Question: The question asks to find a cubic equation given point A (0/18) slope here is 0. Point B only given the x=20 and slope is -0.6.

Please help, been trying to solve it for so long now. :(

1.) y=ax^3+bx^2+cx+d

2.) y'=3ax^2+2bx+c

-0,6=1200a+40b+1c

18=0a+0b+0c+1d

Answer: 
This is one of the most fundamental definitions in all of analysis.

It says that the increment $\Delta f:=f(a+h)-f(a)$ of the function value should in first approximation be a linear function of the increment $h$ attached at the point $a$. In other terms: We want
$$f(a+h)-f(a)=Lh +r(h)\qquad(|h|\ll1)\ ,\tag{1}$$
whereby the error $r(h)$ should be *smaller by magnitudes* than the linear term $Lh$ when $h$ is small. Now in general $|Lh|$ will be of order $|h|$ for "most" $h$. This means that we should require that
$$\lim\_{h\to0}{|r(h)|\over |h|}=0$$
in order to impart any real content to $(1)$. It turns out that this condition determines $L$ uniquely. If it can be satisfied then $f$ is called *differentiable* at $a$, and one denotes the resulting $L$ by $Df\bigr|\_a$, or similar.
I think it's much easier to understand if you write the definition as "$Df\_{|a}$ is the unique linear map $L$ (if it exists) satisfying $f(x) = f(a) + L(x-a) + o(||x-a||)$ when $x\rightarrow a$".

Maybe it's even clearer if you write "$Df\_{|a}$ is the linear part of the unique affine map $A$ (if it exists) satisfying $f(x) = A(x) + o(||x-a||)$ when $x\rightarrow a$". 

So you can see that $A$ is the best possible affine approximation of $f$ near $a$ (because the error you make by replacing $f$ with $A$ is negligible compared to any affine map), and $Df\_{|a}$ is the linear part of this affine approximation.