TI-86

Appendix D Calculus and the TI-86 Calculator
A17
APPENDIX D
Calculus and the TI-86 Calculator
Functions
A. Deﬁne a function, say y1, from the Home
screen
• Press 2nd [quit] to invoke the home screen.
• Press 2nd [alpha] [y] 1 alpha [=] followed by an
expression for the function, and press enter .
B. Deﬁne a function from the function editor
• Press graph f1 to select y(x) = from the
GRAPH menu and obtain the screen for deﬁning
functions; that is, the function editor.
• Cursor down to a function. (To delete an existing
expression, press clear . To create an additional
function, cursor down to the last function and
press enter .)
• Type in an expression for the function. (Note:
Press f1 or x-var to display x. Press f2 or 2nd
[alpha] [y] to display y.)
C. Select or deselect a function in the function
editor
Functions with highlighted equal signs are said to
be selected. The graph screen displays the graphs
of all selected functions, and tables contain a
column for each selected function.
• Press graph f1 to invoke the function editor.
• Cursor down to the function to be selected or
deselected.
• Press f5 , that is, selct, to toggle the state of the
function on and oﬀ.
D. Select a style [such as Line (\), Thick (\\),
or Dot ( . . . )] for the graph of a function
• Move the cursor to a function in the function
editor.
• Press more and then press f3 repeatedly to
select one of the seven styles.
E. Display a function name, that is,
y1, y2, y3, . . .
• Press 2nd [alpha] [y] followed by the number.
or
• Press 2nd [catlg-vars] more f4 to invoke a list
containing the function names.
• Cursor down to the desired function.
• Press enter to display the selected function
name.
F. Combine functions
Suppose y1 is f (x) and y2 is g(x).
• If y3 = y1 + y2, then y3 is f (x) + g(x). (Similarly
for −, &times;, and &divide;.)
• If y3 = y1(y2), then y3 is f (g(x)).
Specify Window Settings
A. Customize a window
• Press graph f2 to invoke the window editor and
edit the following values as desired.
• xMin = the leftmost value on the x-axis
• xMax = the rightmost value on the x-axis
• xScl = the distance between tick marks on the
x-axis
• yMin = the bottom value on the y-axis
• yMax = the top value on the y-axis
• yScl = the distance between tick marks on the
y-axis
Usually xRes = 1. Higher values speed up
graphing, but with a loss of resolution.
Note 1: The notation [a, b] by [c, d] stands for the
window settings xMin = a, xMax = b, yMin = c,
yMax = d.
Note 2: The default values of xScl and yScl are 1.
The value of xScl should be made large (small) if
the diﬀerence between xMax and xMin is large
(small). For instance, with the window settings
[0, 100] by [−1, 1], good scale settings are xScl = 10
and yScl = .1.
B. Use a predeﬁned window setting
• Press graph f3 to invoke the ZOOM menu.
• Press f4 , that is ZSTD, to obtain
[−10, 10] by [−10, 10], xScl = yScl = 1.
• Press more f2 , that is zsqr, to obtain a
true-aspect window. (With such a window setting,
lines that should be perpendicular actually
look
√
perpendicular, and the graph of y = 1 − x2 looks
like the top half of a circle.)
• Press more f4 , that is zdecm, to obtain
[−6.3, 6.3] by [−3.1, 3.1], xScl = yScl = 1. (When
trace is used with this setting, points have nice
x-coordinates.)
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A18
Appendices
• Press more f3 , that is ztrig, to obtain
[−21π/8, 21π/8] by [−4, 4], xScl = π/2, yScl = 1, a
good setting for the graphs of trigonometric
functions.
C. Some nice window settings
With these settings, one unit on the x-axis has the
same length as one unit on the y-axis, and tracing
progresses over simple values.
• [−6.3, 6.3] by [−3.7, 3.7]
• [0, 12.6] by [0, 7.4]
• [−3.15, 3.15] by [−1.85, 1.85]
• [0, 25.2] by [0, 14.8]
• [−9.45, 9.45] by [−5.56, 5.56]
• [0, 63] by [0, 37]
• [−12.6, 12.6] by [−7.4, 7.4]
• [0, 126] by [0, 74]
General principle: (xMax − xMin) should be a
number of the form k &middot; 12.6, where k is a whole
number or 12 , 32 , 52 , . . . , then (yMax − yMin) should
be (37/63) &middot; (xMax − xMin).
Derivative, Slopes, and Tangent Lines
A. Compute f (a) from the Home screen using
der1(f(x), x, a)
• Press 2nd [calc] f3 to display der1(.
• Enter either y1, y2, . . . or an expression for f (x).
