A6
Appendices
APPENDIX B
Calculus and the TI-83/TI-83 Plus/TI-84 Plus Calculators
Functions
A. Define a function
• Press y= to obtain the Y= editor.
• Cursor down to the function to be defined. (Press
clear if the function is to be redefined.)
• Type in the expression for the function. (Note:
Press x,t,θ,n to display X.)
B. Select or deselect a function
(Functions with highlighted equal signs are said to
be selected. Pressing graph instructs the
calculator to graph all selected functions and
pressing 2nd [table] creates a column of the
table for each selected function.)
• Press y= to obtain the Y= editor.
• Cursor down to the function to be selected or
deselected.
• Move the cursor to the equal sign and press enter
to toggle the state of the function on and off.
C. Display a function name, that is,
Y1 , Y2 , Y3 , . . .
• Press vars and then press 1 or enter to
display the list of function names.
• Press the number for the desired function, or
cursor down to the function and press enter .
D. Select a style [such as Line (\), Thick (\\),
or Dot ( . . . )] for the graph of a function
• Move the cursor to the symbol at the left of a
function in the Y= editor.
• Press enter repeatedly to select one of the seven
styles.
E. 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 −, ×, and ÷.)
• If Y3 = Y1 (Y2 ), then Y3 is f (g(x)).
Specific Window Settings
A. Customize a window
• Press window to open the window-setting screen,
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
• Xres = a whole number from 1 to 8
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 zoom to display the list of predefined
settings.
• Either press a number or move the cursor down to
an item and press enter .
• Select ZStandard to obtain [−10, 10] by [−10, 10],
Xscl = Yscl = 1.
• Select ZDecimal to obtain [−4.7, 4.7] by [−3.1, 3.1],
Xscl = Yscl = 1. (When trace is used with this
setting, points have nice x-coordinates.)
• Select ZSquare 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.)
• Select ZTrig to obtain almost [−2π, 2π] 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.)
• [−4.7, 4.7] by [−3.1, 3.1]
• [0, 9.4] by [0, 6.2]
• [−2.35, 2.35] by [−1.55, 1.55] • [0, 18.8] by [0, 12.4]
• [−7.05, 7.05] by [−4.65, 4.65] • [0, 47] by [0, 31]
• [−9.4, 9.4] by [−6.2, 6.2]
• [0, 94] by [0, 62]
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Appendix B Calculus and the TI-83/TI-83 Plus/TI-84 Plus Calculators
General principle: (Xmax − Xmin) should be a
number of the form k · 9.4, where k is a whole
number or 12 , 32 , 52 , . . . , then (Ymax − Ymin)
should be (31/47) · (Xmax − Xmin).
Derivative, Slopes, and Tangent Lines
A. Compute f (a) from the Home screen using
nDeriv(f(x), X, a)
• Press math 8 to display nDeriv(.
• Enter either Y1 , Y2 , . . . or an expression for f (x).
• Type in the remaining items and press enter .
B. Define derivatives of the function Y1
• Set Y2 = nDeriv(Y1 , X, X) to obtain the 1st
derivative.
• Set Y3 = nDeriv(Y2 , X, X) to obtain the 2nd
derivative.
Note: nDeriv( is obtained by pressing math 8.
C. Compute the slope of a graph at a point
• Press 2nd [calc] 6 to obtain dy/dx.
• Use the arrow keys to move to the point of the
graph or type in the x-coordinate of the point.
• Press enter .
D. Draw a tangent line to a graph
• Press 2nd [draw] 5 to select Tangent.
• Use the arrow keys to move to a point of the
graph or type in the x-coordinate of a point.
• Press enter .
Note: To remove all tangent lines, press 2nd
[draw] 1 to execute ClrDraw.
Special Points on the Graph of Y1
A. Find a point of intersection with the graph
of Y2 , from the Home screen
• Press math 0 for the (equation) Solver.
• Press clear to clear the edit screen of the
EQUATION SOLVER.
• Enter either Y1 − Y2 or an expression for the
difference of the two functions to the right of
"eqn:0=".
• Press clear . The equation to be solved will be
on the first line of the screen and the cursor will
be just to the right of "X=".
• Type in a guess for the x-coordinate of a point of
intersection and then press alpha [solve]. (The
word solve is above the enter key.) After a
delay, the value you typed in will be replaced by
the value of the x-coordinate of a point of
intersection.
Note 1: You needn’t be concerned with the last
two lines displayed.
Note 2: If you press enter by mistake, the cursor
will move to the third line of the display. Press alpha [solve] to continue.
• You can now insert a different guess to the right of
"X=" and press alpha [solve] again to obtain
the x-coordinate of another point of intersection.
B. Find intersection points, with graphs
displayed
• Press graph to display the graphs of all selected
functions.
• Press 2nd [calc] 5 to select intersect.
• Reply to "First curve?" by using or (if
necessary) to move the cursor to one of the two
curves and then pressing enter .
