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Math 1314 with Integrated TechnologyMath 1314 with Integrated TechnologyMath 1314 with Integrated TechnologyMath 1314 with Integrated Technology

Calculator GuideCalculator GuideCalculator GuideCalculator Guide

This guide is an introduction to the use of a graphing calculator. It is not a comprehensive guide. The tools illustrated in this guide are limited to the tools needed for Math 1314 with Integrated Technology. The calculator references are to a Texas Instruments TI-84 Plus graphing calculator. The keystroke sequences should be similar on other TI products, although some calculator features (such as key colors) may differ. Students using graphing calculators other than TI calculators are advised to consult the user guide for their calculator brand to make any necessary modifications to the keystrokes described in this guide. Most user guides are available online.

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Getting OrienGetting OrienGetting OrienGetting Orientedtedtedted

The face of the calculator has five distinct parts. Display Screen

The display screen will show all computations and functions that you enter and will display graphs of functions.

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Below the display screen are five buttons which access 10 different menus, each of which will be discussed in detail in this guide. These gray keys are labeled as follows: Y=, WINDOW, ZOOM, TRACE, and GRAPH. There are blue titles above these labels: STATPLOT, TBLSET, FORMAT, CALC, and TABLE. Many of the functions needed for this course will be accessed using these five keys. Menu Keys

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Below these five keys and to the right are arrow keys that will move the cursor up, down, left or right. When looking at graphs, the left/right arrow keys will move the cursor to the left/right on the graph, and the up/down arrow keys will toggle between functions, if more than one function is graphed. Arrow Keys

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The black keys access the most commonly used functions and menus that will be needed in the course. Commonly Used Functions

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The gray and white keys towards the bottom and right of the key pad give the digits from 0 – 9, decimal point, (-) and the usual arithmetic operations. At the bottom right is the ENTER key that you will use to tell the calculator to perform the operations or functions that you have displayed on the screen.

Calculator Keypad and Operations ENTER

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There are two other important keys: The 2nd key and the ALPHA key. Notice that each key on the calculator has blue writing above it (on calculators other than the TI-84 Plus, this may be a different color), and most keys have a green letter or punctuation mark above it. When the function given in blue is needed, press the 2nd key, then the button that has the required function above it. So, for example, to access the square root function, press 2nd 2x . Similarly, to access letters of the alphabet, press ALPHA and the key that has the desired letter above it. 2nd Key ALPHA Key

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Calculator BasicsCalculator BasicsCalculator BasicsCalculator Basics

Note: throughout this guide, the comma key will be noted as Note: throughout this guide, the comma key will be noted as Note: throughout this guide, the comma key will be noted as Note: throughout this guide, the comma key will be noted as

(comma) and the X,T,and the X,T,and the X,T,and the X,T,θ ,n key will ,n key will ,n key will ,n key will be noted as X.be noted as X.be noted as X.be noted as X.

On/Off:

To turn the calculator on, press the ON button (lower left corner). To turn the calculator off, press 2nd ON. Note that the calculator will automatically turn off after a period of disuse.

Making the Screen Lighter/Darker:

To make the screen darker, press 2nd ↑. It may take several applications of 2nd ↑ to achieve the desired result. To make the screen lighter, press 2nd ↓. With weaker batteries, it may be necessary to make the screen darker. With new batteries, it may be necessary to make the screen lighter.

Clear:

To clear the entire main screen, press CLEAR . It may be necessary to press CLEAR CLEAR to get a blank screen.

Previous Entry:

To pull a previous entry to the current line of the main screen, press 2nd ENTER. It may be necessary to do this several times to get to the desired previous entry.

Last Entry: To use the last value on the main screen in future computations, press

2nd (-). The display will show ANS after pressing 2nd (-).

Insert/Delete:

To delete a character, use the arrow keys to put the cursor on the specific character you wish to delete. Then press DEL. To insert a character or characters, use the arrow keys to put the cursor on the character to the right of the desired location for the new character(s). Then press 2nd DEL and enter the new character(s).

Alpha Lock: To lock the calculator for alphabet entry, press 2nd ALPHA .

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Moving Back to the Main Screen: To return to the main screen from any other screen, press 2nd Mode. Error Prompts

Occasionally, the calculator will not understand a command and will return an ERROR prompt. There is usually a description of the error, and sometimes there is an option to GOTO the error so that it can be fixed. Often the correction is easy, such as fixing an erroneous error in the WINDOW menu, or removing an accidental user request to divide a quantity by 0.

