Texas Instruments TI-83 - Plus Graphing Calculator User manual

CONTENTS
Calculator Page
Texas Instruments TI-83 1
Texas Instruments TI-82 30
Texas Instruments TI-85 58
Texas Instruments TI-86 83
30

THE TI-82
Item Page
The Viewing Window 34
Turning the Calculator On and Off 34
Graphing in the Standard Viewing Window 34
The TRACE Feature 35
Changing the Dimensions of the Viewing Window 35
Scale Marks 35
Locating the Graph 36
The ZOOM Feature 36
Deleting a Function 36
Step Size 36
Piecewise Defined Functions and Dot Mode 37
Split Screen 37
The Greatest Integer Function 38
The root Operation 38
The intersect Operation 39
Using STAT and LinReg 40
The TABLE Operation 41
The value Operation 42
Using TRACE to Go Beyond Viewing Window 42
Executing a Program 42
The Program EVAL 43
Drawing a Tangent Line to a Curve 43
The Program SEC 43
The Program nDeriv 44
The Program NDERG 44
The Program nDer2 44
The Program nDer2G 44
The Program RECT 44
The Program SUM 45
The Program NINTG 45
The Program NINT 46
The Program TRAP 46
The Program SIMP 46
The Program VOL 46
Graphing a Sequence 47
31

Item Page
The Program RAND 47
The Program RAND2 48
Histograms 48
Statistics 48
Line Plots and Histograms 49
The Program DFIELD 49
The Program EULER 49
The Program EULERG 50
The Program EULERS 50
Editing a Program 50
Quiting a Program 50
Writing a New Program 50
Using LINK 50
Executing a Program 50
Troubleshooting 51
Keystrokes for the Program SEC 52
A List of Programs 53
32

Introduction
Our goal in this manual is to show you what you need to know to obtain the screens in the text and to do
the Explorations, not to make you an expert in using graphing calculators. We want you spending your time
and energy learning calculus, rather than learning how to use your graphing calculator! We believe you will
discover that very little effort is needed to perform quite remarkable tasks on the graphing calculators.
The manual introduces a feature of the graphing calculator the first time that feature is needed in the text.
Thus, the manual is read together with the text and section headings in this manual correspond to section
headings in the text. Every needed keystroke is given.
In this manual calculator keystrokes are always shown boxed. For example, the ENTER key is designated
as ENTER , the 8 key is designated as 8 , and the B key by B . To actually type the letter on the screen
of your calculator you must use the ALPHA key. Notice that each letter is located above the right side of
some key. For example, on the TI-82 the letter Q is located above the right side of the 9 key. We will refer
to this key as both the 9 key and also the Q key. To write the letter Q on the TI-82 you must press the two
keys ALPHA Q .
In general, we will refer to a key using either the symbols on or above the key.
Above the right side of some of the keys are some symbols other than letters. To write these symbols you
must use the ALPHA key as with letters. All of these keys are self explanatory, except for the symbol t
above the 0 (zero) key. This symbol represents a space.
Above the left side of most of the keys you will see some functions. To access these functions you must use
the 2nd key. For example, above the left side of the x2key is the square root symbol √. To find the
square root of 4 press the keys
2nd √4 ENTER
and the calculator returns a 2.
On the TI-82 there is an important distinction between the two keys −and (−) . The key −is the
subtraction key, while (−) is the negation key. Thus, the expression x−1 is X,T,θ−1 , whereas the
expression −x−1 is (−) X,T,θ−1 .
33