• Type in the remaining items and press enter .
B. Deﬁne derivatives of the function y5
• Set y1 = der1(y5, x, x) to obtain the 1st
derivative.
• Set y2 = der2(y5, x, x) to obtain the 2nd
derivative.
• Set y3 = nDer(y2, x, x) to obtain the 3rd
derivative.
• Set y4 = nDer(y3, x, x) to obtain the 4th
derivative.
Note 1: der1, der2, and nDer are found on the
menu obtained by pressing 2nd [calc].
Note 2: To speed up the display of the 3rd and
4th derivatives, set the value of xRes in the
WINDOW screen to at least 5.
C. Compute the slope of a graph at a point
• Press graph f5 to display the graph of the
function.
• Press more f1 f2 to select dy/dx from the
• Use the arrow keys to move to the point of the
graph and then press enter . (This process
usually works best with a nice window setting.)
Or, type in the value of the x-coordinate of a
point (any number between xMin and xMax), and
press enter .
D. Draw a tangent line to a graph
• Press graph f5 to display the graph of the
function.
• Press more f1 more more f1 to select tanln
• Move the cursor to any point on the graph or type
in the ﬁrst coordinate of a point.
• Press enter to draw the tangent line through the
point and display the slope of the curve at that
point. The slope is displayed at the bottom of the
screen as dy/dx = slope.
• To draw another tangent line, press graph and
then repeat the previous three steps.
Note: To remove all tangent lines, press graph
more f2 more more f1 to select cldrw from
Special Points on the Graph of y1
A. Find a point of intersection with the graph
of y2, from the Home screen
• Press 2nd [solver].
• To the right of &quot;eqn:&quot; enter y1 − y2 = 0 and
press enter . (Note: y1 and y2 can be entered via
F keys and the equal sign is entered with alpha
[=].)
• To the right of &quot;x=&quot; type in a guess for the
x-coordinate of the point of intersection, and then
press f5 . After a little delay, a value of x for
which y1 = y2 will be displayed in place of your
guess.
B. Find intersection points, with graphs
displayed
• Press graph f5 to display the graphs of all
selected functions.
• Press more f1 more f3 to select isect from the
• Reply to &quot;First curve?&quot; by using or (if
necessary) to place the cursor on one of the two
curves and then pressing enter .
• Reply to &quot;Second curve?&quot; by using or (if
necessary) to place the cursor on the other curve
and then pressing enter .
• Reply to &quot;Guess?&quot; by moving the cursor close to
the desired point of intersection (or typing in a
guess for the ﬁrst coordinate of the desired point)
and pressing enter .
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Appendix D Calculus and the TI-86 Calculator
C. Find the second coordinate of the point
whose ﬁrst coordinate is a
From the Home screen:
• Display y1(a) and press enter .
or
• Press a sto x-var enter to assign the value a
to the variable x.
• Display y1 and press enter .
From the Home screen or with the graph displayed:
• Press graph more more f1 to display &quot;Eval
x=&quot;.
• Type in the value of a and press enter . (The
value of a must be between xMin and xMax.)
• If desired, press the up-arrow key, , to move to
points on graphs of other selected functions.
With the graph displayed:
• Press f4 ; that is, trace.
D. Find the ﬁrst coordinate of a point whose
second coordinate is b
• Set y2 = b.
• Find the point of intersection of the graphs of y1
and y2 as in part B above.
E. Find an x -intercept of a graph of a function
• Press graph more f1 f1 to select root from
• Select y2 and deselect all other functions.
• Graph y2.
• Find an x-intercept of y2, call it r, at which the
graph of y2 crosses the x-axis.
• The point (r, y1(r)) will be a possible relative
extreme point of y1.
G. Find an inﬂection point
• Set y2 = der2(y1, x, x) or set y2 equal to the
exact expression for the second derivative of y1.
(To display der2, press 2nd [calc] f4 .)
• Select y2 and deselect all other functions.
• Graph y2.
• Find an x-intercept of y2, call it r, at which the
graph of y2 crosses the x-axis.
• The point (r, y1(r)) will be a possible inﬂection
point of y1.
Tables
• Type in the value of a and press enter .
• If necessary, use or
the desired graph.
to place the cursor on
• In response to the request for a Left Bound, move
the cursor to a point whose ﬁrst coordinate is less
than the desired x-intercept or type in a value less
than the desired x-intercept. Then press the
enter key.
• In response to the request for a Right Bound,
move the cursor to a point whose ﬁrst coordinate
is greater than the desired x-intercept or type in a
value greater than the desired x-intercept. Then
press the enter key.
