On-line Math 21

On-line Math 21

4.5  Curve Sketching

The idea behind curve sketching is to translate the quantitative information you can determine from the formula describing a function into qualitative information, interpreting the quantitative information in terms of the dynamics of the function (expressed as the graph, or picture, of the function). Conversely, given that qualitative information, you can conclude a number of things about the function that have quantatitive meaning. The two ways of understanding a function complement each other.

Because we want to extract the qualitative understanding of the function from the graph, we draw the graph by hand, not using a computer. Of course, Maple can draw all these graphs, very well (although some skill is required to get the domain and range appropriate to see all the features of the function). Certainly, it's a good idea to use Maple to check your work.

4.5.1  The Derivative and Graphing

The primary use of the first derivative in graphing is that, if f¢(x) > 0 , then the function is increasing, and if f¢(x) < 0 , the function is decreasing (as you move from left to right).

So, you can tell whether the curve is sloping up or down (as x increases) just by figuring out whether f¢(x) is positive or negative, which is usually easier, since it's fairly easy to find where f¢(x) = 0 , and between those points, f¢ is either one (positive) or the other (negative). I like to use a simple number-line, with the places where f¢(x) = 0 marked, indicating the sign of the derivative in between:

A few definitions

We're going to deal with a function f . A few points are of special interest, called critical points. As we mentioned earlier, a critical point of f is a value of x for which either:

  1. f¢(x) = 0 ,
  2. f¢(x) doesn't exist, or
  3. x is an endpoint of the domain of the function.

From what we mentioned earlier, it follows that the maximum or minimum value of f occurs at a critical point.

We can, however, refine the idea of a critical point. If, for x < x0 , but x close to x0 , f¢(x) < 0 , and if for x > x0 , but x close to x0 , f¢(x) > 0 , and (of course) f¢(x0) = 0 , then the graph of f is dipping down to f(x0) and rising away from there, that is, f has a local minimum or relative minimum at x0 .

On the other hand, if, for x < x0 , but x close to x0 , f¢(x) > 0 , and if for x > x0 , but x close to x0 , f¢(x) < 0 , and (of course) f¢(x0) = 0 , then the graph of f is rising up to f(x0) and falling away from there, that is, f has a local maximum or relative maximum at x0 .

A point x0 is a global (or absolute) maximum or minimum if that is the point where f has its overall maximum (or minimum) on its domain.

You should try to keep straight that when we talk about a critical point, or a relative minimum point, we refer to the x (or whatever the independent variable is called), not the value of the function. Those are either called critical values (or relative minimum values), or critical numbers (et. cetera).

We can emphasize the idea of what a local maximum or minimum looks like in the following ``test''.

Theorem 1 If, at a critical point x0 , then:

  1. If, for x < x0 , but x close to x0 , f¢(x) < 0 , and if for x > x0 , but x close to x0 , f¢(x) > 0 , f has a local minimum or at x0 .
  2. If, for x < x0 , but x close to x0 , f¢(x) > 0 , and if for x > x0 , but x close to x0 , f¢(x) < 0 , f has a local maximum at x0 .

Graphing Examples

We're going to use the information we just talked about theoretically regarding the first derivative to draw reasonable graphs of the functions. Here are the examples:

Example 1 f(x) = x2-2x+3 .

Solution

Example 2 f(x) = 2x3-3x2-12x .

Solution

Exercise 1 f(x) = 3x4-4x3 .

Answers


   Intercepts


   Increasing/decreasing, and critical points

Here you should describe the range where f¢ is positive, and where it is negative.


   Draw the graph

Well, we can't do that over the net so easily. So, instead, pick which of the following graphs looks most like the graph of that function. Choose the closest:

A
B
C
D

Example 3 f(x) = sin(x)+cos(x) , x Î [0,2p] .

Solution

5.0.2  The Second Derivative, Concavity, and Inflection Points

The second derivative, which measures the rate of change of the slope, tells about the ``curvature'' of the curve. It curves upward (concave up) where f¢¢(x) > 0 , and curves downward (concave down) where f¢¢(x) < 0 .

A place where the concavity changes from concave up to concave down is called an inflection point. It usually will be found where f¢¢(x) = 0 , but can also be so called at a place where the derivative doesn't exist. We do want to restrict to continuous functions, though. The x value, as before, is the inflection point.

