Question $1$

Can we always write the Taylor polynomial in summation notation?

A$.$ Yes, but we may have to consider certain terms $($e$.$g$.,$ the one with j$=0)$ separately.

B$.$ No$,$ you always need to expand it$.$

C$.$ Yes, the summation notation is the original formulation of the Taylor polynomial.

D$.$ Yes, but we have to consider every term separately.

Question $2$

I suggest to use $128$ intervals for plots of Taylor polynomials, because $...$

A$.$ it gives a vector of $129$ points, which is modest in length,

B$.$ it makes the plot of a typical continuous function look smooth,

C$.$ a division by $128$ does not incur any round$-$off error in Matlab$/$Octave$,$

D$.$ all of the above.

Question $3$

In Taylor's theorem involving the Taylor polynomial of degree n$,$ how many times does f$($x$)$ have to be continuously differentiable?

A$.$ n$-1$

B$.$ n

C$.$ n$+1$

D$.$ infinitely often.

Question $4$

The proof for Taylor's theorem uses $...$

A$.$ integration by parts,

B$.$ the Fundamental Theorem of Calculus,

C$.$ the Integral Mean Value Theorem,

D$.$ all of the above.