Section 1.0 in Text
•The Viewing Window
•The Standard Viewing Window
•TRACE
•Dimensions of the Viewing Window
•Scale Marks
•ZOOM
•Deleting a Function
•The Step Size
The Viewing Window
The dashed rectangle in the figure is the boundary of
the viewing screen. The x-axis is marked from Xmin
to Xmax and the y-axis from Ymin to Ymax.
We shall say that this viewing window has dimensions
[Xmin,Xmax] by [Ymin,Ymax]. The viewing window
with dimensions [−10,10] by [−10,10] is referred to as
the standard viewing window.
Graphing in the Standard Viewing Window
Graph y=x4−12x3+x2−2 in the standard viewing
window. (This is the first part of Example 1 in the
text.)
Solution. Press the ON key to turn the calculator
on. (To turn off press the keys 2nd OFF .) Press the
Y= key and you see
To enter Y1=x4−12x3+x2−2, press the keys
X,T,θ∧4−1 2 X,T,θ∧3 +
X,T,θ,n ∧2−2 ENTER
You should have the following.
Press the ZOOM key and you see the ZOOM options
Press the 6 key. This selects the standard viewing
window, graphs, and gives the following (Screen 1.1 in
the text).
34

The TRACE Feature
Find points on the graph of the previous example. Re-
call y=x4−12x3+x2−2.
Solution. After graphing the function press the
TRACE key. A blinking star-like cursor will be sitting
on a point of the graph.
At the bottom you see x= 0 and y=−2. This means
that the point (0,2) lies on the graph. You can move
the cursor by using the and keys. For example,
by pressing the key you obtain
This means that the (approximate) point
(0.21276596,−2.068263) is on the graph.
Change the Dimensions of the Viewing Window
Graph y=x4−12x3+x2−2 using a window with di-
mensions [−5,14] by [−2500,1000]. (This is the second
part of Example 1 in the text).
Solution. Press the WINDOW key and you see
Press the 5key and the blinking cursor moves
down to the −10 after Xmin.
Press the keys (−) 5 . This changes −10 to −5.
(The (−) key is used to change the sign, whereas the
−key is used to subtract two quantities.)
Press the ENTER key. The blinking cursor drops
to the 10 after Xmax.
Press the keys 1 4 . This changes 10 to 14
Notice that Xscl is 1. This means the tick marks on
the x-axis are set at 1 unit. We will not change this.
Press the ENTER key twice. The blinking cursor
drops to the −10 after Ymin.
Press the keys −2 5 0 0 . This changes −10
to −2500.
Press the ENTER key. The blinking cursor drops
to the 10 after Ymax.
Press the keys 1 0 0 0 . This changes 10 to
1000.
Press the ENTER key. The blinking cursor drops
to the 1 after Yscl. We need to set the tick (scale)
marks on the y-axis to a much larger number, say 500.
So press the keys 5 0 0 ENTER .
You should see the following.
35

Now press the GRAPH key. You now see the fol-
lowing (Screen 1.2 in the text).
Locating the Graph
Graph y=x4−2x3+x2+ 10 in the standard viewing
window. If not satisfactory, find another window.
Solution. When we press the Y= key we should still
have y=x4−12x3+x2−2 there. Move the cursor over
the 1 in 12 and press the DEL key. This deletes the
1. (To insert the 1 back, place the blinking cursor over
the 2 in front of x3, and press the keys 2nd INS 1 .)
Now move the cursor to the last minus sign and just
write +10 over the −2. If we graph in the standard
viewing window we see nothing! Press the TRACE
key and obtain the following screen.
We notice that the point (0,10) is on the graph. We
might think to center the viewing window about this
point. So take a screen with dimensions [−10,10] by
[0,20] and obtain the following (Screen 1.3 in the text).
Using the ZOOM Feature
ZOOM about the point (0,10) in the last screen.
Solution. To ZOOM in about (0,10), first press the
ZOOM key and then the 2 key. Place the cursor at
(0,10), press ENTER , and you will obtain the follow-
ing (Screen 1.4 in the text).
Deleting a Function
Remove Y1=x4−2x3+x2+ 10.
Solution. Press the Y= key, place the cursor after
Y1=, and then press the CLEAR key.
The Step Size
Set window dimensions so that TRACE will move in
increments of 0.2 for any graph. Also 0.1.
Solution. There are 95 column of pixels of light on the
TI-82. Thus, there are 94 increments across the screen.
Each increment is then of size (Xmax −Xmin)/94. If
we set Xmax = 9.4 and Xmin = −9.4, each increment
will be of size [9.4−(−9.4)]/94 = 0.2. Try it. If we set
Xmax = 4.7 and Xmin = −4.7, then each increment
will be of size [4.7−(−4.7)]/94 = 0.1. Try it.
36

