Curve sketching

Introduction:

In this section we discuss how to sketch the graph of a function y = f(x) without plotting many points. Indeed, most parts of a curve are routine and uninteresting and plotting them precisely serves very little purpose. What is really important is to learn the exact points features of the curve change and if we plot these points precisely then we will have a reliable as well as useful graph.

Without having to make up a table of values for the function, we can obtain a good qualitative picture of the graph using certain crucial information local maxima and local minima, inflection points, asymptotes, etc. Our aim is not to draw an exact graph, but rather to get an accurate overall picture of the graph and to pinpoint the points where something special happens.

We start by describing the steps to take in curve sketching. For a particular f(x), not all of the steps below will necessarily lead to useful information.

Concavity and Points of Inflection

A graph is called concave upward (CU) on an interval I, if the graph of the function lies above all of the tangent lines on I. A graph is called concave downward (CD) on an interval I, if the graph of the function lies below all of the tangent lines on I.

The second derivative of a function can tell us whether a function is concave upward or concave downward. If

f`(x) > 0 for all x in an interval I, the graph is concave upward on I.

f`(x) < 0 for all x in an interval I, the graph is concave downward on I.

The intervals of concavity will occur between points where f`(x) = 0 or f`(x) is undefined. We test the concavity around these points even if they are not included in the domain of f.

The graphs illustrate the different forms of concavity. Remember that there are two ways in which a graph can be concave upward or concave downward.

A point P on a curve is called a point of inflection if the function is continuous at that point and either

the function changes from CU to CD at P

the function changes from CD to CU at P

Points of inflection may occur at points where f`(x) = 0 or f`(x) is undefined, where x is in the domain of f. We must test the concavity around these points to determine whether they are points of inflection.

The Second Derivative Test

Let f be a continuous function near c. If

f`(c) = 0 and f`(c) > 0, then f has a local minmum at c.

f`(c) = 0 and f`(c) < 0, then f has a local maximum at c.

The graphs containing local maximums and minimums in the Increasing and Decreasing Functions and The First Derivative Test illustrate the second derivative test. When a graph has a local minimum, the function is concave upward (and thus lies above the tangent lines) at the minimum. Similarly, the function is concave downward at a local maximum.