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 [1] 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 [2] 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
[5] 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 [7]. 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