Tuesday, January 16, 2018

TI-74: Extrema of a Cubic Polynomial

TI-74: Extrema of a Cubic Polynomial

Introduction

The program finds the extrema points of a cubic polynomial where the roots A, B, and C are known.  The cubic polynomial is defined as:

y = (x – A) * (x – B) * (x – C)

Expanded to:

y = x^3 – (A + B + C)*x^2 + (A*B + B*C + A*C)*x – (A*B*C)

The extrema can be found by taking the derivative and then solving for x when dy/dx=0.

dy/dx = 3*x^2 – 2*(A + B + C)*x + (A*B + B*C + A*C) = 0

Solving for x:

x = (2*W ± √(4*W^2 – 12*V))/6

Where:

W = A + B + C
V = A*B + A*C + B*C

TI-74 Program: Extrema of Cubic Polynomials

100 PRINT “y=(x-a)(x-b)(x-c)”: PAUSE 1
110 INPUT “a: “;A
112 INPUT “b: “;B
114 INPUT “c: “;C
120 W=A+B+C
122 V=A*B+A*C+B*C
130 X1=(2*W+SQR(4*W^2-12*V))/6
132 X2=(2*W-SQR(4*W^2-12*V))/6
140 Y1=(X1-A)*(X1-B)*(X1-C)
142 Y2=(X2-A)*(X2-B)*(X2-C)
150 IMAGE “######.######, ######.######”
152 PRINT USING 150,X1,Y1: PAUSE
154 PRINT USING 150,X2,Y2: PAUSE
160 END

Example

A = 0, B = 3, C = 5

Results:
4.119633, -4.06067
1.213700, 8.20882

Eddie

This blog is property of Edward Shore, 2018