Resetting the Calculator:

To reset the calculator, press 2nd +. Press 7. Press 1. The following screen cautions that all data and all programs will be erased. To reset, press 2. To leave the reset menu without resetting, press 1. Note: Resetting is a last resort option!! Do NOT reset the calculator until all other options have been exhausted. If there are no other options, programs stored in the calculator can be archived. To do this press 2nd + and select 5. Press ENTER. Now mark the program to archive. Press PRGM and cursor down to the program to archive. Press ENTER. An asterisk will appear next to the program name. When the calculator is reset, the program will not be erased. To unarchive a program, press 2nd + and select 6. Now mark the program to archive. Press PRGM and and cursor down to the program to unarchive. Press ENTER. The asterisk will disappear next to the program name. A program may not be run until it has been unarchived.

Mode:

Press MODE to view the various entries on this menu. For this course, each line should have the left-most entry highlighted. Occasionally, it will be necessary to change the fifth line from CONNECTED to DOT.

Variable Key:

The calculator has one key for use in entering variables, the X key. This key is located below the MODE key. No matter what variable is stated in the problem, the calculator entry will be as x.

Store/Recall:

To store a number as a variable, press (number), STO>, ALPHA, (letter), ENTER.

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To recall a number, press 2nd STO>, ALPHA, (letter) ENTER .

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Basic ComputationBasic ComputationBasic ComputationBasic Computation To perform basic computations, enter the expression using the 10 digits, (-) key, decimal point and the operation symbols on the lower right part of the calculator. To enter π , press 2nd ^ . To enter e (the exponential function), press 2nd ln or 2nd ÷. (The first method automatically opens parentheses so use it to enter ke ; use the second method to enter just 1e ). To make a value negative, use (-) rather than the subtraction symbol. To raise a number to the second power, enter the number and press 2x then ENTER. To raise a number to any other power, enter the number and press ̂ (power) then ENTER, so to raise a number to the fourth power, enter the number and press ̂4 and then ENTER. To find the square root of a number, enter the function first. Press 2nd 2x , then enter the number and close parentheses. Then press ENTER. To find any other root of a number, start by

rewriting the expression as ( )1

ka . Then enter the number and press ^(1/k) , followed by ENTER.

For a more complicated expression, use parentheses liberally, taking care to close any parenthese that you open. To evaluate ( )ka bx+ , enter the expression as ( a + b*x ) ^ k followed by

ENTER. To evaluate ab cd

ef gh

++

, put parentheses around the numerator and the denominator when

entering the numbers: ( a * b + c * d) / (e * f + g * h), then press ENTER. Some functions, such as the square root function and the exponential function, automatically open parentheses. Take care to close these parentheses after entering the argument of the function.

Example: Evaluate: ( )2 22 5 3 2 4⋅ − − .

Enter this exactly as written: 2 * 5 2 – 3 ( 2 – 4 2 ) ENTER

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Example: Evaluate: 2

2

2 3 4 11

3 4 2 3

− −+ ⋅ −

Example: Evaluate: ( )2

3

4 4 3 2.5

7 2e

π − +−

Example: Evaluate: 22 5 3 7

4 7 12

− ⋅ + ⋅− ⋅ +

Note the use of the (-) key for the leading values in the numerator and the denominator rather than the subtraction key.

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Example: Use the store and recall features to evaluate 24AB C

D

− if

9, 7, 12, 2A B C D= = = = − . Press 9, STO>, ALPHA, MATH, ENTER. Press 7, STO>, ALPHA, APPS, ENTER. Press 12, STO>, ALPHA, PRGM, ENTER. Press -2, STO>, ALPHA, 1x− , ENTER

Now press ( 2nd 2x , ALPHA, MATH, * , ALPHA, APPS, ), – ,4, *, ALPHA , PRGM, 2x ,) / , ALPHA

1x−

Press ENTER.

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Entering Functions inEntering Functions inEntering Functions inEntering Functions in Y=Y=Y=Y= To enter a function in the Y= menu, press Y= and select a line for the equation. Press CLEAR to delete an equation that is no longer needed. Use Y1 for most problems. Use the numbers on the keypad to enter coefficients and the X key to enter variables. Even if the problem uses a different variable, use the X key when using the calculator. For fractional

coefficients, use parentheses, so enter (1/2) for 1

2.

Use the operation symbols for addition, subtraction, multiplication and division that appear in the problem. Use (-) for a leading negative rather than subtraction. To raise a variable to the second power, type X and then press 2x . To raise a variable to another power, such as the kth power, type X and then press ^k. For a square root, press 2nd 2x . For a different root, such as the kth root, press ^(1/k). Note that the when working with radicals, the calculator automatically opens parentheses. Take care to close parentheses when needed.

