Question

The linear approximation for the function f(x)=x^5 near x=2 is 32+80(x-2) Check this. need help with B.

A.) What number c will give the best quadratic approximation x^5 approximate to 32+80(x-2)+c(x-2)^2. With the help of myininaya I got c=80.

B.)If this approximation is used for various x's in the interval [2,2.1], can you be certian that the error is no bigger than .05? Explain using Taylor's inequality (the Error Bound).

C.) Graph equation Part A now that we have c. On interval [2,2.1]. note: I don't need someone to graph this, I just want to make sure I have right approach. Would i just pick an x on the interval and set the equation to y and solve? for a point

Answer 1

What I had help with earlier was part A and we ended up taking derivative up to the 2nd degree. and f"(x) was 20x^3 or 160. the taylor poly centered at x=a so f"(a)/2 was 160/2 or 80=c.

Answer 2

Were you able to understand the part before part A?
The linear approximation for the function f(x)=x^5 near x=2 is 32+80(x-2) `Check this.`

Answer 3

x^5=32+80(x-2)
plug in f(2)
32=32+80(0)
32+32

Answer 4

32=32*

Answer 5

Hmm ok.
And part A makes sense maybe? :x

Hmm I'm not sure about this Error Bound nonsense XD lol lemme think..

Answer 6

I'm looking at the formula and explanation in my book. should i type it for you?

Answer 7

Sure :D

Answer 8

note: [ ] will represent absolute value. and *n* is subscript
assume that f^(n+1) (x) exists and is continuous. Let K be a number such that: [f(x)-T*n* (x)] is less than/equal to (K) (([x-a]^(n+1))/(n+1)!) .
where T*n* (x) is the nth Taylor polynomial centered at x+a.

Answer 9

*n* is subscript n

Answer 10

absolute value also on the x-a in the numerator.

Answer 11

Oh ok :x\[\Large \left|f(x)-T_n(x)\right|\quad\le\quad k\frac{|x-a|^{n+1}}{(n+1)!}\]

Answer 12

nice. I have to find out how to do that ha

Answer 13

nice an clean that way

Answer 14

part a was a second degree so i guess it would be T*2*

Answer 15

f(x) =2
n=2?

Answer 16

Grr .. brain.. esploding :c

Answer 17

Hmm I don't think I can make sense of this :(
At least not in a reasonable amount of time, lol.

I found some good videos on the topic though D: Maybe they'll help.
http://www.youtube.com/watch?v=wgkRH5Uoavk
http://www.youtube.com/watch?v=yUUPP70Fhpo

Answer 18

Imma run to the grocery store real quick :D

Answer 19

okay. thank you for the help.