Section 1.1 in Text
•Piecewise Defined Functions and Dot Mode
•Split Screen
•The Greatest Integer Function
We encounter some piecewise defined functions in
this section. The following is Exploration 3.
Graphing Piecewise Defined Functions
Graph y=f(x) on your grapher where
f(x) = 


x2+ 1, x ≤0
3,0<x<3
x−3, x ≥3
This is Exploration 3 in the text.
Solution. You will obtain a better graph if you are in
DOT mode, rather than CONNECTED mode. To get
into DOT mode press the key MODE . You see the
following.
Using the cursor keys move the blinking cursor to
Dot on the fifth line and press the key ENTER .
To enter y= (x < 0) you need to know how to enter
the symbol “<.” To do this press the keys 2nd TEST
and you see the following.
Now press the 5 key. The other inequality symbols
are entered in a similar manner. Thus, to graph the
given function press the Y= key and type the follow-
ing:
( X,T,θ∧2 + 1 ) ( X,T,θ2nd TEST
5 0 ) + 3 ( X,T,θ2nd TEST 3 0 )
( X,T,θ2nd TEST 5 3 ) + ( X,T,θ−
3 ) ( X,T,θ2nd TEST 4 3 )
You should have the following.
Set the window dimensions to [−5,10] by [0,10], graph,
and you will obtain the correct graph.
Split Screen
Reproduce the last graph on a split screen with the Y=
function also displayed.
Solution. To obtain a split screen first press the key
MODE . Then place the blinking cursor over Split
in the last line and press the key ENTER . Graph
and you will see a split screen with the graph in the
top screen. Press the key Y= and you will see the
function Y1.
37

The Greatest Integer Function
Find the greatest integer function, called the int func-
tion on the TI-82. Use this function to write C(x) in
the Warm-Up Exercise 4 of Section 1.1 in the text.
int(x) is the greatest integer less than or equal to x.
Thus, int(5.03) = 5 and int(7.8) = 7. So C(x) =
0.07int(x) + 0.10. This is easier to graph than using
the more complicated piecewise input.
Solution. Obtain the int function by first pressing the
MATH key. Now select the NUM menu by pressing
the key.
Now select the int function by pressing the key 4 .
Section 1.2 in Text
•The root Operation
•The intersect Operation
The root Operation
Find where the function y1 = 0.95x−300,000 is zero.
Refer to Example 2 and Exploration 1 in the text.
Solution. First graph y1 using a screen with dimen-
sions [0,400000] by [−300000,100000]. Now press the
keys 2nd CALC and you see the following.
Now press the key 2 and you obtain the following.
You are asked for a Lower bound. This means place the
cursor to the left of the zero. Press the key ENTER
and obtain the following.
38

You are asked for an Upper bound. This means place
the cursor to the right of the zero. Press the key
ENTER and obtain the following.
You are asked for a guess. Move the cursor near the zero
and press the key ENTER and obtain the following
(Screen 1.6 in the text).
The intersect Operation
Find where the functions y1= 3.84x+ 300,000 and
y2= 4.8xintersect.
Solution. Graph these two functions using a window
with dimensions [0,400000] by [0,2000000] and obtain
the following.
Now press the keys 2nd CALC 5 and obtain the
following.
The blinking cursor is on one of the curves and you are
asked for First curve. Press the key ENTER and you
obtain the following.
The blinking cursor is on the other curve and you are
asked for Second curve. Press the key ENTER and
you obtain the following.
39

