TI85 Programs

## TI-85 Programs

This page contains instructions and examples for the use of the programs commonly made available for the TI-85 calculator in Math 106-107.

### What's New?

The archive of programs will be updated from time to time, with new programs and revisions of existing programs. Changes will be advertised here.

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 .

### CONTENTS

Integration Programs Differentiation Programs Root Finding Programs Miscellaneous Programs

Page Author: John L. Orr

### INTEGRATION PROGRAMS

Riem calculates a numerical value for the Riemann sum approximation of the definite integral of a function. The user can choose the function, right or left-hand sum, upper and lower limits of integration and number of intervals. The user has the option of seeing a graphical demonstration of the area being computed.

To Use:

1. Enter the function as y1 in GRAPH mode.
2. Run the program.
3. Program asks whether you want to run graphics
4. Program will prompt for:
• lower x-value
• upper x-value
5. Program prompts for:
• number of intervals
• right-hand or left-hand sum
6. If graphing mode was selected in 3., the program draws a graph of the function together with the strips whose area gives the Riemann sum. Press ENTER to continue.
7. Program prints the value of the sum.
8. Program gives the option of returning to step 5. or quitting.
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:

1. Enter the function as y1 in GRAPH mode.
2. Run the program.
3. Program asks whether you want to run graphics
4. Program will prompt for:
• lower x-value
• upper x-value
5. Program will prompt for number of intervals, and whether to use inscribed or circumscribed rectangles.
6. If graphical mode is selected, the program will display the graph of the function, break it into the appropriate number of intervals, and show the circumscribed or inscribed rectangles whose area is being calculated.
7. When the user presses ENTER, the program prints the numerical value of the sum.
8. Program gives the option of returning to step 5. or quitting.
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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:

1. Enter the function as y1 in GRAPH mode.
2. Run the program.
3. Program asks whether you want to run graphics
4. Program will prompt for
• lower x-value
• upper x-value
5. Program prompts for number of intervals.
6. If graphing mode was selected in 3., the program draws a graph of the function together with the trapezoidal strips whose area is being calculated Press ENTER to continue.
7. Program prints the value of the sum.
8. Program gives the option of returning to step 5 or quitting.

<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.

### DIFFERENTIATION PROGRAMS

Secant: calculates the difference quotient for a function at a given point. The user can choose the function, point of evaluation, and step size. The user has the option of seeing a graph of the secant line along with the original function.

To Use:

1. Enter the function as y1 in GRAPH mode.
2. Run the program.
3. Program asks whether you want to run graphics
4. If graphics are selected, you have the choice of choosing your own x-range (Set), or letting the calculator pick an appropriate range (auto). Auto will always draw the graph to a scale where the distance between x and x+h occupies the middle half of the screen. Set is faster than auto, and lets you see the big picture better.
5. Program will prompt for x-value
6. Program will prompt for step size
7. If graphics was selected then a graph is drawn of the function and its secant line in a neighborhood of x=a to x=a+h. Press ENTER to continue
8. Program will display the value of the difference quotient, i.e. the slope of the secant line from x=a to x=a+h
9. Program gives the option of rerunning or quitting. There is an additional option of whether you want to change both x and h or just h. .

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:

1. Draw the garph of the function you are interested in.
2. From the graph menu, pick the DRAW option. (You'll have to use the MORE key.)
3. Select LINE and move the cursor to the point on your curve you want the secant line to start at. Press ENTER.
4. When you then move the cursor, it drags a straight line behind it joining the cursor to the point where you pressed ENTER. If you move the cursor to a pint on the graph of your function and trace along it with the cursor keys, you can see the secnt line approaching the curve in real time.

### ROOT-FINDING PROGRAMS

General Iteration Procedure: To iterate the function x_(n+1)= f(x_n), starting at x_0

1. Set y1 = f(x) in GRAPH mode
2. Type the initial value of x and press ENTER
3. Type evalF(y1,x,Ans) and press ENTER reteatedly to iterate

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 type
Ans-(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:

1. Enter the function as y1, its derivative as y2 in GRAPH mode.
2. Run the program.
3. Choose whether or not to run in graphics mode
4. Program will prompt for initial estimate.
5. If the graphics mode was selected, the program will display graph of function and its tangent line at the point of the initial estimate. The graph will include the zero of the tangent line, but may not show the zero of the original function. Press ENTER to continue
6. Program will give the Newton's method approximation.
7. Pressing enter will result in a new approximation using the previous answer as the initial estimate (previous graph with new tangent line will be displayed, press enter again for numerical value).

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.

### MISCELLANEOUS PROGRAMS

Trig: draws a circle of radius 1 and rotates a radius around, drawing the horizontal and vertical lines to the x and y axes, showing the behavior of sine and cosine. Program can be run in dynamic mode, so the radius rotates continuously, or in stepped mode. so it rotates in increments, allowing the viewer to observe the behavior of sine and cosine more closely.

To Use:

1. Run the program
2. Program will prompt run in dynamic or stepped mode.
if dynamic:
1. Program will display the graph of a circle of radius 1, with a radius rotating around it, and vertical and horizontal lines drawn to the axes.
2. When user presses ENTER again, program stops.
if stepped:
1. Program will display the graph of a circle of radius 1.
2. The up and down arrows can be used to control the progression of the radius around the circle. Pressing the up arrow rotates it one increment counter-clockwise (forward) and pressing down makes it rotate one increment clockwise (backwards)
3. The radius will continue to progress around the circle as long as the user pushes the up or down arrows.
4. User can exit program by pressing ENTER.

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:

1. Enter the desired function as y1
2. Run the program
3. Program will prompt for x. Enter the value, and press ENTER. The calculator will display the corresponding value of y1.
4. To re-run the program with a new value of x, press ENTER again.

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.

• eval followed by an argument evaluates all the functions entered for graphing, and returns an ordered list consisting of their values.
• evalF(A,B,C) evaluates the expression A (typically, y1) as a function of the named variable B (typically x) at the value C. So evalF(y1,x,3) returns the value of y1 at 3.
FEVAL is useful for the TI-81 and TI-82, which do not have these functions on them.

Ratios: when given a list of numbers, provides a list of adjacent ratios, x_(n+1) / x_n.