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An excerpt from Appendix E: Graphing Utility Programs, Applied Calculus 6e (Larson);

Programs for graphing calculators are referenced in several sections in the text. This appendix contains translations of these programs for the TI-82, TI-83, TI-85, TI-86, TI-89, TI-92, Casio CFX-9850G, HP38G, and Sharp EL 9200/9300, arranged by calculator model. Similar programs can be written for other brands and models of graphing calculators.
Enter a program in your calculator, then refer to the text discussion and apply the program as appropriate.

Midpoint Rule

This program uses the Midpoint Rule to approximate the definite integral ab ƒ(x) dx. You must store the expression ƒ(x) as f1 before executing the program. The program itself will prompt you for the limits a and b and for the number of subintervals η.

======MIDPOINT======
“LOWER LIMIT”?→ A↲
“UPPER LIMIT”?→ B↲
“N DIVISIONS”?→ N↲
0→S↲
(B-A)÷N→W↲
1→J↲
Lbl 1↲
A+(J-1)W→L↲
A+JW→R↲
(L+R)÷2→X↲
S+Wf1→S↲
J+1→J↲
J≤N⇒Goto 1↲
“APPROXIMATION”↲
S

Simpson's Rule

This program uses Simpson's Rule to approximate the definite integral ab ƒ(x) dx. You must store the expression ƒ(x) as f1 before executing the program. The program itself will prompt you for the limits a and b and for half the number of subintervals you want to use.

======SIMPSON PROGRAM======
“LOWER LIMIT”?→A↲
“UPPER LIMIT”?→B↲
“N÷2 DIVISIONS”?→ D↲
0→S↲
(B-A)÷(2D)→W↲
1→J↲
Lbl 1↲
A+2(J-1)W→L↲
A+2JW→R↲
(L+R)÷2→M↲
L→X↲
f1→L↲
M→X↲
f1→M↲
R→X↲
f1→R↲
W(L+4M+R)÷3+S→S↲
J+1→J↲
J≤D⇒Goto→1↲
“APPROXIMATION”↲
S

Newton's Method

This program uses Newton's Method to approximate the zeros of a function. You must store the expression ƒ(x) as f1 before executing the program. Then graph the function to estimate one of its zeros. The program will prompt you for this estimate.

======NEWTON PROGRAM======
“ENTER APPROXIMATION”?→X↲
1→N↲
X-f1÷d/dx(f1,X)→R↲
Lbl 1↲
R→X↲
X-f1÷d/dx(f1,X)→R↲
N+1→N↲
Abs (X-R)≥1E-10⇒Goto 1↲
“ROOT=”↲
R◢
“ITER=”↲
N

BEST!
SlideRule
I haven't memorized the Simpson's Rule - calculators with calculus functions have spoiled me.
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