Chapter 13: Curve Sketching

In this section we discuss how to sketch the graph of a function 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. The various possibilities will be illustrated later in the examples.

Maxima/Minima

If (p,f(p)) is a point where f(p) is the highest value as long as the x-values vary in a small interval around p, then f(p) is said to be a local maximum value of f(x) . Similarly, f(p) is a local minimum value , if f(p) is the lowest value as x varies in a small interval near p. Collectively, a local maximum or minimum is called a local extremum . The plurals of maximum, minimum and extremum are maxima, minima and extrema respectively.

If the derivative exists at , or, in other words, if the tangent line to the curve exists at , then it can be shown that t he tangent line at that point is horizontal , i.e., .

The idea of the proof of this claim is easy. If then the function is increasing near x = p and there must be values of f(x) bigger than f(p) near (and to the right) of x = p. Similarly, if , then the function is decreasing at x = p and there must be values of f(x) smaller than f(p) near (and to the left) of x = p. So, in either case, the function cannot have a local maximum at x = p. The case of a local minimum is similar.

Points with a horizontal tangent line can tbe found by setting the derivative equal to zero, and solving for x. (There may be one x for which , there may be many, or there may be none.) Once you find the x for which (and the corresponding y-coordinate of each possible max/min point), you have to determine whether it really is a maximum or minimum. In simple examples it might be obvious. A systematic way to tell is by the second derivative test, which will be described below.

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2. A maximum or minimum might occur at a point where there is no well defined tangent line, i.e., where does not exist. For example, has its (local as well as absolute) minimum point at (0,0).

( Note: Generally, we forbid taking fractional powers of negative numbers. However, the function that we are talking about here is the ``real'' cube-root which is well defined even for negative values of x. It can be thought to be the usual cube-root function extended to the negative values by making it an odd function.)

3. Suppose you want to find the absolute maximum of a function f(x) on the interval from a to b. To do this, you have to try:

(a) all points x where ,

(b) all points x where does not exist (as explained above), and

(c) the endpoints a and b (for example, in the above drawing the maximum point is at the endpoint b).

The largest of the function values among these shall come out to be the absolute maximum.

4. Later, we will see examples of points where, even though the slope of the tangent line is 0, we have ( neither ) a maximum nor a minimum there.

The Second Derivative Test

Suppose that x = p is a solution of . The second derivative test is a method that ( in most cases ) will tell us whether the point ( ) is a local maximum or minimum. The test works as follows:

Compute the second derivative of the function at the same value x = p , i.e., . If is positive, that means that is an increasing function at p, i.e., is negative a little to the left of p and is positive a little to the right of p. This is what happens near a local minimum . On the other hand, if is negative, then is positive a little to the left of p and is negative a little to the right of p, which is what happens near a local maximum . If it so happens that = 0 (i.e., if ( it both) and are zero at the same x , then the second derivative test doesn't give you any information. To summarize,

If then:

< 0 implies that p is a local maximum point

< 0 implies that p is a local maximum point

= 0 can't tell if p is a local maximum, local minimum, or neither.

To see how and why this test succeeds or fails, you should always consider
graphs of the functions , , , and their negatives. We have provided a picture below which contains each of these and its derivative. You should analyze the point x = 0 in each case. You should separately consider the function f(x) = 1.

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The First Derivative Test

The Second Derivative Test works like a charm if we can find and it is nonzero. If it is difficult to evaluate or zero, then the first derivative case comes in handy. In fact, it can also be used in place of the Second Derivative Test.

Suppose you are testing a point x = p and . You wish to check if this is a local maximum or local minimum. Evaluate the derivative at a point to the left of p and near p. Also evaluate the derivative at a point to the right of p and near p.

The test says:

If you have only finitely many critical points, then this will be easy to do. If you have infinitely many critical points which cluster around x = p, then this test cannot be applied at all!

See if you can think of a function with infintely many critical points which cluster around a single point. They don't occur naturally but good approximations of them do.

Concavity and Inflection

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There are several ways to describe the difference between upward concavity and downward concavity :

(1) An upward concave function looks like part of a right-side-up bowl (holds water), while a concave downward function looks like part of an upside-down bowl.

(2) If you draw the tangent line to a concave upward function at a point, then the curve sits on the tangent line, whereas in the case of a concave downward function the tangent line rests on the curve.

(3) As you go from left to right, the tangent line to a concave upward curve rotates counterclockwise, while the tangent line to a concave downward curve rotates clockwise.

Thus, inflection points can usually be found by setting and solving for x. Each of the drawings for Examples 13.1 and 13.2 below shows a point of inflection at the origin separating a concave downward part of the curve (on the left) from a concave upward part (on the right).

Warning . Notice that ( ) may not be a point of inflection even though . After finding the points where , you then must check that changes sign. (See Example 13.5 below.)

Second Warning . If (x,f(x)) is an inflection point and exists then it will be zero. However it can happen that (x,f(x)) is a point of inflection, yet does not exist . The simplest example of this is at x=0. The derivative of f(x) is actually which has no derivative at x=0. However the second derivative changes from -2 to 2 as on goes from negative to positive values of x.

Asymptotes and Other Things to Look For

An asymptote of a curve is a line which will be a tangent to the curve if the curve can be analyzed "near infinity". There are precise ways of discussing points at infinity of a curve and making this discussion rigorous. We will, however, take a naive definition and learn how to recognize asymptotes.

An asymptote for us will be a line such that the graph of the curve approaches the line along some branch (i.e. part or component) of the curve going out to the infinite region of the plane.

The easiest to analyze is a vertical asymptote . A vertical line x = p is an asymptote to y = f(x), if f(x) becomes infinite as x nears p. It is possible that the function gets closer to plus or minus depending on our approach to
the point x = p. Usually, the formula for the function has a denominator that becomes zero at x = p. For example, the reciprocal function has a vertical asymptote at x = 0, and the function has a vertical asymptote at x = /2 (and also at x = - /2, x = 3 /2, etc.).

Note that usually f(p) would not be defined - or at least not naturally defined by the formula. It may be artificially defined in the construction of the function.

Vertical asymptotes are found by asking when the denominator of the formula of a function goes to zero.

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On the other hand, a function that satisfies the property is called an `` odd function .'' Its graph is symmetric with respect to the origin. The warning about checking the whole domain should be applied here also.

As before, the main examples of odd functions are: when n is an odd number, and as before this is the reason for the terminology.

Some more examples of odd functions are: sin x, and tan x. Unlike even functions, expressions in odd functions may fail to stay odd. For example the square of an odd function is always even! However, an odd function stays odd if multiplied by a constant and sums of odd functions are odd.

Of course, most functions are neither even nor odd, and do not have any particular symmetry. A striking fact is that every function f(x) can be written as an odd function and an even function . You should try to prove this. You might start by showing that h(x) = f(x) + f(-x) is even. Can you think of a similarly defined odd function?)

Examples

Example 1: Sketch .

First, we set 0 = , which has solutions x = +/- /3 =+/- 0.577. The corresponding y-coordinates are -0.385 and +0.385, i.e., the two ``critical points'' are (0.577,-0.385) and (-0.577,0.385). The second derivative test gives , which is positive for the first point and negative for the second. Thus, the first of the two points (in the fourth quadrant) is a local minimum, and the second is a local maximum. Since >0 when x>0 and <0 when x<0, it follows that the curve is concave upward in the right half of the graph and concave down-ward in the left half. The two halves are
separated by the inflection point at the origin. This curve has no asymptotes. It does have symmetry, however, because it is an odd function. Its graph is shown below.

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