• In response to the request for a Guess, move the
cursor to a point near the desired x-intercept or
type in a value close to the desired x-intercept.
Then press the enter key.
F. Find a relative extreme point
• Set y2 = der1(y1, x, x) or set y2 equal to the
exact expression for the derivative of y1. (To
display der1(, press 2nd [calc] f3 .)
A. Display values of f (x) for evenly spaced
values of x
• Press graph f1 and assign the function f (x) to
y1.
• Press table f2 to invoke the TABLE SETUP
window.
• Set TblStart = ﬁrst value of x.
• Set ΔTbl = increment for values of x.
• Set Indpnt to Auto.
• Press f1 to display the table.
Note 1: You can use or to look at function
values for other values of x.
Note 2: The table can display values of more than
one function. For example, in the ﬁrst step you
can assign the function g(x) to another function
and also select that other function to obtain a
table with columns for x and each of the selected
functions.
B. Display values of f (x ) for arbitrary values
of x
• Press graph f1 , assign the function f (x) to a
function, and deselect all other functions.
• Press table f2 .
and pressing enter .
• Press f1
• Type in any value for x and press enter .
• Repeat the previous step for as many values as
you like.
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Appendices
Riemann Sums
Suppose that y1 is f (x), and c, d, and Δx are
numbers, then sum(seq(y1,x,c,d,Δx))
computes
f (c) + f (c + Δx) + f (c + 2Δx) + &middot; &middot; &middot; + f (d).
The functions sum and seq are found in the
Compute [f (x1 ) + f (x2 ) + &middot; &middot; &middot; + f (xn )] &middot; Δx
On the Home screen, evaluate sum(seq(f (x), x,
x1 , xn , Δx)) ∗ Δx as follows:
• Press 2nd [list] f5 more f1 ( f3 to display
sum(seq(.
• Enter either y1 or an expression for f (x).
• Type in the remaining items and press enter .
Definite Integrals and Antiderivatives
A. Compute
b
a
f (x ) dx
On the Home screen, evaluate fnInt(f (x), x, a, b)
as follows:
• Press 2nd [calc] f5 to display fnInt(.
• Enter either y1 or an expression for f (x).
• Type in the remaining items and press enter .
B. Shade a region under the graph of a
function from x = a to x = b
• Press graph more f1 f3 to select f (x) from
• If necessary, use
the graph.
or
to move the cursor to
• In response to the request for a Lower Limit, move
the cursor to the left endpoint of the region (or
type in the value of a) and press enter .
• In response to the request for an Upper Limit,
move the cursor to the right endpoint of the
region (or type in the value of b) and press enter .
Note: To remove the shading press graph more
f2 more more f1 to select cldrw from the
C. Obtain the graph of the solution to the
diﬀerential equation y = g(x), y(a) = b
[That is, obtain the function f (x) that is an
antiderivative of g(x) and satisﬁes the additional
condition f (a) = b.]
• Set y1 = g(x).
• Set y2 = fnInt(y1, x, a, x) + b. (To display
fnInt, press 2nd [calc] f5 .) The function y2 is
an antiderivative of g(x) and can be evaluated and
graphed.
Note: The graphing of y2 proceeds very slowly.
Graphing can be sped up by setting xRes (in the
GRAPH/WIND menu) to a high value.
D. Shade the region between two curves
Suppose the graph of y1 lies below the graph of y2
for a ≤ x ≤ b and both functions have been
selected. To shade the region between these two
curves, execute the instruction
• Press graph more f2 f1 to display Shade(
• Type in the remaining items and press enter .
Note: To remove the shading, press graph more
f2 more more f1 to execute ClrDraw from the
Functions of Several Variables
A. Specify a function of several variables and
its derivatives
• In the y(x) = function editor, set y1 = f (x, y).
(The letters x and y can be entered by pressing
f1 and f2 .)
• Set y3 = der1(y1, y, y). y3 will be
∂f
.
∂x
∂f
.
∂y
• Set y4 = der2(y1, x, x). y4 will be
∂2f
.
∂x2
• Set y5 = der2(y1, y, y). y5 will be
∂2f
.
∂y 2
• Set y6 = nDer(y3, x, x). y6 will be
∂2f
.
∂x ∂y
• Set y2 = der1(y1, x, x). y2 will be
B. Evaluate one of the functions in part A at
x = a and y = b
• On the Home screen, assign the value a to the
variable x with a sto x-var .
• Press b sto 2nd [alpha] [y] to assign the value b
to the variable y.
• Display the name of one of the functions, such as
y1, y2, . . . , and press enter .
