Texas Instruments TI 83/84 Calculus Tips                        Return to justinsociety.com services

Texas Instrument calculators do many of the functions that are encountered in Calculus. It’s not always apparent what to do from reading the nearly thousand pages of manual that come with the machines.

The best approach in using this calculator is a hybrid approach between the graphics functions and the main data entry screen.  This approach lends to easier and more logical equation manipulation.  Though the most common use of the Y= graph key is for graphing, information entered here is put directly into machine memory as alphanumeric data.  These 10 locations are equations and can also be addressed and used on the main screen to solve problems.  This approach saves a lot of retyping and keystroke errors.  Bear in mind that the calculator does not always “appreciate” and recognize that a function goes beyond the screen when in graphics mode.

GRAPHICS MODE

Finding y for a given x.

Hit the [Y=] key and enter any equation.

Hit graph to view.

Then [2nd] [calc] then select choice  for value.

Enter an x value and the calculator will return the y.

Finding the zeros for a function.

Hit the [Y=] key and enter any equation.

Hit graph to view.

Then [2nd] [calc] select choice  for zero.

Visually locate where it hits the x axis.

Move the blinking locator with the “←”,”→” keys to find a left bound and hit [Enter]

The calculator then asks for a right boundary. Move the cursor to the right and again hit [Enter].

The x value is returned.

(Note that as you move from left to right with the arrows, a corresponding y values is being reported.  Make sure that the x-axis is indeed crossed.  Look for the sign change in the y value.)

Finding a Maximum or Minimum (Choices 3 & 4)

This works much the same way as finding a zero.  Two points are marked and an answer is returned.  Obviously, make sure that you are on either side of the visual value.

Finding intersections

This is option  is still yet different.  You will first be prompted for the “First curve”.  You select the curve by using the up/down arrows to jump from one curve to another.  The upper left corner of the screen will tell you which curve you are “on”.  Select with [enter] and repeat the procedure for the selection second curve.  There are times when there may be many functions of the graph so you may have to use the arrow keys a few times in order to jump to the curves you need considered.  After curve selection, the calculator now asks that you guess at the answer.  (Like you really needed a calculator for that.)  Use the horizontal arrows to go to a location of the guess near where you think the answer is and hit [ENTER].  Like the dy/dx operation, you can just type any x value of your choosing then hit return.  Remember that functions might intersect at more than one place, so choose your guess wisely.

PEARL: There are times when you need to find which x value generated a given y value.  i.e., what x value generated a y value of 3?  The best approach in this case is to graph the horizontal line (y=3).  (Typically, most mathematicians would algebraically or otherwise solve for x in terms of y to get an inverse function.)  On the AP exam you will be given a function that on purpose you will not be able to solve for x and will be purposefully impossible.  This approach allows you to get matching points for the coordinates that you need even if you don’t know what the exact function is.   It gets worse, you might be asked for the corresponding slope of a point on the inverse function that you know very little or nearly nothing about.  Hint! Hint!  When you have a (very) sane moment, look up the relationship of functions, inverse functions and their derivatives.  Caution:  There is a draw function that you can look up which will draw the inverse function for you but the calculator will NOT be able to do any of the mathematical calculations that you may need.

Finding dy/dx

This is a little trickier.  Here the calculator reports the value of a tangent line for a given x value.  Go through the same usual steps as above and select option 6.  Move the cursor back and forth and hit [enter] and when you see the x value that you want is displayed on the screen.  Oh, but wait a second, the cursor doesn’t stop exactly where you may want it to.  You can simply just type in the x value you want and then hit [enter].  (Note this function does not “ask” for an x value like it did when you were finding y values.)

Integration ∫f(x)dx

The “area” under a curve.  From the calc menu select [∫f(x)dx], option .  Once again, use the arrows to jump from curve to curve in order to select the one that you want to use then hit [ENTER].  Just like finding dy/dx as above, you then select the lower then the upper bounds, with the horizontal arrows then hitting enter.  Alternatively, you can just enter the x values that you want used.  Remember that the computed area under the x-axis is “negative” area and the resultant answer may not be the area bound by the function and the x-axis.  In other words, the calculator really computes the definite integral and not for the bound area that might have been actually sought!  If you want the bound area of a function that crosses the x-axis, you then have to break up the integral into two or more parts and add the pieces as if they were all positive.  Integration is better handled form the main entry screen.

A final quick and dirty trick in graphic mode.

A common problem is to find the line equation tangent to a function.  Yes you can go ahead and find the dy/dx for the slope as above.  Then you have to find the matching y coordinate and manually write the point slope equation.  This turns out to be one of the few times that the [DRAW] function (accessed via the blue [2nd]) does the work for you.  Under this menu, select option 5 [Tangent].  Use the up/down arrows to jump to the curve you want to use.  Just like finding dy/dx above, move the cursor back and forth to where you want the tangent to be or better yet, simply enter the desired x value and hit [ENTER].  Let the machine do the work.  The final equation will appear at the bottom.  [2nd][Draw][ClrDraw] removes all drawings leaving only the Y= functions on the screen.

