Different approaches for solving equations

On the impact of the first generation of graphing calculators on the mathematics curriculum at the secondary level.

Different approaches for solving equations

The solution of equations involving algebraic and transcendental functions is a continuous theme throughout secondary school. Students learn different techniques to solve equations depending on the properties of the functions involved. Often, the equations posed are carefully selected with "nice" coefficients and solutions. Graphing calculators expand the degree of difficulty and the scope of the equations that can be posed. These calculators allow for easily testing the correctness of the solution, for greater precision, and for uniform approaches to solutions independently of the type of functions involved. Traditionally, questions on inequalities involved linear, quadratic, and rational expressions. In an informal search of precalculus textbooks in 1992, we found that ninety percent of the books did not include inequalities involving transcendental functions. Since graphing is no longer a time consuming task and inequalities help students to think globally, it may be worthwhile to increase the number of inequalities that we ask.

Example 1

Solve the equation

Solution

Without the use of technology it would not be possible to pose this problem to the students before the Newton-Raphson algorithm is studied in calculus. The versatility of the graphing calculator is illustrated by solving this equation using different methods now accessible to pre-college students.

Let's start by graphing the functions

: and

and to visually determine an interval, say (-2,0), containing the solution (Figures 1.a and 1.b)


Figure 1.a	Figure 1.b

Numerical solution using the Intermediate Value Theorem

Deactivate y1 and y2, and let


Figure 1.c	Figure 1.d

Figures 1.c and 1.d indicate the initial setup and the corresponding table of values. After scrolling down in this table an interval is found where the function values change signs. Now the smaller subinterval found containing the zero is used to redefine the Tblset, thus we let TblStart=-0.1 and DTbl=0.01. The process continues until the degree of precision sought is reached. Students can be challenged to determine numerically, i.e., without looking at the graph, the initial interval containing a zero. Notice that a simple program can be written to emulate the Table feature in the older graphing calculators.

Numerical solution using the bisection method

Setting the independent variable on the table setup on Ask, the table feature provides the function values corresponding to any value or expression for x that it is entered in the command line at the bottom of the screen. Thus, if (a,b) is the least subinterval containing the zero, the student can enter

and choose the new subinterval, say (a,c), so that the condition

is satisfied (Fig 1.e).

Figure 1.e

It is not uncommon to find that when the solution is a negative real number, a surprisingly large number of students make mistakes. These mistakes do not seem to be caused by misconceptions on how the algorithm works but rather by a lack of familiarity with the ordering of the real numbers.

Iteration on a fixed point (Picard's method)

Initialize x in Home Screen. Then, proceed by successively assigning to x the value of the function evaluated at the previous value of x until the difference of two consecutive results is less than or equal to the precision sought.

Figure 1.g

Newton's Method

Let

: .

First initialize x with a value close to the root, then recursively calculate

and assign the result to x. Students appreciate how fast this method converges as compared with the previous ones.

Figure 1.h

Using the solver
Figure 1.i Figure 1.j

From the graph screen select root of y3 as in Figure 1.i., or intersect of y1 and y2. Alternatively Figure 1.j shows the use of the solver from Home Screen with an appropriate initial value.

Example 1
Solution
Numerical solution using the Intermediate Value Theorem
Numerical solution using the bisection method
Iteration on a fixed point (Picard's method)
Newton's Method
Using the solver

Quesada - 5 OCT 1996

There is a PDF file available for this paper.

Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.

Mail comments to Pedro Gómez: pgomez@uniandes.edu.co


Figure 1.i	Figure 1.j