Theorem 2 [The Second Derivative Test]. If f(x) has a critical point at c , then:

  1. If f¢¢(c) > 0 , then f(x) has a relative minimum at c ,
  2. If f¢¢(c) < 0 , then f(x) has a relative maximum at c , and
  3. If f¢¢(c) = 0 , then you can't tell whether f(x) has a relative maximum, or minimum, or neither, at c .

Proof.

Examples

Find the inflection points and concavity; then sketch the curve, showing these points as well as slope and critical points.

Example 4 f(x) = x1/3 .

Solution

Example 5 f(x) = x4+4x3+4x2-2 .

Solution

Exercise 2 f(x) = x4-2x3 .


   Intercepts

Find all intecepts


   Increasing/decreasing, critical points

Find where the curve is increasing/decreaing, and find all critical points and values.


   Concavity and points of inflection

Find where the curve is concave up or concave down, and find all inflection points (sometimes, finding the values at the critical points is messy, but do what you can).

6.0.3  Complete Curve Sketching

There are a number of techniques to apply to produce a reasonable qualitative picture of the graph of a function. The idea is to be able to translate analytic properties of the function to the qualitative behavior illustrated in the graph. Here are the points (pun unintended) you should make sure to include in any sketch.

Domain: Your graph should give an idea of the domain of the function. This does not mean that you have to include x 's out to +¥, but certainly if the domain is restricted, the picture needs to illustrate that.

Intercepts: Finding the y -intercept of a graph, where the function crosses the y -axis, gives you a good place to start drawing the graph. Finding the x -intercepts, where the function has value 0 , is often harder, but it is a good thing to include if you can find them.

Symmetry/Periodicity: Look for whether the function is even f(-x) = f(x) or odd f(-x) = -f(x) . Even functions are symmetric about the y -axis, so whatever the graph is for x > 0 is mirrored for x < 0 . Odd functions are upside-down symmetric about the y -axis, or are ``symmetric about the origin''.
Also look for periodicity, such as with trigonometric functions, that is, for some number a , called the period of the function, for which f(x+a) = f(x) . Then, the graph is simply repeated period after period.

Asymptotes: An asymptote is a line, vertical, horizontal, or slanted, which the graph of the function approximates ``near infinity''. This was already discussed somewhat back in Chapter 1, when we were dealing with infinite limits.

More About Asymptotes

Increasing/Decreasing Find the intervals where f¢(x) is positive, and where it is negative, and the critical numbers. This would also imply checking which are local extrema, using the first derivative test. Draw a number line with the sign of f¢(x) indicated.
ConcavityFind the intervals where f¢¢(x) is positive, and where it is negative, and the inflection points. Draw a number line with the sign of f¢¢(x) indicated.

Then sketch the curve

  1. The first thing I use is the asymptotes, if any. They give the tails of the graph. With a little care, checking on which side of the lines the graph lies on, you can learn a great deal of information from the asymptotes alone. Remember that, while a graph can never cross a vertical asymptote, it can cross a horizontal one.
  2. Find the intercepts that you can, which gives a start. Find endpoints, and the value of the function there. Check for symmetry.
  3. Compute the derivative, and find where it is positive/negative. Find all critical points, and the values at those points. Draw each of them, with a horizontal line to indicate that it has a horizontal tangent.
  4. Find the second derivative, and again find its sign, and find inflection points.

Then draw the curve, showing these points.

Examples

Example 6
f(x) = 2x+3
x+1

Solution

Exercise 3
f(x) = ex
x
 .


   Asymptotes


   Intercepts


   Increasing/decreasing, and critical points


   Concavity, and inflection points

Exercise 4
y = x
x+1
 .


   Asymptotes

Look for a slant asymptote for this one.


   Intercepts


   Increasing/decreasing, and critical points


   Concavity, and inflection points

Example 7
y = Öx-   ___
Öx-1
 
 .

Solution

Exercise 5
y = 2cosx+sin(2x).


   Asymptotes


   Intercepts


   Increasing/decreasing, and critical points


   Concavity, and inflection points

Example 8
y = xe1/x.

Solution

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Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 21 Dec 2000, 00:47.