Section 4.3 The Derivative in Graphing and Applications- “Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents”

All graphics are attributed to: Calculus,10/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

Introduction In this section we will discuss procedures for graphing rational functions and other kinds of curves. In many problems, the “properties of interest” in the graph of a function are: Some of these properties may not be relevant in certain cases; for example, polynomials do not have asymptotes. symmetries x-intercepts relative extrema intervals of increasing and decreasing asymptotes periodicity y-intercepts concavity inflection points end behavior

Graphing Rational Functions Recall that a rational function is a function in the form f(x) = P(x)/Q(x) which is the ratio of polynomials P(x) and Q(x). Before graphing, check to make sure whether or not P(x) and Q(x) have any common factors. If they do, there will be holes in the graph. After that, you will be looking at many of the properties listed on the previous slide. You will find more specific instructions on the next slide.

Example – Steps 1-3

Example continued – Steps 4-7

Example with Oblique or Curvilinear Asymptotes There are examples on the bottom half of page 258 where the rational functions have a higher degree polynomial in the numerator than in the denominator. These result in other kinds of asymptotes, such as slant (oblique) asymptotes or curvilinear asymptotes. We will not do these this year.

Graphs with Vertical Tangents and Cusps We commonly find points in our graphs where there is a vertical tangent line, therefore, the function is not differentiable at those values of x. There are four examples of this occurring on the next slide. In the first two graphs, a&b, there is an inflection point with the vertical tangent line. In the second two graphs, c&d, there is a cusp (where the limit approaching from the left does not equal the limit approaching from the right).

Example with Vertical Tangents Follow the same steps as the previous example, but don’t forget to look for vertical tangents. Steps 1-3, 5

Example with Vertical Tangents con’t Step 6 (this is where the extra work comes in)

Example with Vertical Tangents con’t Steps 4 & 7

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