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

Apr 2010
57
1
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
,
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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.

\(\displaystyle -4-1(x-8) + (3/2)(x-8)^2\) 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.
 
May 2010
20
8
\(\displaystyle p_{2}(x)=g(8)+\frac{g'(8)(x-8)}{1!}+\frac{g''(8)(x-8)^2}{2!}\)

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

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

the nth degree adds the term...

\(\displaystyle \frac{g^n(x)(x-8)^n}{n!}\)
 
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Apr 2010
57
1
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
,