward above x equals 0,, it has a negative second derivative.. or in other words,, even though the rate of change is 0, at that point, the rate of change itself is decreasing around that point.. specifically,, since its derivative is negative sine of x,, its second derivative is negative cosine of x, and at x equals 0,, that equals negative 1..-now in the same way that we wanted the derivative of our approximation to match that of the cosine, so that their values wouldn't drift apart. needlessly quickly, making sure that their second derivatives match will ensure that they curve at the same rate, that the slope of our polynomial doesn't drift away from the slope of cosine x any more quickly than it needs to.. pulling up the same derivative we had before, and then taking its derivative, we see that the second derivative of this polynomial is exactly 2 times C2. so to make sure that this second derivative also equals negative 1 at x equals 0, 2 times C2 has to be negative 1, meaning c2 itself should be negative 1 half. this gives us the approximation 1 plus 0x minus 1 half x squared.. To get a feel for how good it is, if you estimate cosine of 0.1 using this polynomial, you'd estimate it to be 0.995, and this is the true value of cosine, of 0.1.. it's a really good approximation.! take a moment to reflect on what just happened.. you had 3 degrees of freedom with this quadratic approximation, the constants C0, C1, and C2. C0 was responsible for making sure that the output of the approximation matches that of cosine x at x equals 0, C1 was in charge of making sure that the derivatives match at that point, and C2 was responsible for making sure that the second derivatives match up. this ensures that the way your approximation changes as you move away from x equals 0, and the way that the rate of change itself changes, is as similar as possible to the behaviour of cosine x, given the amount of control you have. you could give yourself more control by allowing more terms in your polynomial and matching higher order derivatives.. for example, let's say you added on the term C3 times x cubed for some constant C3. in that case, if you take the third derivative of a cubic polynomial, anything quadratic or smaller goes to 0. as for that last term, after 3 iterations of the power rule, it looks like 1 times 2 times 3 times C3.. on the other hand, the third derivative of cosine x comes out to sine x, which equals 0 at x equals 0. so to make sure that the third derivatives match, the constant C3 should be 0. or in other words, not only is 1 minus ½ x2 the best possible quadratic approximation of cosine, it's also the best possible cubic approximation. you can make an improvement by adding on a fourth order term, C4 times x to the fourth. the fourth derivative of cosine is itself, which equals 1 at x equals 0. and what's the fourth derivative of our polynomial with this new term? well, when you keep applying the power rule over and over, with those exponents all hopping down in front, you end up with 1 times 2 times 3 times 4 times C4, which is 24 times C4. so if we want this to match the fourth derivative of cosine x, which is 1, C4 has to be 1 over 24. and indeed, the polynomial 1 minus ½ x2 plus 1. 24 times x to the fourth, which looks like this, is a very close approximation for cosine x around x equals 0. in any physics problem involving the cosine of a small angle, for example, predictions would be almost unnoticeably different if you substituted this polynomial for cosine of x. take a step back and notice a few things happening with this process. So you don't simply set the coefficients of the polynomial equal to whatever derivative you want, you have to divide by the appropriate factorial to cancel out this effect.. for example, that x to the fourth coefficient was the fourth derivative of cosine, 1, but divided by 4 factorial, 24. the second thing to notice is that adding on new terms, like this C4 times x to the fourth, doesn't mess up what the old terms should be, and that's really important. for example, the second derivative of this polynomial at x equals 0 is still equal to 2 times the second coefficient, even after you introduce higher order terms. And it's because we're plugging in x equals 0, so the second derivative of any higher order term, which all include an x, will just wash away. and the same goes for any other derivative,, which is why each derivative of a polynomial at x equals 0 is controlled by one and only one of the coefficients.. if instead you were approximating near an input other than 0, like x equals pi,, in order to get the same effect, you would have to write your polynomial in terms of powers of x minus pi, or whatever input you're looking at.. this makes it look noticeably more complicated, but all we're doing is making sure that the point pi looks and behaves like 0, so that plugging in x equals pi will result in a lot of nice cancellation that leaves only one constant. and finally,, on a more philosophical level,, notice how what we're doing here is basically taking information about higher order derivatives of a function at a single point, and translating that into information about the value of the function near that point.. you can take as many derivatives of cosine as you want.. it follows this nice cyclic pattern,, cosine of x, negative sine of x, negative cosine,, sine, and then repeat. Derivative Informationat at a point ->Output information near that point The Taylor series formula for a function f(x) centered at a point a is: f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ... This represents an infinite sum of terms, where each term involves a derivative of f(x) evaluated at a , multiplied by a power of (x-a) and divided by the factorial of the derivative's order. The series approximates the function f(x) near the point a. Whether the series converges to the function's actual value depends on the function and the interval around a. Matching higher-order derivatives in a Taylor polynomial approximation leads to a more accurate representation of the original function near the point of expansion. Each derivative term captures a different aspect of the function's behavior (slope, concavity, etc.), so including more terms (higher-order derivatives) provides a more detailed approximation and reduces error. The greater the number of matching derivatives, the closer the polynomial's behavior resembles the original function's behavior near the expansion point. then for the polynomial approximation,, the coefficient of each x to the n term should be the value of the nth derivative of the function, evaluated at 0, but divided by n factorial..