Least-Squares Approximations
A. Obtain the equation of the least-squares
line
Assume the points are (x1 , y1 ), . . . , (xn , yn ).
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Appendix D Calculus and the TI-86 Calculator
• Press 2nd stat f2 to obtain a table for entering
the data.
• If necessary, clear data from columns xStat and
yStat as follows. Move the cursor to xStat and
press clear enter . Repeat for the yStat column.
• To enter the x-coordinates of the points, move the
cursor to the ﬁrst blank row the xStat column,
type in the value of x1 , and press enter . Repeat
with x2 , . . . , xn .
• Move the cursor to the ﬁrst blank row of the yStat
column and enter the values of y1 , . . . , yn .
• Place a 1 in each of the ﬁrst n entries of the fStat
column. (The other entries should be blank.)
• Press 2nd [quit] to display the home screen.
• Press 2nd stat f1 f3 enter to obtain the values
of a and b where the least-squares line has
equation y = a + bx.
• If desired, the straight line (and the points) can be
graphed with the following steps:
(a) First press graph f1 and deselect all
functions.
(b) Press 2nd stat f4 to invoke the statistical
DRAW menu. (If any graphs appear, press
more f2 to clear them.)
(c) Press the button for drreg (either f1 or
more f1 ) to draw the least-squares line and
press more f2 to draw the n points with
SCAT.
B. Assign the least-squares line to a function
• Press graph f1 , move the cursor to the function,
and press clear to erase the current expression
for the function.
• Press 2nd [stat] f5 more more f2 to assign the
equation (known as RegEq) to the function.
C. Display the points from part A
• Press graph f1 and deselect all functions.
• Press 2nd [stat] f3 to select STAT PLOTS menu.
• Press f1 to select PLOT1.
• Move the cursor to ON and press enter .
• Select scat as the Type, select xStat as the Xlist
Name, select yStat as the Ylist Name, and select
any symbol for Mark.
• Press graph f5 to display the data points.
Note 1: Make sure the current window setting is
large enough to display the points.
Note 2: When you ﬁnish using the point-plotting
feature, turn it oﬀ. Press 2nd [stat] f3 f1 , move
the cursor to OFF and press enter .
D. Display the line and the points from part A
• Press graph f1 and deselect all functions except
for the function containing the equation of the
least-squares line.
• Carry out all but the ﬁrst step of part C.
The Differential Equation y' = g(t, y)
A. Carry out Euler’s method, with a, b, y0 ,
and h as given in Section 10.7
• To invoke diﬀerential equation mode, press 2nd
[mode], move the cursor down to the ﬁfth line,
move the cursor right to DifEq, and press enter .
• Press graph more f1 and select Euler from the
4th row and FldOff from the 5th row.
• Press graph f1 and enter the diﬀerential
equation. Use Q1 (or Q2, Q3, . . . ) instead of y1
(or y2, y3, . . . ). The letters t and Q can be
entered with the keys f1 and f2 . Up to nine
diﬀerential equations, with function variables Q1,
Q2, . . . , can be speciﬁed.
• Press graph f4 and set x = t and y = Q.
• Press graph f2 to invoke the window-setting
screen.
• Set tMin and xMin to a, set tMax and xMax to b,
and set tStep to h. (Leave the values of tPlot and
EStep at their default settings of 0 and 1.)
• Set the values of xScl, yMin, yMax, and yScl as
you would when graphing ordinary functions.
• Press graph f3 and set the initial value (denoted
by QI1, QI2, etc.) to y0 . Alternately, you can
simultaneously graph solutions for several diﬀerent
initial values, call them v1 , v2 , . . . , by setting the
initial value to {v1 , v2 , . . . }. Braces are entered
• Press graph f5 to see a graph of the Euler’s
method of solution of the diﬀerential equation.
Note 1: To make the graph more accurate (and
the graphing slower), decrease the value of tStep.
Note 2: A table of values can be displayed by
setting TblStart to a, ΔTbl to h, and Indpnt
to Auto.
Note 3: The next-to-last step shows one way to
obtain a family of solutions of a diﬀerential
equation. Another way is presented in item B
below.
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Appendices
B. Graph a family of solutions of a diﬀerential
equation
• To invoke diﬀerential equation mode, press 2nd
[mode], move cursor down to the ﬁfth line, move
the cursor right to DifEq, and press enter .
• Press graph more f1 and select Euler from the
4th row and FldOff from the 5th row.
• Press graph f1 and enter the diﬀerential
equation. Use Q1 (or Q2, Q3, . . . ) instead of y.
The letters t and Q can be entered with the keys
f1 and f2 .
• Press graph f2 to invoke the window-setting
screen.
