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Section3.5Curve Sketching

We have been learning how we can understand the behavior of a function based on its first and second derivatives. While we have been treating the properties of a function separately (increasing and decreasing, concave up and concave down, etc.), we combine them here to produce an accurate graph of the function without plotting lots of extraneous points.

Why bother? Graphing utilities are very accessible, whether on a computer, a hand-held calculator, or a smartphone. These resources are usually very fast and accurate. We will see that our method is not particularly fast — it will require time (but it is not hard). So again: why bother?

We are attempting to understand the behavior of a function f based on the information given by its derivatives. While all of a function's derivatives relay information about it, it turns out that “most” of the behavior we care about is explained by fand f. Understanding the interactions between the graph of f and fand fis important. To gain this understanding, one might argue that all that is needed is to look at lots of graphs. This is true to a point, but is somewhat similar to stating that one understands how an engine works after looking only at pictures. It is true that the basic ideas will be conveyed, but “hands-on” access increases understanding.

Key Idea 3.5.1 summarizes what we have learned so far that is applicable to sketching graphs of functions and gives a framework for putting that information together. It is followed by several examples.

Key Idea3.5.1Curve Sketching

To produce an accurate sketch a given function f, consider the following steps.

  1. Find the domain of f. Generally, we assume that the domain is the entire real line then find restrictions, such as where a denominator is 0 or where negatives appear under the radical.

  2. Find the critical values of f.

  3. Find the possible points of inflection of f.

  4. Find the location of any vertical asymptotes of f (usually done in conjunction with Item 1).

  5. Consider the limits lim and \lim\limits_{x\to\infty}f(x) to determine the end behavior of the function.

  6. Create a number line that includes all critical points, possible points of inflection, and locations of vertical asymptotes. For each interval created, determine whether f is increasing or decreasing, concave up or down.

  7. Evaluate f at each critical point and possible point of inflection. Plot these points on a set of axes. Connect these points with curves exhibiting the proper concavity. Sketch asymptotes and x- and y-intercepts where applicable.

Example3.5.2Curve sketching

Use Key Idea 3.5.1 to sketch f(x) = 3x^3-10x^2+7x+5\text{.}

Solution
Example3.5.5Curve sketching

Sketch \ds f(x) = \frac{x^2-x-2}{x^2-x-6}\text{.}

Solution
Example3.5.8Curve sketching

Sketch \ds f(x) = \frac{5(x-2)(x+1)}{x^2+2x+4}\text{.}

Solution

In each of our examples, we found a few significant points on the graph of f that corresponded to changes in increasing/decreasing or concavity. We connected these points with straight lines, then adjusted for concavity, and finished by showing a very accurate, computer generated graph.

Why are computer graphics so good? It is not because computers are “smarter” than we are. Rather, it is largely because computers are much faster at computing than we are. In general, computers graph functions much like most students do when first learning to draw graphs: they plot equally spaced points, then connect the dots using lines. By using lots of points, the connecting lines are short and the graph looks smooth.

This does a fine job of graphing in most cases (in fact, this is the method used for many graphs in this text). However, in regions where the graph is very “curvy,” this can generate noticeable sharp edges on the graph unless a large number of points are used. High quality computer algebra systems, such as Mathematica and Sage, use special algorithms to plot lots of points only where the graph is “curvy.”

In Figure 3.5.11, two graph of y=\sin(x) is given, generated by Sage and Mathematica. The small points represent each of the places where each CAS sampled the function. Notice how at the “bends” of \sin(x)\text{,} lots of points are used; where \sin(x) is relatively straight, fewer points are used. (In the Mathematica plot, many points are also used at the endpoints to ensure the “end behavior” is accurate.)

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(a)Sage output
(b)Mathematica output
Figure3.5.11CAS plots of y=\sin(x) illustrating the sample points

How does Sage know where the graph is “curvy”? Calculus. When we study curvature in a later chapter, we will see how the first and second derivatives of a function work together to provide a measurement of “curviness.” Sage employs algorithms to determine regions of “high curvature” and plots extra points there.

Again, the goal of this section is not “How to graph a function when there is no computer to help.” Rather, the goal is “Understand that the shape of the graph of a function is largely determined by understanding the behavior of the function at a few key places.” In Example 3.5.8, we were able to accurately sketch a complicated graph using only five points and knowledge of asymptotes!

There are many applications of our understanding of derivatives beyond curve sketching. The next chapter explores some of these applications, demonstrating just a few kinds of problems that can be solved with a basic knowledge of differentiation.

Subsection3.5.1Exercises

In the following exercises, practice using Key Idea 3.5.1 by applying the principles to the given functions with familiar graphs.

In the following exercises, sketch a graph of the given function using Key Idea 3.5.1. Show all work; check your answer with technology.

In the following exercises, a function with the parameters a and b are given. Describe the critical points and possible points of inflection of f in terms of a and b\text{.}