• Reply to "Second curve?" by using or (if
necessary) to move the cursor to the other curve
and then pressing enter .
• Reply to "Guess?" by moving the cursor near
the point of intersection (or typing in an
approximate value of the x-coordinate of the point
of intersection) and pressing enter .
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,t,θ,n 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 2nd [calc] 1 to select value.
• Type in the value of a and press enter .
Note: The value of a must be between Xmin and
Xmax.
• If desired, press to move to points on graphs of
other selected functions.
With the graph displayed:
• Press trace .
• Type in the value of a and press enter .
D. Find the first coordinate of a point whose
second coordinate is b
From the Home screen:
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A8
Appendices
• Select the EQUATION SOLVER screen as in part A
above, and set "eqn:0" = Y1 − b.
• Press clear to switch to another screen and
clear the initial value of X.
• Type in a guess for the value of X and then press
alpha [solve].
With the graphs displayed:
• 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 (that is, a zero of Y1 )
• Graph Y1 .
• Press 2nd [calc] 2 to select zero.
Note: If more than one graph is displayed press until the expression for Y1 appears at the top of
the screen.
• Move the cursor to a point just to the left of a
zero (or type in a number less than a zero) and
press enter .
• Move the cursor to a point just to the right of the
zero (or type in a number greater than a zero) and
press enter .
• Move the cursor to a point near the zero (or type
in a number near the zero) and press enter .
F. Find a relative extreme point
• Set Y2 = nDeriv(Y1 , X, X) or set Y2 equal to the
exact expression for the derivative of Y1 . [To
display nDeriv( press math 8.]
• 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 = nDeriv(Y1 , X, X) and
Y3 = nDeriv(Y2 , X, X), or set Y3 equal to the
exact expression for the second derivative of Y1 .
• Select Y3 and deselect all other functions.
• Graph Y3 .
• Find an x-intercept of Y3 , call it r, at which the
graph of Y3 crosses the x-axis.
• The point (r, Y1 (r)) will be a possible inflection
point of Y1 .
Tables
A. Display values of f (x ) for evenly spaced
values of x
•
•
•
•
•
•
Press y= , assign the function f (x) to Y1 .
Press 2nd [tblset].
Set TblStart =first value of x.
Set ΔTbl =increment for values of x.
Set both Indpnt and Depend to Auto.
Press 2nd [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 Y2 and also select
Y2 to obtain a table with columns for X, Y1 , and
Y2 .
B. Display values of f (x ) for arbitrary values
of x
• Press y= and assign the function f (x) to Y1 , and
deselect all other functions.
• Press 2nd [TblSet].
• Set Indpnt to Ask by moving the cursor to Ask
and pressing enter .
• Leave Depend set to Auto.
• Press 2nd [table].
• Type in any value for X and press enter .
• Repeat the previous step for any other values of X.
Riemann Sums
Suppose 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) + · · · + f (d).
The function sum is in the LIST/MATH menu and
the function seq is in the LIST/OPS menu.
Compute [f (x1 ) + f (x2 ) + · · · + f (xn )] · Δx
On the Home screen, evaluate sum(seq(f (x), X,
x1 , xn , Δx)) ∗ Δx as follows:
• Press 2nd [list] and move the cursor right to
MATH
• Press 5 to display sum(.
• Press 2nd [list] and move the cursor right to
OPS.
• Press 5 to display seq(.
• Enter either Y1 or an expression for f (x).
• Type in the remaining items and press enter .
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Calculus and Its Applications 12/E 11/20/2008 Page 8
Appendix B Calculus and the TI-83/TI-83 Plus/TI-84 Plus Calculators
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 math 9 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 and find its area
• Press 2nd [calc] 7 to select f(x) dx.
• 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 2nd [draw] 1
to execute the ClrDraw command.
C. Obtain the graph of the solution to the
differential equation y = g(x), y(a) = b
[That is, obtain the graph of 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. [Press math 9 to
display fnInt(.] 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 speeded up by setting xRes (in
the WINDOW screen) 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 instructions
Shade(Y1 ,Y2 ,a,b) as follows:
• Press 2nd [draw] 7 to display Shade(.
• Type in the remaining items and press enter .
Note: To remove the shading, press 2nd [draw] 1
enter to execute the ClrDraw command.
Functions of Several Variables
A. Specify a function of several variables and
its derivatives
• In the Y= editor, set Y1 = f (X, Y). (The letter Y
is entered by pressing alpha [y].)
• Set Y3 = nDeriv(Y1 , Y, Y). Y3 will be
∂f
.
∂x
∂f
.
∂y
• Set Y4 = nDeriv(Y2 , X, X). Y4 will be
∂2f
.
∂x2
• Set Y5 = nDeriv(Y3 , Y, Y). Y5 will be
∂2f
.
∂y 2
• Set Y6 = nDeriv(Y3 , X, X). Y6 will be
∂2f
.
