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)

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

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. ,

Answered by Calvin Khor on January 7, 2022