Now you are asked for a guess. Move the blinking cur-
sor to a point near the intersection and press the key
ENTER and you obtain the following.
Section 2.1 in Text
Using STAT and LinReg
Use the STAT the LinReg operations to find the equa-
tion of the best-fitting straight line through the three
points (0,0), (1,2), and (2,1). (This is the Warm Up
Exercise in Section 2.1)
Solution. First press the keys 2nd STAT PLOT
and obtain the following.
Now press the key 1 and obtain the following.
Place the blinking cursor over On and press ENTER .
Make sure the other items are highlighted as shown in
the previous screen. Now set the dimensions of the
window to [−1,3] by [−1,3]. It is best to press the
Y= key and delete any functions. Now press the
keys STAT ENTER and obtain the following.
Input the data setting L1(1) = 0, L1(2) = 1, L1(3) = 2
in the first column and L2(1) = 0, L2(2) = 2, and
L2(3) = 1 in the second column. You obtain the fol-
lowing.
Now press the keys STAT 5 and you should see
the following.
40

Now press the key ENTER and obtain
Shown is the equation we are seeking. To graph press
the key Y= VARS 5 7 . This places the
equation into Y1. Now press GRAPH and obtain the
following, which is the screen found in the text).
Clearing Data in STAT
To clear data in STAT press the keys STAT 4 L1
, L2 ENTER .
Sections 2.2-2.7 in Text
To obtain the best fitting quadratic follow all the in-
structions found in STAT above except select QuadReg.
That is, press the keys STAT 6 .
Section 3.1 in Text
•The TABLE Operation
•The value Operation
•Using TRACE to Go Beyond Viewing Window
•Executing a Program
•The Program EVAL
We can easily create the tables found in Section 3.1
in the text by using the TABLE operation on the TI-82.
The TABLE Operation
Use the TABLE operation to make a table of values for
the function y= 3x+ 1 found in Table 3.1 in the text.
Solution. Press the keys 2nd TblSet to obtain the
table setup menu. You should see the following.
Move the blinking cursor to the Ask on the next to the
last line and press the ENTER key. Ask will then be
highlighted. Make sure that Auto is highlighted on the
last line. Now, using the Y= key, enter the function
Y1= 3x+ 1. To obtain the table press the keys 2nd
TABLE . You see a column headed by xand a column
headed by Y1.
41

At the bottom you see x=. Type 2.1 and press the
ENTER key. Then 2.1 appears on the first row under
the x-column and 7.3 appears under the Y1-column.
Repeat by inputting 2.01, 2.001, 1.999, 1.99, and 1.9.
You should see the following.
The other tables can be created in a similar manner.
Notice that Y1is calculated with limited accuracy
in the TABLE operation. Higher accuracy can be ob-
tained using the evaluate operation given next, or us-
ing the EVAL program given in the next section. If
accuracy is required, we recommend using the EVAL
program, rather than the evaluate operation, because
the evaluation operation has some inconveniences.
The value Operation
Use the value operation to evaluate the function y=xx
at some positive values close to zero. (Refer to Example
5 in the text).
Solution. Input Y1=xx, use a window of dimen-
sions [0,2] by [0,4], and graph. This gives the following
(Screen 3.3 in the text).
Now select the value operation by pressing the keys
2nd CALC 1 . At the bottom of the screen appears
Eval x=
Input .1 and press the ENTER key. You then see
x=.1Y=.79432823
Now press the keys 2nd CALC 1 again and input
.01 and press the ENTER key. You then see
x=.01 Y=.95499259
Continue in this manner to complete the table.
To use the evaluate operation we must select an xfor
evaluation in the interval [Xmin,Xmax]. Sometimes,
after some experimentation, we find that we need to
evaluate xoutside this interval, requiring us to reset
Xmin or Xmax. This inconvenience can be avoided by
using the program EVAL mentioned in the next section.
Using TRACE to Go Beyond Viewing Window
Graph y=xxusing a window with dimensions [0,2] by
[0,4] and then use TRACE to go beyond the window.
Solution. Input the function, graph, and press the
TRACE key. Notice that you can use the cursor to
continue beyond x= 2. The graph will change as the x
values move beyond x= 2. You can also reset Xmax to
be, say, 4 and graph again and use the TRACE feature
again. This process can be repeated as many times as
you wish.
Executing a Program
To execute a program press PRGM and you will see
a list of your programs.
See pages 59 and 60 on the two ways to place pro-
grams into your calculator.) Move the cursor next to
the desired program and press the key ENTER and
the program is executed.
The TABLE operation to evaluate functions is ex-
tremely convenient, but lacks accuracy. The evaluation
operation gives high accuracy but has some inconve-
niences. We now give the EVAL program, which gives
accuracy without the inconveniences of the evaluation
operation.
42

