Question

For the equation f(x)=arcsinx
find the Taylor Polynomial of degree 3 of at c=1/2

Answer 1

sin^-1 derivatives.... thats gonna be messy i think

Answer 2

4 iterations of it eh...

Answer 3

well, 3

Answer 4

y = sin^1(x)

(y' = 1-x^2)^(-1/2)

Answer 5

f(x)=arcsin(x)
f'(x)=1/Sqrt[1-x^2]
f''(x)=x/(1-x^2)^(3/2)

Answer 6

f(x)+f'(x)(x-c)+f''(x)((x-c)^2)/2!

Answer 7

y'' = x(1-x^2)^(-3/2) yeah, that was easier than i expected :)

Answer 8

y''' = 3x^2 (1-x^2)^(-5/2) right?

Answer 9

so all you are doing is taking the derivative of the the equation to the 3rd prime?

Answer 10

\[f(x)+f'(x)\frac{(x-c)}{1!}+f\text{''}(x)\frac{(x-c)^2}{2!!}+f\text{''}(x)\frac{(x-c)^3}{3!!}\]

Answer 11

\[\sin^{-1}(x) + (1-x^2)^{-1/2} [x-(1/2)] + 2x(1-x^2)^{-3/2} [x-(1/2)]^2 + 3x^2 (1-x^2)^{-5/2}[x-(1/2)]^3\]

Answer 12

i forgot the factorials..... i loathe typing these things out lol

Answer 13

That's why I typed them in mathematica and copy them here

Answer 14

whats the difference between your two answers?

Answer 15

well; mine is missing factorials under them for starters lol

Answer 16

actually same, amistreo actually plug in for f(x),f'(x)

Answer 17

sin−1(x)

+(1−x^2)^(−1/2) [x−(1/2)]

+2x(1−x^2)^(−3/2) [x−(1/2)]^2
----------------------------
2!

+3x^2 (1−x2)^(−5/2) [x−(1/2)]^3
-------------------------------
3!

Answer 18

Oh ok makes sense thanks so from here how do i determine the accuracy of this polynomial?

Answer 19

htat i got no idea about; i just read the material on HOW to do them the other day :)

Answer 20

There is actually a formula that's pain to solve, let me find it

Answer 21

ok

Answer 22

It would be the accuracy at x=\[\sqrt{2}/2\]

Answer 23

Oh, that's easy if you know x

Answer 24

sorry to make you scramble

Answer 25

arcsin(Sqrt[2]/2)

Answer 26

sin of what angle is Sqrt[2]/2

Answer 27

?

Answer 28

i dunno its part of the same question...first it askes to find the taylor polynoial like you did than it said find the accuracy of the polynomial at x=sqrt2/2

Answer 29

So sin(45degree or radian pi/4) is Sqrt(2)/2 which means
arcsin(Sqrt(2)/2) is 45degree or radian pi/4.

Answer 30

To find accuracy
\[\left| pi/4 - what u got up there \right|\]

Answer 31

oh ok gotcha

Answer 32

There are other kind of question where they ask you to find error bound, which is pain to solve

Answer 33

oh yea we haven't gotten that far yet.

Answer 34

oh you will and you will hate it