Question 4: (7) If a function f(x) has derivatives of all...

Question 4: (7) If a function $f(x)$ has derivatives of all orders in $R$, proof that we can write $\infty$ $f(x) = \sum_{n=0} \frac{f^{(n)}(x_0)}{n!} \cdot (x - x_0)^n$ Justify your answer with an example.

Transcript

00:01 Here's our function, which is the product of two functions that have derivatives that all orders.

00:06 So to get the first derivative, we are going to use the chain rule.

00:10 I'm not going to write the of x just for conciseness.

00:13 So we take the derivative of f and then multiply it by g.

00:17 And then we'll take the derivative of g and multiply it by f.

00:21 And that's our first derivative.

00:23 Then to get the second derivative, we're going to take basically the pro.

00:28 It's going to be four terms.

00:30 Then we might not be able to combine like terms.

00:32 So it's the second derivative of f times g plus we'll differentiate with respect to g.

00:40 And then we'll take this second term here.

00:42 So basically that goes with that.

00:45 And then for this term, we have the derivative of g.

00:51 And then we'll take their derivative with respect to, i'm sorry, with f and then with respect to g.

00:57 And then these two terms in the middle, because it's just multiplication, it doesn't matter what order we write the f and the g.

01:04 So that's equivalent to two times the derivative of f times the derivative of g.

01:09 And therefore we have the same expression that's given in the problem and we've shown that that's true.

01:14 Then we'll take the third derivative.

01:19 We'll use the simplified expression.

01:23 So let me just write it in blue just so we're clear on all the terms.

01:34 I'm basically just copying the terms over.

01:38 Okay.

01:39 So we have the third derivative of f times the derivative with respect to g, i mean times g, and then we'll take the derivative with respect to g.

01:50 That's term number one.

01:52 And then we have two differentiate with respect to f plus two times differentiate with respect to g plus two more terms.

02:09 Differency with respect to g and then with respect to f.

02:16 Like so...

Question

Question 4: (7) If a function $f(x)$ has derivatives of all orders in $R$, proof that we can write $\infty$ $f(x) = \sum_{n=0} \frac{f^{(n)}(x_0)}{n!} \cdot (x - x_0)^n$ Justify your answer with an example.

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