Using the EVAL Program
Use the EVAL program to create a table of values for
the function f(x) = xx. (Refer to the table in Exam-
ple 5 in the text.)
Solution. We assume that you already have placed
the EVAL program into your calculator. Input Y1=xx
and execute the EVAL program. The program requests
x=. Input 0.1 and press the ENTER key. The pro-
gram returns .7943282347 and requests another x=.
Continue in this manner to complete the table.
Section 3.2 in Text
•Drawing a Tangent Line on a Curve
•The Program SEC
Draw a Tangent Line Using DRAW
Graph y= 16x2on a screen with dimensions [0,1.88] by
[0,64]. Draw the tangent line to the graph of the curve
when x= 1 and note the slope. What is the instanta-
neous velocity when x= 1? Refer to Exploration 2 in
the text.
Solution. We can draw the tangent line from the
DRAW menu on the TI-82. Enter Y1= 16x2into the
function menu. Set the dimensions of the window to
[0,1.88] by [0,64] and graph. Now press the keys 2nd
DRAW 5 . Move the cursor so that X= 1. Then
press ENTER . A tangent line is drawn at x= 1. The
slope is shown on the bottom as dy/dx = 32.
The Program SEC
The program SEC given in the Program section of this
manual graphs y=f(x), the tangent line at the indi-
cated point, and a sequence of secant lines for values of
happroaching zero and thus demonstrates Figure 3.34
of the text. Use this program on the function y=x2
at x= 1 with a window of dimensions [.8,2] by [0,4].
Solution. Input Y1=x2. Use a window with dimen-
sions [.8,2] by [0,4]. Execute the SEC program.
The program requests a value for x. Press the keys
1 ENTER . The program requests a value for N.
(This will be the number of secant lines drawn.) Press
the keys 1 0 ENTER . Graphs of y=x2and the
tangent line at x= 1 are drawn.
Now each time you press the ENTER key a secant
line will be drawn with hsmaller. You see the secant
lines getting closer and closer to the tangent line.
43

Section 3.3 in Text
•nDeriv
•NDERG
nDeriv
Find nDeriv p(2), the numerical derivative, where
p(x) = 1/x. Refer to Examples 2 and 3 and Explo-
ratio 1 in the text.
Solution. To find nDeriv p(2), use the built-in nDeriv
function on your TI-82. Press MATH 8 and you will
see nDeriv( on the screen. Now press the keys
1÷X,T,θ,X,T,θ,2 ) ENTER
and obtain
Your answer is −.2500000625.
NDERG
nDERG draws a graph of the numerical derivative. Use
NDERG for f(x) = x1/3.
Solution. No programming is required for NDERG
on the TI-82, just use the built-in nDeriv program. Set
Y1= nDeriv(X(1/3),X,X) and graph using a window
with dimensions [0,1] by [0,10].
Section 5.2 in Text
•nDer2
•NDER2G
nDer2
nDer2 gives a numerical second derivative. Find
nDer2f(1) if f(x) = x4.
Solution. To find nDer2f(1), input Y1=x4and ex-
ecute the program NDER2. The program asks for c.
Input 1 and press the ENTER key. The program re-
turns 12.000008.
NDER2G
NDER2G graphs a numerical second derivative. With
f(x) = x4use NDER2G to obtain a graph of y=
nDerf(x).
Solution. Input Y1=x4and use a window with
dimensions [0,1] by [0,12]. Execute the program
NDER2G and you obtain the following
Section 6.3 in Text
•The Program RECT
•The Program SUM
The Program RECT
The program RECT draws the graph of a function,
the rectangles associated with the left- and right-hand
sums, and the values of these sums. Obtain the left-
and right-hand sums for y=x2and n= 5. (You will
obtain Figures 6.7 in the text.)
Solution. First input Y1=x2and then set the di-
mensions of the window to [0,1] by [0,1]. Execute the
program RECT. A value of nis requested. Press 5
44