-this whole rather abstract formula is something you'll likely see in any text or course that touches on Taylor polynomials.., and when you see it,, think to yourself, that the constant term ensures that the value of the polynomial matches with the value of f., the next term ensures that the slope of the polynomial matches the slope of the function at x equals 0., the next term ensures that the rate at which the slope changes is the same at that point,, and so on,, depending on how many terms you want.. And the more terms you choose,, the closer the approximation, but the tradeoff is that the polynomial you'd get would be more complicated.., And to make things even more general,, if you wanted to approximate near some input other than 0,, which we'll call a,, you would write this polynomial in terms of powers of x minus A., and you would evaluate all the derivatives of f at that input, A.. this is what Taylor polynomials look like, in their fullest generality.. changing the value of a changes where this approximation is hugging the original function,, where its higher order derivatives will be equal to those of the original function..-one of the simplest meaningful examples of this is the function e to the x around the input x equals 0.. computing the derivatives is super nice,, as nice as it gets,, because the derivative of e to the x is itself,,. so the second derivative is also e to the x, as is its third, and so on.., so at the point x equals 0,, all of these are equal to 1.., and what that means is our polynomial approximation should look like 1 plus 1 times x plus 1 over 2 times x squared, plus 1 over 3 factorial times x cubed,, and so on,, depending on how many terms you want..-these are the Taylor polynomials for e to the x. That statement, "Height is equal to slope * base," is only true for a specific type of triangle within a specific context. It's derived from the relationship between the slope of a line and the coordinates of points on that line. Consider a right-angled triangle formed by a line segment on a graph. The slope of the line is the ratio of the vertical change (height) to the horizontal change (base). Therefore, height = slope * base. This relationship doesn't hold for triangles that aren't right-angled or aren't formed by a line segment on a graph. Slope formula: Slope = (Change in Vertical Direction) / (Change in Horizontal Direction) In this context: "Change in Vertical Direction" is the height, and "Change in Horizontal Direction" is the base. Equation: Height = Slope * Base the base of that little triangle is that change?, x-a, and its height is the slope of the graph times x-a.. since this graph is the derivative of the area function,, its slope is the second derivative of the area function, evaluated at the input a.. so the area of this triangle, 1 half base times height,, is 1 half times The second derivative of this area function,, evaluated at a, multiplied by x-a2.. And this is exactly what you would see with a Taylor polynomial.. if you knew the various derivative information about this area function at the point a,, how would you approximate the area at the point x?, well, you have to include all that area up to a, f of a, plus the area of this rectangle here,, which is the first derivative,, times x-a, plus the area of that little triangle, which is 1 half times the second derivative,, times x-A2.. I really like this,, because even though it looks a bit messy, all written out, each one of the terms has a very clear meaning that you can just point to on the diagram.. if you wanted,, we could call it an end here, and you would have a phenomenally useful tool for approximating these Taylor polynomials.. But if you're thinking like a mathematician, one question you might ask is whether or not it makes sense to never stop and just add infinitely many terms. in math,, an infinite sum is called a series., So even though one of these approximations with finitely many terms is called a Taylor polynomial,, adding all infinitely many terms gives what's called a Taylor series.. you have to be really careful with the idea of an infinite series, because it doesn't actually make sense to add infinitely many things, you can only hit the plus button on the calculator so many times.. But if you have a series where adding more and more of the terms, which makes sense at each step,, gets you increasingly close to some specific value,, what you say is that the series converges to that value.. is called the radius of convergence for the Taylor series.. there remains more to learn about Taylor series,. there are many use cases,, tactics for placing bounds on the error of these approximations,, tests for understanding when series do and don't converge.,. And for that matter,, there remains more to learn about calculus as a whole, and the countless topics not touched by this series.., the goal with these videos is to give you the fundamental intuitions that make you feel confident and efficient in learning more on your own, and potentially even rediscovering more of the topic for yourself.? in the case of Taylor series,, the fundamental intuition to keep in mind as you explore more of what there is,, is that they translate derivative information at a single point to approximation information around that point.. thank you, once again, to everybody who supported this series.. the next series, like it will be on probability., and if you want early access, as those videos are made,, you know where to go.. thank you.. Taylor series find applications in diverse fields: Physics: Solving differential equations that lack analytical solutions, approximating solutions to problems in mechanics and electromagnetism. Engineering: Analyzing and designing control systems, modeling complex systems (e.g., nonlinear oscillations), and simulating physical phenomena. Computer Science: Developing numerical algorithms, approximating functions in computer graphics and simulations, and creating efficient computational methods. Economics and Finance: Modeling economic growth, pricing derivatives, and forecasting market trends. The applications are broad and depend on the need for accurate function approximations. trying to solve are beyond the point here.,. But what I'll say is that this cosine function made the problem awkward and unwieldy, and made it less clear how pendulums relate to other oscillating phenomena.., but if you approximate cosine of theta as 1 minus theta squared over 2, everything just fell into place much more easily.. if you've never seen anything like this before, an approximation like that might seem completely out of left field.. if you graph cosine of theta along with this function, 1 minus theta squared over 2, they do seem rather close to each other, at least for small angles near 0,. But how would you even think to make this approximation, And how would you find that particular quadratic? the study of Taylor series is largely about taking non-polynomial functions and finding polynomials that approximate them near some input.. the motive here is that polynomials tend to be much easier to deal with than other functions, they're easier to compute, easier to take derivatives, easier to integrate, just all around more friendly.. So let's take a look at that function, cosine of x, and really take a moment #FYNEMAN AI -> notewave Powerful tool for approximating function Taylor Series is all about approximating non polynomial function into polynomial function as polynomial are easy to deal with