1. Assume that there are two bowling balls in a ball-return machine. They are touching each other. At the point at which they touch, what can you say about their respective normal vectors? What about their tangent planes? What would happen if the bowling balls were of different sizes?
2. A weeble is a doll that is roughly egg-shaped. It is an ideal toy for little children, because weebles wobble but they don’t fall down.
A Russian weeble is a hollow weeble, with one or more weebles inside it. Picture two nested hollow eggs as shown.
At the point at which two nested Russian weebles touch each other, what can you say about their respective normal vectors? What about their tangent planes?
3. Now picture your two favorite differentiable surfaces that touch at exactly one point. What can you say about their normal vectors at the point where they touch? What about their tangent planes?
15.7 MAXIMUM AND MINIMUM VALUES
TRANSPARENCY AVAILABLE
#46 (Figures 7–9)
SUGGESTED TIME AND EMPHASIS 1 class Essential material
POINTS TO STRESS
1. The contrast between optimization problems in single-variable calculus (relatively few cases) and in multivariable calculus (many possible solutions)
2. Critical points and local maxima and minima 3. The Second Derivative Test.
4. Absolute maxima and minima
QUIZ QUESTIONS
• Text Question: Can a differentiable function f have a local maximum at a point (a, b) with fx(a, b) = 3?
Answer: No
• Drill Question: Can you give an example of a function f with the property that fx(a, b) = 0, fy(a, b) = 0, and f does not have a local maximum or minimum at (a, b)?
Answer: f (x, y) = xy at (0, 0).
MATERIALS FOR LECTURE
A good way to introduce this topic may be to have the students do Group Work 1: Foreshadowing Critical Points and Extrema.
• Stress geometric interpretations: If f is differentiable at a local maximum or minimum, then the tangent plane must be horizontal. Note that there are critical points at which there is no local maximum or minimum. For example, examine the saddle points at the origin for f (x, y) = xy and g (x, y) = x2− y2.
SECTION 15.7 MAXIMUM AND MINIMUM VALUES
• Illustrate the idea behind the Second Derivatives Test using f (x, y) = 12
ax2+ by2
. Note that D = ab means that
• D > 0, a > 0 gives b > 0 and hence f (x, y) has a local minimum at (0, 0) [See Picture (a)]
• D > 0, a < 0 gives b < 0 and hence f (x, y) has a local maximum at (0, 0) [See Picture (b)]
• D < 0, a > 0 gives b < 0 and hence f (x, y) has a saddle point at (0, 0) [See Picture (c)]
(a) Minimum (b) Maximum (c) Saddle Point
If there is time, discuss the ideas of the proof given in the text.
• Describe some of the ways saddle points can occur for functions of two variables. (For example, see Figures 3, 4, 7, and 8.) Contrast with the single-variable case, where there are fewer possibilities.
• Discuss local and absolute maxima and minima for f (x, y) = xy + 1/ (xy).
WORKSHOP/DISCUSSION
• Use f (x, y) = x4+ y4, g(x, y) = x4− y4, h(x, y) = −
x4+ y4
to show that no information is given about local extrema when D = 0.
• Consider the problem of finding the distance between a point and a plane. Contrast this chapter’s approach to the method used in Example 8 in Section 13.5.
• Pose the problem of finding the maximum of f (x, y) = ax +by +c on the set of points x2+ y2≤ 4. Note that the gradient of f is never zero, so the maximum and minimum values must occur on the boundary.
One way to find these maximum and minimum values is by parametrizing the boundary x2 + y2 = 4 by r(θ) = 2 cos θ, 2 sin θ, where θ is the angle made by the position vector with the x-axis, and then optimizing the function g(θ) = f (r (θ)).
• Illustrate that it is often easier to optimize fn(x, y) instead of f (x, y) for a function f that is always positive. Point out that fn has the same maxima as f for any n. One good example to use is
f (x, y) = /
(x − 1)2+ (y + 1)2+ 101/3
. Another example is the problem of finding the point on the surface z2 = xy + x2+ 1 that is closest to the origin. [Answer:1
7, −47,3√72 ]
CHAPTER 15 PARTIAL DERIVATIVES
GROUP WORK 1: Foreshadowing Critical Points and Extrema
This group work is best done just before this section is covered. First present the single-variable definitions of local and global maximum and minimum. (This was done in single-variable calculus, but the students have probably forgotten the technical definitions by this point.) Then put the students into groups and ask them to come up with good multivariable definitions of the same concepts. They should present their definitions and discuss them. At the end of the activity, look up the definition presented in the text, and compare it with the student definitions.
If there is time, do a similar activity for the various types of critical points. Graph y = x2, y = −x2, y= x3, y = −x3, y = |x|, and y = − |x| on the board to show different types of critical values at x = 0. Then have the students try to come up with the variety of types that can occur for functions of two variables.
GROUP WORK 2: The Squares Conjecture
Note that calculus is not needed to solve this problem; students should be able to get the answer intuitively.
They can be asked to use the techniques of this section to verify that their intuition was true.
Answer: x = y = z =√3
100 (other answers are possible)
GROUP WORK 3: Strange Critical Points
In this case, fx and fy do not exist at the critical point(1, −1) and so the students cannot use the Second Derivative Test. Acceptable answers include graphing the surface or recognizing that it is an elliptic cone.
Answers: 1.(1, −1) 2. The absolute minimum is 2, and it occurs at(1, −1).
EXTENDED LABORATORY PROJECT: The Genetic Algorithm
The use of “genetic algorithms” for finding maxima and minima for functions of several variables has become popular in recent years. Usually this technique is used to optimize functions of hundreds of variables, but we’ll look at the simpler case of functions of two variables.
Although we don’t intend to give a complete description of how genetic algorithms work, an outline is as follows:
Suppose you want to maximize a function of several variables. Start by selecting several arbitrary points (at random or otherwise) from your domain. Select two points among these which give the two largest values of your function. Now choose several more arbitrary points close to these selected points. Continue to repeat this process until you have what seems to be a maximum value.
SECTION 15.7 MAXIMUM AND MINIMUM VALUES
We will study this process for the complicated function 100e−(|x| + 1)(|y + 1| + 1)sin(y sin x)
1+ x2y2 . Let D be the square [−3, 3] × [−3, 3].
(i) Use your computer program to select 5 points at random in this square and then evaluate the function at these 5 points.
(ii) Select the points which give the two largest values for f (x, y) and then select 4 points at random close to each of these points. Again, selecting the points at random near these points isn’t so trivial. Evaluate the function at the 10 points you now have. Select the two points among these which give the largest value for f (x, y). Repeat (b) until it appears that you have a maximum.
(iii) Is the value you found in (ii) likely to be an absolute maximum?
HOMEWORK PROBLEMS
Core Exercises: 3, 6, 13, 21, 39, 44, 50
Sample Assignment: 1, 3, 6, 8, 13, 18, 19, 21, 23, 26, 31, 35, 39, 41, 44, 48, 50, 53
Exercise D A N G
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