and ENTER . A graph of y=x2appears on the in-
terval [0,1] together with the rectangles associated with
the right-hand sums.
Press ENTER and the value of the right-hand sum
is given. Press ENTER and a graph of y=x2ap-
pears on the interval [0,1] together with the rectangles
associated with the left-hand sums.
Press ENTER and the value of the left- and righthand
sums is given.
Press ENTER again and another nis requested, and
so on.
The Program SUM
The program SUM finds the left- and right-hand sums,
but does not draw the rectangles as done in the program
RECT. Use the program SUM to find the left and right-
hand sums for Z1
0
x2dx when n= 50.
Solution. First input Y1=x2. Execute the program
SUM. A lower limit (of integration) is requested. Press
the 0 key and the ENTER key. An upper limit (of
integration) is requested. Press the 1 key and the
ENTER key. Now Nis requested. Press the keys 5
0 ENTER . Wait several seconds and you then see
Pressing ENTER will start the program again.
Section 6.5 in Text
•NINTG
•NINT
The Program NINTG
The program NINTG graphs a numerical approxima-
tion to the integral Zx
a
f(x)dx on a prescribed interval.
Use NINTG to graph an approximation to Zx
0
t2dt.
Solution. First set the dimensions of the view-
ing window to [−2,2] by [−3,3]. On the TI-
82 NINTG is a built-in function. Use the func-
tion fnInt(. This function requires you to fill in
fnInt(expression,variable,lower,upper). So press the
keys
Y= MATH 9 X,T,θ∧2,X,T,θ,0
,X,T,θ)
You will see the following.
45

Press the GRAPH key and the graph will appear.
The Program NINT
NINT finds a numerical approximation to Zb
a
f(x)dx.
Use NINT to find an approximation to Z2
1
4x3dx.
(This is Exercise 1 in Section 6.5 of the text.)
Solution. We wish to evaluate Z2
1
4x3dx. Press the
following keys
MATH 9 4 X,T,θ∧3,X,T,θ,1,
2 ) ENTER
You will see
The answer is 15.
Section 7.3 in Text
•The Program TRAP
•The Program SIMP
The Programs TRAP and SIMP
Use the programs TRAP and SIMP to find
the trapezoidal and Simpson approximation for
Z2
0p4−x2dx =πfor n= 100, 200, 400. Compare
the results. (Refer to Example 4 in the text.)
Solution. First input Y1=√4−x2. Execute the pro-
gram TRAP. A lower limit (of integration) is requested.
Press the keys 0 ENTER . An upper limit (of inte-
gration) is requested. Press the keys 2 ENTER . The
value of Nis requested. Press 1 0 0 . Wait about
15 seconds and you see
T=
3.140417032
Done
Press ENTER to start the program again.
The program SIMP works in a similar manner.
Section 8.6 in Text
The Program VOL
Find the Riemann sums for the functions f(x, y) =
6xy2using the domain consisting of the rectangle R=
{(x, y) : 0 ≤x≤2,0≤y≤1}, for n= 4, 8, and 100
by taking the points (xi, yi) at the lower left and then
the top right of each subrectangle. (Refer to the table
in Section 8.6.)
Solution. Execute the program VOL. The program is
set to use the domain R={(x, y) : A≤x≤B, C ≤
y≤D}. Therefore, when requested set A= 0, B= 2,
C= 0, and D= 1. The program then request N.
Select an N. The calculator will need to run many
minutes for N= 100.
Section 9.2 in Text
Reproduce Screen 9.1 in the Text
Solution. Set the mode functions to radian, para-
metric, and simultaneous. Set the range variables to
Tmin = 0, Tmax = 6.28, Tstep = 0.01. Use a view-
ing window with dimensions [−1,6.26] by [−2,2]. Set
x1= cos t,y1= sin t(giving the unit circle), and
x2=t,y2= sin t, and graph. As the graph slowly
46

