I need to find the Polynomial with these properties (apparently using Taylor Series)

[sleek vine]

I need to find the Polynomial with the degree n that has the properties listed in the picture.
Thanks for your time!

[winged wharf]

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[forest summit]

So is that the polynomial such that for a particular point a, it satisfies those conditions?

[limber ice]

Well use Taylor ‘s formula for polynomials or if you don’t know it it’s simply an application of the formula of the coefficients for Taylor series

$f(x)=\sum_{n=0}^{\infty} a_n (x-a)^{n}$

[swift smelt]

Rotor 😑

[limber ice]

Whats $a_n$ for any n?

[swift smelt]

Rotor 😑

[limber ice]

I’m guessing so

[sleek vine]

Yes

I have an example for a particular case:

But for n the professor recommended using Taylor Series. But... don't have any Idea.

[limber ice]

Like I said use this

[limber ice]

.

Whats $a_n$ for any n?

[swift smelt]

Rotor 😑

[limber ice]

@sleek vine maybe the general case is too general if P is a polynomial then you can write $P(x)=\sum_{k=0}^{n} a_k (x-a)^{k}$ and you can actually find each $a_k$

[swift smelt]

Rotor 😑

[limber ice]

With respect to the successive derivatives of P at a

[limber ice]

Polynomial of degree n

[sleek vine]

Aight

[limber ice]

So you know how to ?

[sleek vine]

Yes, I think I found a way, thanks!

+close