Estimate $f(b)$ using Taylor Expansion for $f'(x) = cos(x^2)$

Mathematics Asked by brucemcmc on January 7, 2022

I am using Taylor Expansion for the following problem, but for some reason I am getting wrong solutions from a program I am running it on. Can someone please help me find $f(b)$ for $b=2,3$?? Thank you so much for your time and help!

Suppose $f'(x) = cos(x^2)$ and $f(0) = 1$. I wish to estimate $f(b)$ for the following:

$b=0 : f(b) approx 1$

$b=1 : f(b) approx 1.90452$

$b=2 : f(b) approx 1.975$ (which is incorrect, and I do not understand why)

$b=3 : f(b) approx ??$

One Answer

You're just not using enough terms to get a good approximation so far away from the point that you are Taylor expanding around. Taylor expansions around $0$ are most accurate near $0$.

In the following Desmos graph the function $f(x) = 1 + int_0^x cos(s^2), text ds$ is plotted in $color{red}{text{red}}$. A $pm0.1$ tolerance region is plotted around $f$ in $color{darkgreen}{text{green}}$. The first 5 terms, i.e. the 4th order Taylor approximation, plotted in $color{blue}{text{blue}}$ leaves the green region at about $x=1.9$, which explains the bad quality of your approximation at $x=2$. If you want to approximate $f$ at $x=3$ to $0.1$ error, you will need to use at least 12 terms; this is the $color{purple}{text{purple}}$ graph. Graph,

Answered by Calvin Khor on January 7, 2022

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