unfolds note carefully the y-coordinates. You obtain
Screen 9.1. Repeat, if needed, and use Tstep =.03 to
speed up the graphing.
Section 10.3 in Text
Reproduce Screen 10.12
Solution. Set your calculator on parametric and dot
mode. Set x1=t,y1= (2t+ 1)/t. For the range
variables set Tmin =0,Tmax = 100, Tstep = 1, xmin = 0,
xmax = 100, ymin = 1, ymax = 3. Also set x2=t,
y2= 2. Graph, obtaining Screen 10.12.
Section 10.4 in Text
Graphing a Sequence
By graphing the partial sums on your graphing calcu-
lator give support to the conjecture that the series
1 + (0.5) + (0.5)2+ (0.5)3+···
converges to the sum 2. (Refer to Exploration 1 in the
text.)
Solution. Set the mode of your calculator to param-
eter and dot. Do this by pressing the MODE key.
Move the blinking cursor so that the word Par is high-
lighted and then press the ENTER key. Now move
the blinking cursor so that the word Dot is highlighted
and then press the ENTER key. Now press the Y=
key and set x1T=T,y1T= sum seq((.5)J,J,0,T,1).
The keys 2nd LIST 5 gives sum, while the keys
2nd LIST 5 gives seq. We have
sum seq(aJ, J, I, N, K) =
N
X
J=I
aJ
The Kin the expression
sum seq(aJ, J, I, N, K) means that Jis incremented by
Kin the sum.
Now press the WINDOW key and set Tmin= 0,
Tmax= 20, Tstep= 1, Xmin= 0, Xmax= 20, Ymin= 0,
Ymax= 3.
Graph. Notice that the sequence of partial sums ap-
pears to be converging to 2. Using the TRACE feature,
determine the values of the partial sums, and notice
that these numbers are approaching 2.
Section 11.1 in the text
•The Program RAND
•The Program RAND2
•Histograms
•Statistic
The Program RAND
The program RAND simulates a fair coin being flipped,
and graphs the relative frequency every Pflips. Exe-
cute this program. (Refer to Exploration 1 in the text.)
Solution. Set the dimensions of the viewing window
to Xmin= 0, Xmax= 94, Ymin= 0.4, Ymax= 0.6, and
Yscl= 0.1. Execute the program RAND. The program
requests P. Use P= 100 or perhaps P= 50 for faster
action. The relative frequencies are graphed after every
P-flips. The following is one such graph, but every one
should be different.
If you wish, you can also keep track of the cumulative
number of “heads” and the numerical relative frequency
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by executing RAND2. Follow the same procedure as for
RAND. The program RAND2 pauses after displaying
the number of heads and the relative frequency and
also after displaying the graph. You must press the key
ENTER for the program to continue.
Histograms
Reproduce the histogram in the following figure (Figure
11.1 in the text).
Solution. First set Xmin= 2, Xmax= 13, Xscl= 1,
Ymin= 0, and Ymax= 7.
Press the keys STAT 1 and a screen will appear
with 6 columns headed by
L1,...,L6, although you will only see the first three
columns.
A dark rectangle appears under L1. Input the data 2,
3, ... , 12 in the first column by pressing 2 ENTER
3 ENTER and so on. When this is completed, press
the key and the dark rectangle will move to under
the second column. In the same way as before enter
the frequency data (NOT the probabilities) 1, 2, 3, 4,
5, 6, 5, 4, 3, 2, 1.
Now press the keys 2nd STATPLOT and then the
key 1 . Place the blinking cursor over the word On in
the first line and press the key ENTER . Now move
