Question

How to decide whether a taylor polynomial will give you an over or under estimate @ganeshie8

Answer 1

Error: |E_n(x)| = |f^(n + 1)(z)(x - a)^(n + 1)/(n + 1)!|, where z is in 0 < z < a.

Answer 2

The remainder is:
R_n(x) = f^(n + 1)(z)(x - a)^(n + 1)/(n + 1)!, where z is in 0 < z < a.

If R_n(x) > 0, then the polynomial gives an underestimate; if R_n(x) < 0 then the polynomial gives an overestimate.

Answer 3

What you basically want is to compute the distance between the taylor polynomial and the real function, and then tell whether this distance is positive or negative

Answer 4

so if you have this

Answer 5

then you need to evaluate the third degree polynomial at x=.1. which is greater than 0. @P0sitr0n

Answer 6

compute the taylor polynomial of degree 3. Lets say it gives you r*
compute sin(0.1). Lets say it gives you r_.
Then compute r*-r_=R
if R>0, then its an overestimate, R<0 it is underestimate

Answer 7

\[P_3(x)=.1-\frac{ .1^3 }{ 3! }+\frac{ .1^5 }{ 5! }=.0998\]

Answer 8

x=.1, so it would be an overestimate

Answer 9

looks you have computed 5th degree polynomial

Answer 10

thats the taylor series for sin(x) third degree

Answer 11

i think

Answer 12

@P0sitr0n any guidance

Answer 13

siths my dood, you are mad smart. u can guide me in the right direction

Answer 14

if it just asks for \[P_3(.1)\] would you stop when the degree is 3

Answer 15

so it would be n=1

Answer 16

@SithsAndGiggles

Answer 17

@Zarkon

Answer 18

you should stop when the exponent is 3 :
\[P_3(x)=x-\frac{x^{\color{red}{3}} }{ 3! }\]

\[P_3(0.1)=0.1-\frac{0 .1^{\color{red}{3}} }{ 3! }\]

Answer 19

it is still underestimate
http://www.wolframalpha.com/input/?i=%280.1+-+0.1%5E3%2F3%21%29+-+sin%280.1%29

Answer 20

it is an underestimate because the answer is greater than 0? @ganeshie8

Answer 21

your estimation of sin(0.1) using 3rd degree taylor polynomial is
\[P_3(0.1)=0.1-\frac{0 .1^{\color{red}{3}} }{ 3! } \approx 0.0998333\]

actual value of sin(0.1) :
\[\sin(0.1)\approx 0.0998334\]

Answer 22

Clearly your estimation is less than the actual value of sin(0.1)
so yours is an underestimate

Answer 23

while if i were to have this question on a test, and cant use a calculator to see the actual number of sin(.1), how would i reason this?

Answer 24

i dont remember these sorry
@Zarkon @P0sitr0n

Answer 25

for extremely small values of x, sinx ~ x.

Answer 26

so the difference between the taylor polynomial of degree n about x=a. and f(a)=R, where if r is negative its an underestimate, positive overestimate