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Find the Taylor polynomial of order $3$ at the point $a$ for function $f$ given below and estimate/bound the error by approximating the function $f$ by its Taylor polynomial at point $b$. The function is as follows:

$f(x)=tan(x)$ and $a=\pi/4$, $b=3/4$.

My work:

I found the Taylor polynomial to be :

$p_3(x)=1+2(x-\pi/4)+2(x-\pi/4)^2+(8/3)(x-\pi/4)^3$

Now, the thing that is confusing me is how estimate and find the bound. To find the error, I found error function:

$R_3(x)=tan(x)-p_3(x) = tan(x)-1-2(x-\pi/4)-2(x-\pi/4)^2-(8/3)(x-\pi/4)^3$

Then, this evaluated at $b$, $R_3(b)$, is my error.

From here, I need to find the bound so by assumption of Taylor's theorem, we know that the bound $M$ has to satisfy $\left | f^4(x) \right |\leq M$.

So at point $b$, $\left | f^4(b) \right |\leq M$ so let the bound be $\left| f^4(b) \right |$.

Hence, $\left | R_3(x) \right |\leq\frac{\left| f^4(b) \right|\left | x-\pi/4 \right |^4}{4!}$.

Is this correct?

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