Welcome to the video on the Taylor theorem and Taylor polynomials and we’ve actually already touch on this. When we did the videos on approximating function with polynomials we used McLauren’s theories which is actually a special case of the Taylor polynomial or the Taylor theorem and we just pick or approximate the function around x equals zero when we did the McLauren. But in general you could approximate around any value and if you do around zero if kind the more general case and where dealing with the Taylor polynomials. What is that? Let me just write the definition down and will do couple of example and we will graph it to get the intuition. Taylor polynomial said if I have differentiable function f of x and I want to approximate it with the polynomials at c. At some value of x equal c I want to approximate this function. Let’s me just draw quick and dirty one and draw the accurate one later. Let say that’s my axis this is my function f of x so I could pick some value c some value x equal to c maybe its right there. So that’s c and I would like to approximate it I would like to create a polynomial that can approximate the function around this point and Taylor theorem tells as the Taylor polynomial to approximate this is and I will give you the intuition for it and the second p of x and its look complicated when you some examples see its not so bad p of x is equal to f of x plus f prime of c time x minus c plus f prime prime of c they say two factorial which is just two but all right two factorial they do that to see the pattern that is merges this is over one factorial really and this is zero over factorial really time x minus c squared plus f the third derivative of the function that evaluated that c over three factorial time x minus c to the third and you can just keep adding terms you could go long like this for infinity.
Let me give the intuition of what this is I could let me just show you just to hit the point then you could plus the fourth derivative of the function evaluated in c times over four factorial time x minus c to the fourth factorial. Now what’s in intuition first of all what happen to the polynomial at c. So what is p of c? p of c is equal to well if p of c everywhere you see next to you put c this term would be c minus c that could be zero or would be zero this term c minus c be zero, this term would be c minus c this could be zero and this term will be c minus c be zero and all you be left is f of c. Great we already know that at the value of c the polynomial is equal to the function so it will intersect this line. And actually if we just had a Taylor polynomial with just this first term will it look like or just be a horizontal line right there. It will be pretty bad approximation but what is the second term doing us? Because we know that the value of c all this other term drop out now what they do for? Well the second term actually in shores that the derivative of this polynomial value is c is equal to derivative of this function of the evaluated of this c. what do I mean there? What’s the prime of x? P prime of x is equal to but this is just a constant term it might look like a function but it’s a function of value in it and c so this is just a constant term. And so that is zero and what is this? What’s the derivative of this? Well the derivative we can use this is a constant term this is derivative this is just one you could almost use f prime of c times x minus f prime of c times c which is a constant. The derivative of this expression is f prime of c, f prime c and then plus the derivative of this expression and that’s equal to what? Two divided by two factorial which is one f prime prime of c times x minus c and then plus three over three factorial its three over six will just have a two in the denominator f prime prime prime of c over two times x minus c to the square and don’t have to worry about all of this we can just keep going but I just want to show you one thing what is p prime at c? What is the derivative of this polynomial when you evaluated that c? Well when you put c in derivative function all this other term will just drop off and you’re going to be left at this one you just left with this one right. If you put the c here it drop out so the derivative of this function that evaluate c is equal to f prime of c. as you could see Taylor polynomial is equal to the function had c its derivative equal to the function c. At second derivative is equal to the function of c and every term you add to the Taylor polynomials actually makes it so that you know that term derivative at the polynomial evaluated its c is equal to the function. Hope I didn’t confuse you.
The big picture is the whole thinking behind I guess what Taylor thought of was you know I was construct if this function is infinitely differentiable meaning that I can take the first second third fourth you know all the way infinity derivative of this function I could construct a polynomial like this I could keep going by adding more terms so that this polynomials you know zero derivative the function zero first second third fourth all of this polynomials derivatives are going to the equal to the function at least around that point. Actually the whole class of function is that the Taylor polynomial if you where take the infinite series is equal to that function at all points. I talk a little bit about that one I proved that either the pi is equal to negative one which to me was the most amazing result in mathematics whatever. This might have been little confusing for you so lets do a particular example.