Example: Enter 3 22 3( ) 5 9

3 4f x x x x= − − + on the Y1 line on the Y= menu.

Example: Enter 7

( ) 43

xf x x

x

+= − + −−

on the Y2 line on the Y= menu.

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Example: Enter 3 2( ) 2 5 4f x x x= − + on the Y3 line on the Y= menu.

First rewrite the function as ( )1

2 3( ) 2 5 4f x x x= − + .

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TablesTablesTablesTables The calculator has two different formats for viewing tables of values. In the first format, the user sets the starting point and the increment value value for x, and the calculator automatically generates a table of values. To set up this type of table, press 2nd WINDOW. This is the resulting screen.

In setting up the table, the user can select the start point (TblStart ) and the increment ∆ Tbl. Use the up and down arrow keys until the character to the right of TblStart= is highlighted. Enter the starting number. Arrow down to the next line and enter the desired increment. An increment of one is often useful, but at times a larger or smaller increment may work better. To finish TABLE SETUP for an automatic table of values, make sure the word Auto is highlighted on both the Indpnt: and Depend: lines. To view the table of values, press 2nd GRAPH. Example: Using the functions that were entered in the Entering Functions in Y=Entering Functions in Y=Entering Functions in Y=Entering Functions in Y= section, create a table of values starting at x = 0 with an increment of 1. This is the TABLE SETUP:

Here is the table of values. Note that there are several columns. The values for the function entered in Y1 are recorded in the Y1 column; the values for the function entered in Y2 are recorded in the Y2 column, and so on.

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To view the values for the function entered in Y3, use the right arrow and cursor over until the Y3 column comes into view.

Notice that the table can display a limited number of decimal places. To view an entry to more decimal places, move the cursor to the desired value and view the number to 12 or 13 significant digits at the bottom of the screen. Note also that the calculator will display ERROR if the function is not defined at a listed value for x. In the screen shot displayed above, the function entered in Y2 is not defined when x = 3. The table currently only displays the y values associated with x values from 0 to 6. However, by scrolling up and down the list of x values, other y values will be displayed in the table. In the table below, y values associated with x values from 26 to 32 are displayed.

The other table format allows the user to select the specific values for x that are to be used in the table. Go to TBLSET and arrow down to the word AUTO on the Indpnt: line. Right arrow over so that the work ASK is highlighted.

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Press ENTER. Now press 2nd GRAPH to view the table. The table that was there before has disappeared. The calculator is waiting for x values to be entered.

Now, enter the desired values for x in the X column, and corresponding values for Y1, Y2 and Y3 will appear in those columns. Example: Use the functions that were entered in the Entering Functions in Y=Entering Functions in Y=Entering Functions in Y=Entering Functions in Y= section, create a table of values that gives only the function values when x = 1, x = -10 and x = 12.5. Put the cursor in the top of the X column. Press 1, ENTER, -10 ENTER, 12.5 ENTER. This is the result.

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Graphing Functions and Setting the WindowGraphing Functions and Setting the WindowGraphing Functions and Setting the WindowGraphing Functions and Setting the Window Once a function is entered in Y=, graph the function by pressing GRAPH. If more than one equation is entered in Y=, the calculator will graph all of them, unless the user turns off the functions that should not be graphed. To disable or “turn off” an equation, move the cursor so that it is on the equal sign of the graph to turn off and press ENTER. The equal sign will not be highlighted. To enable the function again, move the cursor onto the equal sign and press ENTER. The equal sign will be highlighted again, and will show up when graphing. Of the three functions listed on the screen shot below, only Y1 will appear on the graph, since the other two functions have been disabled.

Press GRAPH to view the graph of the function.

Without modification, the calculator will use a standard 10 unit by 10 unit viewing window. This means that the minimum x value is -10 and the maximum x value is 10, and the minimum y value is -10 and the maximum y value is 10. This is sometimes written as [-10, 10, -10, 10], indicating [Xmin, Xmax, Ymin, Ymax]. The graph above shows that the standard viewing window is sometimes not sufficient to view all of the features of the graph of the function. In this case, the turning point of the graph in the fourth quadrant is not shown in this viewing window. To change the viewing window, press WINDOW and change the appropriate values. To revert to the standard viewing window, press ZOOM, 6.

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Example: Select an appropriate viewing window for the function graphed in this screen shot.

The graph seems to show enough graph in the x direction, and there seems to be enough graph in the positive y direction. Change the y minimum to get a better view of the function.

