Thread: What is the Taylor polynomial of degree 2 for g near 8?

1. What is the Taylor polynomial of degree 2 for g near 8?

Suppose is a function which has continuous derivatives, and that , , . (a) What is the Taylor polynomial of degree 2 for near ?

I got -4-1(x-8), but it's wrong.

(b) What is the Taylor polynomial of degree 3 for near ?

I got -4-1(x-8) + (3/2)(x-8)^2, but it's wrong.

(c) Use the two polynomials that you found in parts (a) and (b) to approximate .
With ,
I got -4-1(.1), but it's wrong
With ,

2. Do you understand what "degree" means? A polynomial is of degree "n" if and only if the highest power in the polynomial is n (I bet you learned that long ago!).

4- 1(x- 8) is wrong because it is of degree 1, not 2.

$-4-1(x-8) + (3/2)(x-8)^2$ is wrong because it is of degree 2, not 3.

You seem be thinking that "degree" is the number of terms in the polynomial- it isn't. A polynomial of degree n typically has n+1 terms.

3. $p_{2}(x)=g(8)+\frac{g'(8)(x-8)}{1!}+\frac{g''(8)(x-8)^2}{2!}$

$p_{2}(x)=-4-1(x-8)+\frac{4(x-8)^2}{2}$

$p_{3}(x)=p_{2}(x)+\frac{g'''(8)(x-8)^3}{3!}$

the nth degree adds the term...

$\frac{g^n(x)(x-8)^n}{n!}$

4. Thanks! do you know how to do part c?

c) Use the two polynomials that you found in parts (a) and (b) to approximate .
With ,
With ,