When drawing function, compute $f',~f''$ first or vertical asymptotic first?

Question

In Calculus, some functions don't have vertical asymptotics (e.g. $f(x)=2x+\cos x$), while some functions, say $f(x)=x+1/x$, may have vertical asymptotics. In the former case, we just compute $f'$ and $f''$ as always. However, in the latter case, some teachers tend to compute vertical asymptotic before computing $f'$ and $f''$, which interrupt the habit that students accustomed to earlier. So the natural question to ask here, which thing to compute first is better? Any concrete example is welcomed.

Answer 1

People generally check for vertical asymptotes first because a vertical asymptote can immediately and easily be sketched into a graph. Also the location of vertical asymptotes can help flesh out the meaning of derivatives.

On the other hand, if we started with derivatives, they might imply asymptotes but we would probably still want to check for asymptotes before graphing.

Answer 2

If in the denominator some terms are noticeable which go to zero then compute roots (called poles) of that term at first, to fix the column in which your curve can fit in.

Next setting derivative of $f(x)$ to zero find corresponding max/min values. Look to $x$ values when $y=0$ or $y$ values when $x=0$.

When sketching curve in area between asymptotes choose $x$ values to fill in U shaped deep cups and humps with steep cliffs.