the blinking cursor to the histogram in the second line
(the last figure) and press the key ENTER . Move
the cursor over L1 in the third line and press the key
ENTER . Move the cursor over L2 in the last line and
press the key ENTER . You should have the following.
Go back to STAT PLOTS and make sure that Plot
2 and Plot 3 are set to Off. Finally press the key
TRACE and the histogram will appear.
Statistics
Find the mean and standard deviation for the data
(1,10), (2,15), (3,16), (4,7), (5,2). (Refer to Exam-
ple 6 in the text.)
Solution. Press keys STAT 1 and enter the data 1,
2, 3, 4, 5 under the first column. Then enter the data
10, 15, 16, 7, 2 under the second column. You should
see.
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Now press the keys STAT 1 L1 , L2
ENTER and you see
The first number you see is the mean ¯x= 2.52.
Further down the list is the standard deviation σx=
1.081480467.
Section 11.2 in Text
Line Plots and Histograms
Reproduce Figure 11.7 in the text.
Solution. Set the dimensions of the viewing window
to [64,76] by [0,1500], with Xscl= 1. After pressing
the keys STAT 1, enter the data 64.5, 65.5, 66.5,
67.5, ... in the first column, and enter the frequency
(NOT probability) data 385, 439, 794, 1070, ... in the
second column. Press the keys 2nd STATPLOT
1 . Place the blinking cursor over On in the first line
and press ENTER . Place the blinking cursor over
the histogram (last item) in the second line and press
ENTER . Place the blinking cursor over L1 in the
third line and press ENTER . Place the blinking cur-
sor over L2 in the fourth line and press ENTER . Press
the key TRACE and you obtain Figure 11.6 in the
text.
Now press the keys 2nd STATPLOT 2 . Place
the blinking cursor over On in the first line and press
ENTER . Place the blinking cursor over the line plot
(second item) in the second line and press ENTER .
Place the blinking cursor over L1 in the third line and
press ENTER . Place the blinking cursor over L2 in
the fourth line and press ENTER . Place the blink-
ing cursor over the last item in the fifth line and press
ENTER . Press the key TRACE and you obtain Fig-
ure 11.7 in the text.
Section 12.3 in Text
•The Program DFIELD
•The Program EULER
•The Program EULERG
The Program DFIELD
The program DFIELD draws a direction field for the
differential equation dy
dx =f(x, y). Find the direction
field for dy
dx = 4 −y. (Refer to Example 1 in the text.)
Solution. Input Y1= 4 −Y. Use a window with
dimensions [−2,2] by [0,6]. Execute the program
DFIELD. A sketch of the direction field will appear.
The Program EULER
The program EULER executes Euler’s method . Exe-
cute the program EULER for the differential equation
dy
dx =−y+x+ 1 with initial condition y(0) = 1. (Refer
to Example 3 in the text.)
Solution. We want to use the program EULER to
approximate the solution to the initial value problem
dy
dx =−y+x+ 1, y(0) = 1,on the interval [0,1] by
taking h= 0.2.
Input Y1=−Y+X+ 1. Execute the program EU-
LER. The value x0, or x0is requested. Press the keys
0 ENTER . The value y0, or y0is requested. Press
the keys 1 ENTER . The value His requested. Press
the keys . 2 ENTER . The value Bis requested.
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