TITLE: Intuition behind Taylor/Maclaurin Series
QUESTION [5 upvotes]: ** This is a different question than Intuition explanation of taylor expansion? **
I understand some of the intuition behind a Taylor/Maclaurin expansion. More specifically, I understand that adding higher and higher degree polynomials will add more 'turning points' on a graph to better represent the curves of the function you wish to approximate.
I don't understand why
a.) you add the terms; shouldn't adding terms shift the graph left/right, up/down?  In addition to the question of shifting the graph, I just don't understand why you would add more terms, rather than just change your first term accordingly.
I now understand the above, thanks to microarm15 and Nicholas Stull.  I now just do not understand part b of this question
b.) the terms added are the successive derivatives of the function.  What does adding successive derivatives mean/give you?
Any help on the matter is greatly appreciated.
Thanks!

REPLY [2 votes]: b.) the terms added are the successive derivatives of the function.
  What does adding successive derivatives mean/give you?
Any help on the matter is greatly appreciated. Thanks!

Look at the $N$th order Taylor polynomial for $f(x)$ at $x=a$:
$$T_N(x):=\sum_{n=0}^N {f^{(n)}(a)\over n!}(x-a)^n.$$
You say "adding successive derivatives of the function". Well, not exactly. What is happening is that as we add more terms in the Taylor polynomial for $f$, we are adding on higher and higher powers of $x$. The coefficients of those particular terms are related to $f$ via the formula above. But we are not adding on "derivatives of $f$", rather higher order polynomial terms. 
The reason this is helpful is because if you think back to how the $N$th degree Taylor polynomial was derived: $T_N(x)$ was the polynomial of degree $N$ that matched up through the first $N$ derivatives of $f$ at $x=a$. That is, $${d^j\over dx^j}T_N(x)\Bigg|_{x=a}={d^j\over dx^j}f(x)\Bigg|_{x=a}\forall j=0,1,\dots,N$$.
So for example, $T_1(x)$ is the best linear approximation of $f$ at $x=a$ in the sense that $T_1(x)$ matches $f$ in function value and derivative at $x=a$. 
Similarly, $T_2(x)$ is the "best quadratic approximation" of $f$ at $x=a$. And $T_3(x)$ is the best cubic approximation of $f$ at $x=a$. Geometrically, as we had more and more terms to the Taylor polynomial, the result is a better and better approximation (in the sense described above).
Here's a picture of $T_N(x)$, $N=0,\dots,12$ for $f(x)=\sin x$ at $a=0$:

Finally, to connect this back to Taylor series, just let $N\to\infty$. You might find this post helpful.