Lesson 3.4: Curve Sketching

Curve sketching is one of the most conceptually important applications of the derivative. We have already seen that a point at which the derivative is equal to zero is often a local extreme; combining that with the knowledge we acquired about the second derivative’s meaning, we can determine whether the point is a local minimum, maximum, or neither.

Now it’s time to extend this knowledge so we can discover even more about a function’s graph by analyzing its derivatives. Stewart’s book presents excellent examples on page 243 of the analytic definition of a function’s concavity, but like limits, an intuitive understanding is sufficient for the present.

The concavity of a function is determined by its second derivative: for a continuous function $ f$ on an interval $ (a,b)$, $ f$ is concave upward if $ f”(x)>0$ for $ x\in(a,b)$, and $ f$ is concave downward if $ f”(x)<0$ for $ x\in(a,b)$. Since we have already shown that an inflection point is usually where $ f”(x)=0$, the inflection point is equivalently where the function changes concavity (since it’s continuous).

When attempting to sketch a function’s graph, it’s usually a good idea to make a table of relevant intervals of the graph, along with the sign of the function itself and its first two derivatives on each interval. Then, plot the endpoints of these intervals and synthesize the information in the table into a reasonable sketch of the graph. It will never be perfect, but along with extrema, note where the tangent line must be vertical (because $ f'(x)\to\pm\infty$) and all sign changes of the function and its first two derivatives.

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