∂x ∂y
• Set Y2 = nDeriv(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,t,θ,n .
• Press b sto 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 ).
• Press stat 1 for the EDIT screen, and obtain a
table used for entering the data.
• If necessary, clear data from columns L1 and L2 as
follows. Move the cursor to the top of column L1
and press clear enter . Repeat for column L2 .
• To enter the x-coordinates of the points, move the
cursor to the first blank row of column L1 , enter
the value of x1 , and press enter . Repeat with
x2 , . . . , xn .
• Move the cursor to the first blank row of column
L2 , and enter the values of y1 , . . . , yn .
• Press stat for the CALC menu, and press 4
to place LinReg(ax + b) on the Home screen.
• Press enter to obtain the slope and y-intercept of
the least-squares line.
B. Assign the least-squares line to a function
• Press y= , move the cursor to the function, and
press clear to erase the current expression.
• Press vars 5 to select the Statistics variables.
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A10
Appendices
• Press to select the EQ menu, and press 1
for RegEQ (Regression Equation), which then
assigns the least-squares line equation to the
function.
C. Display the points from part A
• Press y= and deselect all functions.
• Press 2nd [stat plot] enter to select Plot1,
press enter to turn Plot1 ON.
• Select the first plot from the six icons for the types
of plots. This icon corresponds to a scatter plot.
• Press graph to display the data points.
Note 1: Press zoom 9 to ensure that 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 plot] enter
to select Plot1. Then press to move to OFF
and press enter .
D. Display the line and the points from part A
• Press y= 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)
Obtain a table for the approximation given
by Euler’s method, with a, b, y0 , h, and n
as given in Section 10.7
Note: We will use N to represent the value of n in
order to avoid confusion with the variable n in the
sequence Y = editor.
•
•
•
•
•
With the calculator in sequence mode, the values
for t0 , t1 , t2 , . . . are stored as the sequence values
u(0), u(1), u(2), . . . , and the values for
y0 , y1 , y2 , . . . are stored as the sequence values
v(0), v(1), v(2), . . . .
To invoke sequence mode, press mode , move the
cursor down to the fourth line, move the cursor
right to Seq, and press enter .
Press y= to obtain the sequence Y= editor.
Set nMin = 0.
Clear u(n), if necessary, and set
u(n) = u(n − 1) + h. [In Seq mode, pressing 2nd
[u] (the second function of 7 key) generates u and
pressing x,t,θ,n generates n. Here h refers to the
value for the length of the subintervals.]
Set u(nMin)=a.
• Set v(n) = v(n − 1) + g(u(n − 1), v(n − 1)) ∗ h. [In
Seq mode, pressing 2nd [v], the second function of
the 8 key, generates v. To form
g(u(n − 1), v(n − 1)) ∗ h, in the function g(t, y)
replace t with u(n − 1) and replace y by v(n − 1),
and then multiply it by the value of h.]
• Set v(nMin)= y0 .
• Press 2nd [format], move on the first row to uv
and press enter .
• Press window and set nMin = 0, nMax = N ,
Xmin = a, and Xmax = b. Set the values of Xscl,
Ymin, Ymax, and Yscl as you would when
graphing ordinary functions. Leave PlotStart and
PlotStep set at their default values of 1.
• To display a graph of the solution, press graph .
(You may use trace to examine the coordinates
of points on the graph, where the first coordinate
is denoted X and the second coordinate denoted
Y.)
• To display a table of the points on the solution
given by Euler’s method, first press 2nd [tblset],
and set TblStart=0, ΔTbl = 1, and the other
items to Auto. Then press 2nd [table]. The
successive values of t and y are contained in the
u(n) and v(n) columns, respectively.
Note: After completing using Euler’s method,
reset the calculator to function mode by pressing
mode , moving to Func in the fourth line, and
pressing enter .
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,t,θ 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
nDeriv(Y1 ,X,X). In this case, the successive
approximations will differ slightly from those
obtained with Y2 equal to f (x).
Copyright © 2010 Pearson Education, Inc.
Goldstein/Schnieder/Lay/Asmar:
Calculus and Its Applications 12/E 11/20/2008 Page 10
Appendix B Calculus and the TI-83/TI-83 Plus/TI-84 Plus Calculators
Sum a Finite Series
A11
Note: At most 999 terms can be summed.
Compute the sum
f (m) + f (m + 1) + · · · + f (n)
On the Home screen, evaluate
sum(seq(f(X),X,m,n)) as follows:
• Press 2nd [list] and move the cursor right to
MATH.
• Press 5 to display sum (.
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.
• Enter an expression for f (x).
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.
• Type in the remaining items and press enter .
C. To clear the Home screen, press clear twice.
• Press 2nd [list] and move the cursor right to
OPS.
• Press 5 to display seq (.
Copyright © 2010 Pearson Education, Inc.
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