The standard programs have been rewritten so that they're quicker and easier to use.
GRIEM and RIEM have been combined into one new program Riem which offers graphics as an option.
SECANT has been rewritten to calculate the slope of a secant line, and provide pictures of the line. This can either be a tool for calculating tables of difference quotients or for illustrating how the secant line approximates the slope of a graph.
Memory Usage:The entire set of programs described here take up roughly 7.6K on the TI-85. This leaves about 20.5K free.
Alternatively, all the programs decsribed here can be downloaded in binary TI-85 form, by clicking here .
Page Author: John L. Orr
Last Modified: Aug 9, 1995
To Use:
Example: Suppose you are interested in approximating the area under the graph of x^2 between x=0 and x=1. First you need to go to GRAPH mode and enter the function x^2 for y1. Then exit graph mode, and select the program RIEM. When the program asks for a lower x-value enter 0; when it asks for an upper x-value enter 1. When asked to enter the number of intervals, start by trying 10. The program will prompt for left or right; press the appropriate key. You can start by trying the right-hand sum. The calculator will give you an answer of 0.385. It will then ask if you want to change options (this means number of intervals and right or left sum). You can run it again, changing the number of intervals to 100. It now gives an answer of 0.33835. Changing the options again, you can calculate a left-hand sum with 100 intervals. It will give you the answer 0.32835.
Circumscribe: calculates an approximation of the definite integral of a function using circumscribed or inscribed rectangles. In other words, it uses rectangles which either always lie above the curve (circumscribed), or always lie below the curve (inscribed). The user can choose the function, limits of integration, and number of intervals. The user has the option of seeing a graphical demonstration of the area being computed.
To Use:
Example: Suppose you are interested in approximating the area under the function 25 - (x - 5)^2 between x=0 and x=10. Start by going to GRAPH and entering the function as y1. Then exit graph mode, and select the program CIRCUM. Select the graphics option when prompted.When the program asks for a lower x-value enter 0; when it asks for an upper x-value enter 10. When asked to enter the number of intervals, start by trying 10 and choose inscribed rectangles. The calculator will first display the graph of 25 - (x - 5)^2. It will break up the graph into 10 rectangles, the top of each being bounded by the highest functional value within it. When you press enter, the numerical value of the sum is given, 189.999... . You then have the option of changing the number of intervals. This time you might choose to estimate the area using 20 rectangles. Again, the appropriate graph will be displayed and after pressing ENTER you will get a numerical value for the sum, ... . You will have the option to change the intervals again or quit.
Trap: Uses the trapezoidal rule to estimate the area under the graph of a function. The user can choose the function, upper and lower x-values, and the number of partitions. The user has the option of seeing a graphical demonstration of the area being computed.
To Use:
<Example: Suppose you are interested in measuring the area under the graph of sin(x) from zero to p. First, go to GRAPH and enter sin x as y1. Then exit graph mode and select the program TRAP. If you are interested in running the program quickly, decline the graphics option. When it asks for a lower x value, enter 0; when it asks for an upper x value enter p. When it asks for number of partitions, start by trying 10. It will display the estimate of the area under the graph, 1.9835... . You will have the option of changing the number of partitions. This time try 20. The calculator will give you a new estimate, 1.9958... . Finally, you have the option of changing the partitions again or quitting.
Integ: This comprises all three integration methods of the previous programs. The first option the user chooses is which of the previous methods should be used: thereafter the program run exactly as the corresponding program above.
To Use:
Example: Suppose you are interested in the slope of the graph of e^x at the point x=3. Start by going to GRAPH and enter the function e^x as y1. Then exit graph mode and select the program SECANT. When the program asks for an x-value, enter 3. When it asks for the step size, start by trying -2. The calculator will give you the value of the difference quotient.Experiment with the program in each of the two graphics modes. Try keeping x=3 but trying h=-2,-1,-0,5,-0,25. (Try h=0!) Note that the secant line overshoots the points x and x+h a little, but that the two points are marked with dots on the graph.
Note: This isn't really a program, but the following is a good classroom demonstration of taking the limit of secant lines:
Alternatively, omit step 1, and in place of step 3, enter the function f, as a function of Ans.
Example: To find the root of x^2-2 using Newton's Method, type 50 and ENTER to set the starting value and then typeAns-(Ans^2-2)/(2 Ans) Presssing ENTER repeatedly, iterates Newton's method, which eventually converges to root2.
Newt: Uses Newton's method to approximate the root of a function. User chooses function, enters its derivative, and a starting estimate for the root. The user has the option of a graphical demonstration.
To Use:
Example: Suppose you are interested in the root of x^2-2. Start by going to GRAPH and entering x^2-2 as y1, and 2x as y2. Then exit graph mode, run the program NEWT, and select to run in graphics mode. It will ask for an initial estimate. For a good demonstration, set xMin=0 and xMax=8. Take the first estimate to be 7. The graph of x^2-2 with a tangent line at x=7 will be displayed. Press ENTER and the program will give the new estimate. Press ENTER again and you will see the graph you just saw with another tangent line, this time at x = 3.6. Press ENTER yet again and you will get the numerical value of the new estimate. You can keep pressing ENTER until you are satisfied with the estimate.NOTE: The effectiveness of Newton's method as a root finding algorithm makes graphical demonstrations difficult. The method converges so quickly that the screen can only show one or two iterations before the tangent is so close to the root that it is invisible.
To Use:
Example: Suppose the class is discussing the behavior of the sine and cosine functions. The program can be used to enhance the discussion since the rising and falling of the vertical line represents the behavior of sin, and the changes in the horizontal line represent cosine. You might begin by running the program in the stepped mode, to show how sine rises and falls. The radius can be rotated around the circle infinite times, just as the periodic behavior of the sine function continues to infinity.
FEVAL: This program evaluates the function y1 at the given input x.
To Use:
NOTE: This program is duplicated on the TI-85 by the functions eval and evalF. If one programs these two functions onto the CUSTOM keys then they can be reached very quickly.
Ratios: when given a list of numbers, provides a list of adjacent ratios, x_(n+1) / x_n.
To Use:
Example: Suppose you are interested in the relationship between the first five numbers of the sequence 1,1,2,3,5,... . Start by selecting the program RATIO. The program will ask for a list of numbers. Enter 1 then 1, then 2, then 3, then 5, pressing ENTER after each number. Press ENTER again and the program will give you the adjacent ratios, 1,2,1.5,1.667.