The particular are always the more fan. I think if you see doing example you’ll see that its not so bad I mean erase fourth term. Lets approximate sine of x so lets say that f of x is equal to let me do this in different color we want approximate f of x is equal to sine of x and lets pick some arbitrary number, let’s not pick number works well in trigonometric functions. Let’s pick around or let say c is equal to two or one. Where going to approximate sine of x around one. What is the Taylor approximation or the Taylor polynomials? We can just chug through this one, p of x put it in yellow p of x is equal to f of c so the function evaluated c is just sine of one plus f prime of c. What is derivative of sine of x its minus sine of x minus sine of x and we have to evaluated c so its minus sine of one, c is one that is our approximating around times x minus c and then plus the second derivative plus the second derivative of x. What was the second derivative? What’s going to be the derivative of minus sine which is minus sine of x so it’s minus sine but where evaluating c the section to be number right, so c is one sine of one over two, two factorials is two time x minus one time x minus one squared lets just keep going plus the third derivative plus what the third derivative of sine? Well it’s the derivative of minus sine so that’s plus sine, so plus sine evaluated as one divided by three factorials so that’s six over six times x minus three to the third and then lets do one more term just for fun. So then where going to take the fourth derivative which is the derivative of the third derivative so the third derivative was positive sine plus sine evaluated one over four factorial.
What is four factorial? It’s three factorial plus four so over twenty four times x minus one to the fourth we can just keep going the fifth derivative over five evaluated in one over five factorial times x minus one over the fifth and just keep adding then will take as forever. What this thing look like? I want to do I want to show you how this polynomial develop as we add terms. Lets see I have this graphing calculator that I so this thing o got from to just give some credit its my.hrw.com and this is the graph of sine of x just the first term here sine of one if we where just graph the first term of this polynomial what it is look like? I’m just typing sine one and graph it. There you go the first term of the polynomial if all of this terms this other terms where not here the polynomial will be constant. Sine of one and its pretty bad approximation but at least it equals the function at this point give as something. Let’s have some terms done it so what was the second term. Sine of one minus sine of one time x minus one so let me add that. Minus sine of one times x minus one graph it there you go. This is neat when you just add two terms the function will equal the polynomial will equal the function at x equals one and the slope is also equal to the function. The slope of the polynomial is also equal to the slope of the function and x is equal to one this is the better approximation at least if we stay pretty close to our chosen c we’re pretty—it’s a decent approximation for the function.
Obviously if we get far away out here this is horrible approximation for the function lets keep adding terms. As you can see I will show you that I’m just typing in the actual terms. Let me type next term just to see you believe that I’m doing it. The next term will have to see it let me type it in so the next term is minus sine of one divided by two times x minus one squared and graph it. Know just to show I jut type in the second term and I’ll look at the graph know this is neat. So the first term got us a horizontal line that just intersected the point at sine of one it’s a really bad approximation then the second term made 13:56 at least the first term was the same and so then we the line was just tangent line where only have two terms. The third terms make sure that the second derivative of our polynomial at x equals one is equal to the second derivative of polynomial of the function. Notice that this green graph is con cave downwards which means and so it’s the function at one. This is pretty neat where getting a little bit so it’s kind a approximating the curve here it’s getting a little bit better remember when far out on the left start to approximate the function better here closer at least. The last time the line just went up here was a really bad approximation. Let’s add another term lets have our third term I can see it right there so plus sine of one divided by six times x minus one to the third power. Just to show you I just type it in right there. It’s minus one to the third power let me graph it. That is neat just with three terms in our polynomial actually that was. The first term was anyway you got the point. But where already start the approximate this pretty well. Know the third derivative is equal to the third derivative of the polynomial is equal to the third derivative of the function at the point x is equal to one. And we have even studied this just kind the like the connectivity of derivative or whatever.
As you can see the approximate the function even better obviously we go further away its start to break down again pretty close if this fall from here hard to tell apart. Let’s have the last term we calculated. This should be pretty neat, the last term plus sine of one divided by twenty four as of every term the scaling factor one then one half six one twenty fourth it becomes a smaller impact on it and its starts to matter as you move really really far away from you chosen c, in this case one right the further you have to go when your close to your point that you pick these other term don’t matter much because your doing one twenty fourth then five factorial but if get further and further away these term becomes more significant as x gets further, further away from one and then that’s the way you plane in approximation.
Now let me graph it so sine of one divided by twenty four times x minus one to the fourth. Let me graph it even neater if you have spare time you have keep adding terms to this. That’s all the Taylor polynomial is. Now I realize that this is probably longest videos I’ve done I’m pushing seventeen minutes. It’s a little confusing at first cause it give you huge formula and they give you the c and how do I take the derivative section. All I’m saying constructing a polynomial that add some points c that we pick these polynomials zero first second third fourth fifth and so on to the derivative is going to be equal to our function I actually if we did you know if we did ten term or all the derivatives this start equally each other. Hopefully did not confuse you I know its kind the see the formula first it can be daunting and especially sometimes it’s even more daunting when someone even explains it to you. But hopefully that gave you some intuition, if it didn’t, ignore this video. See you soon
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