Monday, September 03, 2012

Calculator Help: The Calculate Menu – Part 2

I am using a TI-84 Plus Silver Edition graphing calculator for demonstrations; consult your calculator's manual for help finding menus and buttons.

In the previous post, we learned how to use the value, zero, minimum, and maximum commands found in the Calculate menu on a graphing calculator (click here to view that post). Today, we will look at the last three commands: intersect, derivative, and integral. In an Algebra class, you will probably use the intersection command frequently, but you will not use the derivative and integral commands until you reach Calculus. Even then, many students do not realize the advantage those two little commands offer. Let's get started.

To begin, refer to the first set of instructions in the previous post linked above to set up your calculator with our function and window, so that you can see the graph I'll be working with throughout this post.

HOW TO USE THE "INTERSECT" COMMAND

The "intersect" command allows you to determine the point of intersection of two functions. For this section, input the following function into the second slot in the Y= menu.
Press GRAPH to graph both functions.
Navigate to the Calculate menu (press 2ND then TRACE) and choose option 5: intersect. Take notice of a few things. The calculator has placed a cursor on the graph of the first function at a random point (circled in blue below) and has opened with the first prompt (circled in red): "First curve?"
We want to know the intersection point located in the bottom right, circled in green in the photo above. To answer the first prompt, move the cursor close to the intersection point and press ENTER.
Since there are only two functions graphed here, the calculator automatically jumps to the second equation—in the upper left corner of the screen, the second equation is displayed and the cursor has "jumped" from the first curve to the second. Since this point is close to the intersection, we can press ENTER here without moving the cursor. If it were not very close to the intersection we desire to find, we would arrow closer to it. Press ENTER again to skip through the guess screen and obtain the intersection answer.
There are a couple of things to note here. If there were many functions graphed on the screen, we could toggle through them by pressing the UP or DOWN arrow key in order to identify which curves we want to find the intersection for. Also, the cursor must be near the intersection you desire to find when you press ENTER for each curve. The calculator will find the intersection point closest to the cursor points you entered. Use the "intersect" command to find the other two points of intersection for these two curves. You should get the following answers: (0.89m -2.66) and (-1.47, 4.42).

The "intersect" command is also helpful for finding all x-values that yield a particular y-value. For example, if we want to know at what points our first function is equal to -10, we can do the following:
Change the second function to y = -10.
Then press GRAPH. Use the "intersect" command to find the three points of intersection of the two curves. You should get the following answers: (-2.25, -10), (1.49, -10), and (4.76, -10).

HOW TO USE THE "DERIVATIVE" COMMAND

Clear the second function from your calculator, but keep the first one and graph it in the same window we've been using. The "derivative" command allows you to determine the derivative of the function at any given x-value in the window you have set. Open the Calculate menu and choose option 6: dy/dx. You can choose an x-value in one of two ways: move the cursor along the curve to the point you desire or type in the x-value you want to evaluate. Let's evaluate the derivative at x = 5. When you type a number, an input line will appear automatically.
Press ENTER after you input the value and the calculator will return the value of the derivative at that point. We would round this answer to dy/dx = 28.
This command is very helpful when you first learn about derivatives. You will have to find derivative values manually in the beginning, but this command allows you to check those answers.

HOW TO USE THE "INTEGRAL" COMMAND

I cannot input the "integral" symbol here, but the final Calculate command is the integral. The integral over an interval is the area under the curve (between the curve and the x-axis) over that interval. To practice, let's find the integral of this function over the interval (1, 5). With the same function graphed on your screen, open the Calculate menu and choose the last option. The calculator will place a cursor on the screen (circled in blue below) and opens with the first prompt (circled in red below): "Lower Limit?"
The lower limit we want to consider is x = 1. You can enter limits one of two ways: move the cursor to the point on the graph or directly input the x-value. I prefer to directly input the x-value because it is not usually possible to place the cursor exactly on the desired limit. Type "1" and an input line will automatically appear at the bottom of the screen.
Press ENTER to set the lower limit. You can probably guess the next prompt: "Upper Limit?" As for the lower limit, enter the upper limit, which in this case is 5.
Press ENTER (there is no guess screen this time) and the calculator does two things. It shades the integral for a visual representation and also gives the numerical value, which is -69.33 in this case.
In order for this command to work, you must choose limit values that are contained in the window you have set. This is a fantastic way to check your integration answers for Calculus class!

There you have it. We have now covered all of the commands found in the Calculate menu. I hope you now have the ability and confidence to use these commands to help you perform well in your math classes. Most math teachers/professors require you to show your steps on homework, but these commands allow you to very easily check your answers before turning in your work. If you have any questions or comments, please leave them below!