Question

Use the first three non-zero terms of a Taylor series to approximate the following integral ∫.1 (cos(x^2))dx

Answer 1

okay do you know the Taylor series?

Answer 2

how do I find Taylor series again? I know I have to expand it right?

Answer 3

okay i 'm not sure about my answers well this notes will help i hope...
http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx
http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx

Answer 4

So the taylor series is \[\sum_{k=0}^{}\] (-1)^k(x^2)^2k/2k!

Answer 5

and then do I plug .1 in for x?

Answer 6

okay wait a sec please..

Answer 7

okay check your question is it \[\int\limits_{}^{}(1+\cos ^{2}x)dx\]

Answer 8

no it's just\[\int\limits_{0}^{.1}\cos(x^2)\]

Answer 9

okay so here we f(x)=cos(x^2) so can you find Taylor series for it?

Answer 10

i meant we have f(x)=cos(x^2)

Answer 11

okay do you really need Taylor series to do this question

\[∫cos(x2)dx\]

you can do it by simple integration..

or have they asked you do this by using taylor series?

Answer 12

it says to use the first three nonzero terms of a taylor series

Answer 13

if you have to then find the taylor series about x=1 first and then x=0 for f(x)=cos^2 then subtract it

Answer 14

how do i do that?

Answer 15

can you fnd the taylor series about x=1 for f(x)=cos(x^2)

Answer 16

so i plug 1 into the taylor series?

Answer 17

can you show me your taylor series of f(x)=cos(x^2) ? because you have to plug x=1 in taylor series of f(x)=cos(x^2)

Answer 18

-1^k*x^4k/2k!
\[\sum_{k=0}^{infinity}\]

Answer 19

okie here we have a prob.. you have shown me taylor series of f(x)=cosx

and we want to find taylor series of cos(x^2)

i don't what it would be but you can always find it see this notes if need help

http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx

so find it and then plug x=1 find summation of the first three terms

then again plug x=0 in our taylor series of cos(x^2) see the first three terms you get take summation of this terms and subtract it from the summation of the three terms you got when you had put x=1

if when you put x=0 if the series becomes zero.. then no worry

because then your answer would be the summation of first three terms of taylor series of cos(x^2) when you put x=1

[as (summation of first three terms when x=1) - (summation of first three terms when x=0) =summation of first three terms when x=1] assuming we have (summation of first three terms when x=0 is zero

Answer 20

sorry if i elaborated it a lot but i just wanted to be straight and simple