Name: ___________________

Math 122 Lab 1: Introduction to the TI-89

Graphing Graphing is one of the most important tools in calculus. A graph is often the first step in solving a problem. You should be able to use the TI-89 to get good graphs of complicated functions. The terms in bold are used throughout the course.

We will start by graphing a cubic, in particular y = x³ + 4x² - 5x - 1. Type this function into your y= list. Notice that the calculator reformats the function after you enter it; always check to see that there are no typos. Now do a standard zoom: press F2 for zoom and 6 for ZoomStd. Sketch the result on this paper.

It should be clear that part of this graph is missing. To find the rest of the graph, you need to adjust the graphing window. There are several ways to do this. First, press Green F2 to get the Window menu. Notice that the standard zoom shows both x and y between -10 and 10. If you know the values of x and y that you want, type them in here. An emphasis of chapter 3 is to compute these values, but you can also use educated guesses. Since the previous graph ran off the screen, we will zoom out. To use an automated zoom, press F2 then 3 then Enter (to leave the center at the origin). Sketch the result here.

In this graph, it looks like there is a local minimum at (0,0); how do you know that this is incorrect?

In the original graph, we could see y-values for x’s between about -6 and 3. We now do a zoom fit. Go to the Window page. (While you’re here, notice that the zoom out command multiplies both the x and y-ranges by 4.) Enter -6 for xmin and 3 for xmax. Then press F2 and then A. The calculator computes all y-values for the given range of x’s and chooses ymin and ymax accordingly. Sketch the result here. Use Window to see what the ymin and ymax are: write them here.

Let’s try a different graph, y = x³ - 20x² - x + 19. Sketch the standard graph here. What is missing?

Calculus techniques tell us where the rest of the graph is. To find intercepts, extrema and inflection points, you solve equations. That is the next lab topic.

Solving equations As with graphing, the calculator is tremendously powerful at equation solving, but must be used intelligently. We will find all solutions of the equation x³ - 20x² - x + 19 = 0. One issue with any equation is how many solutions there are. Be careful: theory says that there are at most 3 solutions; there may be fewer. Since the function x³ - 20x² - x + 19 is already in your y= list, look at this graphically. From the standard zoom, you should be able to see 2 solutions of the equation x³ - 20x² - x + 19 = 0. Use the graph to estimate the values of the solutions.

The Zero command works from the graph. Press F5 for Math and then 2 for Zero. When the calculator asks for “Lower Bound” use the arrow keys to move the cursor to the left of the solution you want. Press Enter. Move the cursor to the right of the solution for “Upper Bound” and press Enter. The calculator will return its best approximation of the zero. Do that for the two solutions that you can see and write down the results here.

Now, press 2^nd Esc to quit the graphing screen. Without a graph, we can solve equations using the calculator’s Solve command. Press F2 for Algebra and 1 or Enter for Solve. Enter the equation and a comma x (this tells the calculator what to solve for). The command should look like

solve( x³ - 20x² - x + 19 = 0, x )

Enter this. After a short pause, the calculator will give you all 3 solutions. Write them down here.

Note: since we already had the function entered as y1, we could have saved some typing by entering the command solve(y1(x)=0,x) instead.

Now, find all solutions of the equation 2 cos x = 2 - x. Note that algebra is useless here. Try entering this into the Solve command and write down the solutions here. If the calculator takes too long on this, press the On key to interrupt the current calculation.

Check the bottom of the screen and write down any warnings that are there.

Let’s look at this graphically. Enter 2 cos(x) for y1 and 2 - x for y2. Press F2 for Zoom and 7 for ZoomTrig. Identify the feature of this (double) graph that corresponds to solutions of the equation 2 cos x = 2 - x and indicate how many solutions there are. Did the Solve command get them all?

Approximate one solution using the Intersection command. This command works just like the Zero command except that it asks you two extra questions – just hit Enter on the first two questions.

Another property of the Solve command is important to understand. Use Solve to approximate solutions of the equation x³ - x + 1 = 0. Write down the result here. (This answer is given in decimal form. That means that the calculator could not find an exact answer, but found the best approximation it could.)

Computing function values A third tool that is constantly used in calculus is computing function values. For f(x) = 2 cos x, some values like f(0) and f(p) should be easy, but most like f(2.2) or f(2) are too hard. We will look at two ways of using the TI-89. First, on the main screen, enter 2cos(2.2) and write the result here. (Note: if you got 1.9985--- you got the wrong answer because your calculator is in degrees mode. Press MODE, scroll down to the line for Angle and change back to radians mode.)

Now, try 2 cos(2). You can type this all from scratch or use the arrow keys and backspace key to edit the previous command. Write your “answer” here.

The TI-89 gives exact answers whenever it can. The “answer” it just gave is correct, isn’t it? But we want a decimal approximation. There are multiple ways to get that. Enter 2 cos(2.) and the decimal you enter will kick the calculator into decimal mode. Or type in 2 cos(2) and press Green Enter. Or, if the function is stored as y1, you can just enter y1(2.) and avoid some typing. Write the decimal approximation here.

Derivatives The TI-89 has remarkable symbolic (CAS) capabilities. The biggest “gee whiz” feature is exact calculations. To fill up a screen, try 80! (The easiest way to get the factorial is Green divide. Move the cursor up and over to see all the digits.) Something that’s more useful in calculus is derivatives. To get the derivative of xsinx, enter d(x*sin(x),x) . The d is Yellow 8 and you need the times between x and sinx. If you get an answer involving 180 and π, your calculator is in degrees mode and you need to change to radians. Write the answer here.

To get a second derivative, edit the current line to read d(x*sin(x),x,2) . Write your answer here.

Integrals We will use the integration feature of the TI-89 regularly. Enter the following:

∫(x^2+2x,x,1,3) Write down the result and identify the integral evaluated.

Now try the same command without the 1 and 3: ∫(x^2+2x,x). What is being evaluated here?

Finally, try ∫(e^(x^2),x,1,3). Notice first that it takes a long time to respond, then it gives the answer in a decimal form instead of a fraction form. Try to explain why this occurs.

To Turn In:

Turn in this sheet filled out and separately a brief report on the exploration on the next page.

You can tear off this page and turn in the previous pages now, if you want

Neatly write up a report of your work on the following problem. You do not need to word process this. It is due at the beginning of class tomorrow. Papers turned in later Friday lose 1 point, papers turned in Monday lose 4 points, and papers turned in after that lose 4 points per day. A good report should include an introduction (briefly describe what you are doing), brief explanations of the purpose of each graph or calculation and a summary of what you learned. A good report does not include scratch work or unnecessary calculations. I recommend working it out on scratch paper first, then starting your report.

Recall that some functions have elementary (simple) antiderivatives and many do not.

First, try ∫(e^(x^2),x) -- does have an elementary antiderivative?

Next, try ∫(x*e^(x^2),x) -- does have an elementary antiderivative? What technique is used to find it?

Now, try ∫(x^2*e^(x^2),x) -- does have an elementary antiderivative?

Now, try ∫(x^3*e^(x^2),x) -- does have an elementary antiderivative? What techniques are used to find it?

Based on this evidence and other tests (that is, try other integrals if you want more evidence; include this in your report if you do), for which positive integers n does have an elementary antiderivative?