Main Screen Funtions

From the main screen hit [MATH], selections 8 [nDeriv(] and 9 [fnInt(] (are by far the most powerful tools in this machine in calculus.  The general format of these functions is:

nDeriv(argument1,x,argument2)

nfnInt(argument1,x,argument2,argument3)

where:

à      Argument1 is typically your function like 4x^3+2x+1.

followed by a comma

à      x will always be x

followed by a comma

(Yes, you do have to actually type in “x” even though it’s implied in argument1.  I think the reason for this is that Texas Instruments wants logical consistency with their other machines that they manufacture.  Other devices allow for x,y and z variables and in those machines, a variable absolutely must be specified.)

à      Argument 2 (and 3) are typically a numbers.  These are the range(s) for the preceding variable (which was x).

Examples:

nDeriv(4x^3+2x+1,x,3) [ENTER] will return the value of dy/dx when x = 3

fnInt(4x^3+2x+1,x,3,5) [ENTER] will return the value of the definite integral of the above function between x=3 and x=5.

Pearls – (a string of them)

There are many times that the same function is integrated again and again which many can many times be messier than above.  Here’s where the hybrid approach comes into play.  Hit the [Y=] key and enter your equation in one of the locations, say Y1.  You could put ten different equations into the calculator if you really wanted to. These equations can be used from the main screen.

Examples:

For the first problem,

Put 4x^3+2x+1 in the Y1 location.

Go back to the main screen

Type [Math][ nDeriv(] but instead of typing in the equation, hit [VARS] and scroll to the right with the arrow select Function which brings you to the [Y column] and then scroll down to [Y1] then hit [ENTER].  The screen will now look like; nDeriv(Y1,.  Type in the rest of the stuff as done before.  The screen should show nDeriv(Y1,x,3).  Hit [ENTER] to retrieve your answer.  This looks a lot neater than the original mess and makes much more logical sense.

The same approach can be used for the integration function and will look like this on the screen fnInt(Y1,x,3,5).  In this case, the range of integration is from 3 to 5.

But remember when we wanted to find the bound area between the curve and the x-axis when the function crossed the line.  This is different than computing the definite integral.  Here’s the sweet part.  Before you select Y1, use the abs function first (found in the math section) to obtain this: fnInt(abs(Y1.  Complete typing as before.  The screen should now look like this: fnInt(abs(Y1),x,3,5).

Think of fnInt(abs(Y1),x,3,5) to say:  integrate the function in Y1 for x from 3 to 5 and report the absolute value.

But wait, there’s more!

Bound area between two curves would read like: fnInt(Y1-Y2,x,3,5).

You are allowed to add and subtract equations like you do in real math.  Y1 and Y2 are retrieved for the Y-vars menu as listed above.

Use the abs function if the functions intersect and you want the bound area.  Again, you do not need to find the point of intersection as you would have to do if this was to be done on paper.

(FYI. When you think of it, when calculating the area under a curve to the x axis Y2 is really y=0.)

Mathematical manipulations can be done for any of the arguments.

Volume of a function rotated around the x axis would look like:

fnInt(pi(Y1)^2,x,3,5) or if you wish pi(fnInt(Y1^2,x,3,5))

Volume rotating one curve around another:

fnInt(pi(Y1-Y2)^2,x,3,5) or pi(fnInt(Y1-Y2)^2,x,3,5)

The list is endless.  Trig functions will also accept Y1 variables as an argument in graphing mode.  i.e. sin(Y1).

Now you know why the main screen is recommend over using the Integration ∫f(x)dx in graphics mode.  This is so much neater and way more powerful and avoids much rekeying.

One final neat trick. Graphing the Derivative.

Suppose you were given a function and wanted to graph it along with the graph of its derivative.  Also suppose that the function was too obnoxious to derive or maybe you were just too lazy to figure it out.

Enter your equation into the Y1 slot.

In the Y2 slot, enter: nDeriv(Y1,X,X).  Think of this as saying, give me the derivative with respect to X for (all) X.  Enjoy the graphs.

Playing with known functions will give you a better understanding of what happens to a function and derivatives on the AP exam, one of their favorite multiple choice questions.  On the AP exam you will be asked to determine which picture is a function and which is the derivative.

What do you think that you would enter into the Y3 for the second derivative?  (ans. Deriv(Y2,X,X))

One final fatal error to avoid:

24/4/3 =

What does this mean to the calculator?

Which are you telling the calculator to do?

(24/4)/3 or 24/(4/3).

24/4/3 means (24/4)/3 to the calculator.  Be careful with calculator order of operations and parenthesis.

Same goes for 24/4*3.  The calculator thinks (24/4)x3 not 24/(3x4).

Hope you find these tips useful on the AP Exam.

Justin