Putting Constraints in Optimization for First-Year Calculus StudentsAuthor(s): Kelly BlackSource: SIAM Review, Vol. 39, No. 2 (Jun., 1997), pp. 310-312Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2133113 .

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SIAM REV. X) 1997 Society for Industrial and Applied Mathematics Vol. 39, No. 2, pp. 310-312, June 1997 007

PUTTING CONSTRAINTS IN OPTIMIZATION FOR FIRST-YEAR CALCULUS STUDENTS*

KELLY BLACKt

Abstract. A method of presenting optimization to first-semester calculus students is demon- strated. The method is based on the Lagrange multiplier approach, and the geometry of a given problem becomes the central focus of the effort to find a solution. The method proposed here deviates from the traditional approach in that the student is asked to examine a constrained optimization problem and the geometry of the problem remains the central theme.

Key word. optimization

AMS subject classifications. 49-00, 49-01

Pll. S0036144595294813

1. Introduction. Within the calculus curriculum, the idea of optimization is presented early in the first semester. Optimization is presented as a simple exercise, and the student is asked simply to take the derivative of a function and set it equal to zero, usually without constraints. In this process, the student is not asked to identify the cost and the constraint; the student simply reduces the problem to an unconstrained optimization problem. Despite its simplicity, the method proposed here is not usually introduced into the first-semester calculus course. However, we have successfully introduced it into a calculus course, and it has been used to demonstrate the connections between calculus and the geometry of a given problem.

Optimization can be an exciting topic and can represent an opportunity to bring together other topics that are taught in the calculus classroom. For example, students are taught implicit differentiation early in the curriculum but are not asked to apply it in any other setting. Optimization can be used to introduce many exciting applica- tions into the early mathematics course. Moreover, by emphasizing the relationship between the cost and the constraint, the first-year calculus student can be introduced to some of the most basic principles used by applied mathematicians, such as simple conservation laws and energy principles.

The method proposed here is simply the Lagrange multiplier method [1, 2] re- stricted to the plane. It is analogous to the way that linear programming is introduced into an optimization course. Because it is a simple restriction to the plane, the use of normal vectors is not needed, and they would only complicate the problem. Because it is easier to find the tangent lines, the tangents to the curves are used in the same way that normal vectors are used in the Lagrange multiplier approach [1, 2], which is taught later in the multivariate calculus course.

2. Optimization. The goal is to minimize a cost function subject to some con- straint:

(2.1) minimize f(x,y) subject to g(x, y) = 0.

*Received by the editors June 21, 1995; accepted for publication (in revised form) October 16, 1995.

http://www.siam.org/journals/sirev/39-2/29481.html tDepartment of Mathematics, University of New Hampshire, Durham, NH 03824 (black@

vidalia.unh.edu).

310

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CLASSROOM NOTES 311

Another way to express this is to say that we want to find the smallest number c such that

(2.2) f(x,y) = c,

and at the same time we must meet a constraint

(2.3) g(x,7y) = 0.

The goal is to find the values of x and y that will satisfy equations (2.2) and (2.3).

2.1. A simple example. To motivate the problem a simple example is pre- sented:

(2.4) minimize x ? y2 (2.4) subjectto x+y-1=0.

The cost function is defined by a family of circles whose radii is given by the square root of the cost. By plotting the two relationships we see that we want to make the radius of the circle as small as possible and still intersect the line x + y = 1. Specifically, we want to decrease the cost function until it touches the constraint at only a single point. At this point the slopes of the two curves are the same. By examination, the minimum occurs when

2 21 (2.5) x +y

From the constraint this occurs when x = and y =

2.2. Using calculus to solve the problem. Looking at the slopes of the two curves, we can also use calculus to find the minimum through the same ideas stated above. The slopes of the two curves are found and set equal to each other. To do this, implicit differentiation is utilized to find the slopes of the cost and the constraint, and the two derivatives are set equal. From this new relationship and from the constraint the minimum is found.

We first need to find the slopes of the tangents of the two curves:

cost: 2x + 2ydy = 0, (2.6)

d

constraint: 1 + ? = 0. dx

Isolating the derivatives and setting them equal yields a new equation that must be met at the extrema:

cost: dy -y x

dyx _

(2.7) constraint: d!= =1,

-x = -1 y

y=x.

From the constraint we have two equations and two unknowns:

(2.8) y =x

x + y = 1,

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312 CLASSROOM NOTES

which yields one solution:

(2.9) (x,y)=

In other more complicated problems several solutions may be found and Newton's method might also be required to find the roots. To decide which ones are local maxima and local minima, the results must be substituted into the cost function and compared. The student must also consider the endpoints, which requires that the extremes defined by the physical system be examined.

For a given problem, the method can be distilled into the following approach: 1. Identify the cost and the constraint. 2. Use implicit differentiation to find an expression for the slopes of the tangents

of the two curves. 3. Set the slopes of the tangents of the two curves equal to one another. Use

this new expression and the constraint to find the solution. While the student may concentrate on the later steps, the most difficult part of a given problem can be identifying the cost and the constraint. Most importantly, in the identification of the cost and the constraint the student may be explicitly asked to cross the line between mathematics and the physical world.

3. Conclusion. The traditional approach to teaching optimization in the calcu- lus classroom centers on finding a derivative of a function and setting it equal to zero. Such an approach ignores the ideas behind the use of Lagrange multipliers, which are introduced later into the calculus sequence, and ignores the importance of the geometry and the interplay between the cost and the constraint. The approach pro- posed here centers on the geometry of a given problem and is essentially a Lagrange multiplier approach restricted to the plane.

By focusing on the geometry of the problem, optimization problems can be used to demonstrate to the student why the slope of the tangent line to the graph of a function is an important concept. Furthermore, it is an excellent demonstration of how the simple idea of a derivative can be used as a powerful tool. While we do not propose to do away with the traditional approach, the approach presented here offers another way to emphasize how physical principles are used in mathematics and is done quite early in the students' college education.

REFERENCES

[1] J. D. FAIRES AND B. FAIRES, Calculus, 2nd ed., McGraw-Hill, New York, 1992. [2] J. M. H. OLMSTED, Advanced Calculus, Prentice-Hall, Englewood Cliffs, NJ, 1961.

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