• Set tMin and xMin to 0.
• The values of tMax and xMax must be set by trial
and error. Try setting them both to values like 2,
5, or 10 and see if the solutions look like those in
Section 10.6.
• Set tStep to a small value, such as .1 or the
default (≈ .13). The smaller the value of tStep,
the better the accuracy will be and the longer the
time required.
• Leave the values of tPlot and EStep at their
default settings of 0 and 1.
• Set the values of xScl, yMin, yMax, and yScl as
you would when graphing ordinary functions. The
value of yMin should be less than the smallest
constant solution, and the value of yMax should
be greater than the largest constant solution.
• Press graph f3 and set the initial value to y0 .
• Press graph f5 to graph the solution of the
diﬀerential equation.
• Press graph more f5 to explore solutions with
other initial values.
• Move the cursor up or down to another place on
the y-axis and press enter to see the graph of the
solution beginning at that point.
• Repeat the process in the previous step as often as
desired.
Note: To erase one of the graphs drawn, select
C. Graph the slope ﬁeld of a diﬀerential
equation
• To invoke diﬀerential equation mode, press 2nd
mode , move the cursor down to the ﬁfth line,
move the cursor right to DifEq, and press enter.
• Press graph more f1 and select SlpFld from
the sixth line.
• Press graph f1 and enter the diﬀerential
equation. Use Q1 (or Q2, Q3, . . . ) instead of y1
(or y2, y3, . . . ). The letter t and Q can be entered
with the keys F1 and F2.
• Press graph f4 and set fldRes = 15. To draw
more line segments in the slope ﬁeld, enter a
larger value for fldRes.
• Press graph f2 to invoke the window-setting
screen.
• Set tMin and xMin to 0. Enter all other settings
as you would when graphing solutions of a
diﬀerential equation. For example, set tStep to a
small value, such as .15, or use the default (≈ .13).
Set tMax and xMax to values like 3, 5, or 10, and
see if the slope ﬁelds look like those in Section
10.1.
• Press graph f3 and set an initial condition for
Q1. This will add a solution curve with the
speciﬁed initial condition to the slope ﬁeld. If you
do not specify the initial condition, then only the
slope ﬁeld will be graphed.
• Press graph f5 to graph the slope ﬁeld and a
solution curve (if you have speciﬁed an initial
condition) or the slope ﬁeld only.
• To display the slope ﬁeld with diﬀerent solution
curves, press graph f3 , change the initial value
for Q1, then press graph f5 .
The Newton–Raphson Algorithm
Perform the Newton–Raphson Algorithm
• Assign the function f (x) to y1 and the function
f (x) to y2.
• Press 2nd [quit] to invoke the home screen.
• Type in the initial approximation.
• Assign the value of the approximation to the
variable x. This is accomplished with the
keystrokes sto x-var enter .
• Type in x − y1/y2 → x. (This statement
calculates the value of x − y1/y2 and assigns it to
x.)
• Press enter to display the value of this new
approximation. Each time enter is pressed,
another approximation is displayed.
Note: In the ﬁrst step, y2 can be set equal to
der1(y1,x,x).
Copyright &copy; 2010 Pearson Education, Inc.
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Calculus and Its Applications 12/E 11/20/2008 Page 22
Appendix D Calculus and the TI-86 Calculator
Sum a Finite Series
Compute the sum
f (m) + f (m + 1) + &middot; &middot; &middot; + f (n)
On the Home screen, evaluate
sum(seq(f(x),x,m,n,1)) as follows:
• Press 2nd [list] f5 more f1 ( f3 to display
sum(seq(.
• Enter an expression for f (x).
• Type in the remaining items and press enter .
Note: About 6000 terms can be summed.
Miscellaneous Items and Tips
A. From the Home screen, if you plan to reuse a
recently entered line with some minor changes,
press 2nd [entry] until the previous line appears.
You can then make alterations to the line and
press enter to execute the line.
B. If you plan to use trace to examine the values of
various points on a graph, set yMin to a value
that is lower than is actually necessary for the
graph. Then, the values of x and y will not
obliterate the graph while you trace.
C. To clear the Home screen, press clear twice.
D. When two menus are displayed at the same time,
you can remove the top menu by pressing exit .
(The remaining menu can be removed by pressing
exit again.)
E. To obtain the solutions of a quadratic equation, or
of any equation of the form p(x) = 0, where p(x)
is a polynomial of degree ≤ 30, press 2nd [poly],
enter the degree of the polynomial, enter the
coeﬃcients of the polynomial, and press f5 .
(Some of the solutions might be complex
numbers.)
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