To find an appropriate Ymin value, use trial and error, or scroll down the TABLE to see the smallest y value in the list between x values of 2 and 7. The table below shows a y value of -36.42, so a Ymin of -50 will display the whole graph and will allow you to see the information at the bottom of the screen without covering the graph.

Here is the resulting graph of the function.

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The Xscl and Yscl values on WINDOW set the scale for each of the axes. These numbers tell the increment for placing tick marks on the axes. On the graph shown, the Xscl is 1 and the Yscal is 10, so every mark along the x axis is 1 unit and every mark along the y axis indicates 10 units. With a scale of 0, the graph will display no tick marks. Example: Graph the function entered at Y2. First, go to Y= to turn off Y1 and turn on Y2.

Press GRAPH.

The WINDOW is still set for the function entered at Y1. To return to the standard viewing window, press ZOOM, 6.

More detail in quadrant 2 is needed, and more of the quadrant 4 portion of the graph should be displayed. Unfortunately, it may not be possible to get a good view of the entire graph on one window. Change the viewing window to [-20, 12, -8, 5] and check the result. Change Xscl to 4 and Yscl to 5.

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Here’s the result:

This does not provide a good view of the graph and shows one of the limitations of relying exclusively on a graphing utility. To view the portion in quadrants 2 and 3 in greater detail, change the window to [-5, 2, -1, 6] with scales of 1 on both axes.

To view the quadrant 4 portion in greater detail, change the window to [0, 12, -12, -4] with scales of 1 in both directions.

The original graph gives an overview of the graph of the function. The last two provide the details of the graph of the function.

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Example: Graph the function entered in Y3. Disable the functions in Y1 and Y2 and enable Y3.

Then press GRAPH.

The modified window was still in place, so the display shows no part of the graph of the function entered in Y3. Revert to the standard viewing window by pressing ZOOM, 6.

The display now shows a good view of the function.

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Graphing a Function on a Limited DomainGraphing a Function on a Limited DomainGraphing a Function on a Limited DomainGraphing a Function on a Limited Domain

To enter a function on a limited domain, put parentheses around the function; then press /; then enter the limitation on the domain and put parentheses around the domain.

Example: Enter the function 3 22 3( ) 5 9

3 4f x x x x= − − + on the interval [-1, 5].

Put parentheses around the function when entering it into Y=. Then press the division key to separate the function from the domain. The interval [-1, 5] is the same as the inequality 1 5.x− ≤ ≤ To enter this, press ( (-)1, 2nd , MATH, 6, X, 2nd, MATH, LOGIC, and, X, 2 nd, MATH, 6, 5).

Press GRAPH to view the graph. Note that it may be necessary to adjust the viewing window to see an appropriate display of the graph of the function. The window used in the screen shot below is [-2, 8, -40, 10].

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Graphing a Piecewise Defined FunctionGraphing a Piecewise Defined FunctionGraphing a Piecewise Defined FunctionGraphing a Piecewise Defined Function

Example: Graph the piecewise defined function:

2

2 3, 1( )

2, 1

x xf x

x x

+ ≤= − >

Enter the function as follows Y=: (2 X + 3) * (X 2nd, MATH, 6, 1) + ( X, 2x , -, 2) * ( X 2nd, MATH, 3, 1).

The graph of the function will look like this:

An alternate means of entering a piecewise defined function on a TI84 calculator is to enter as Y1 the top line of the function (in parentheses) followed by its domain (in parentheses) with no operation symbol separating the parentheses and then to enter as Y2 the second line of the function and its domain.

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Here’s the graph:

Either method generates the same graph. Some older TI calculators can require different operation symbols for graphing piecewise defined functions. If the multiplication symbol between the function and its domain does not work, try the division symbol.

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Format, Format, Format, Format, Trace and ZoomTrace and ZoomTrace and ZoomTrace and Zoom

There are several options for formatting the display screen when graphing functions. To access these, go to the FORMAT menu. Press 2nd ZOOM . The default settings are all of the entries in the left column. To change any of these, arrow down to the line that needs to be changed, arrow over and then press ENTER. Experiment with these to see the changes that can be made to the graphing window.

Graph a function by entering the function in Y=. The graph shown is Y1 as entered in the earlier examples. When TRACE is activated, the calculator places a cursor on the screen and reports the x and y coordinates of the location of the cursor at the bottom of the spring. Also, the function that is graphed is displayed at the top of the screen (if the default ExprOn is activated). To move the cursor around the graph, use the left and right arrow keys.

If two functions are graphed, use the up and down arrow keys to toggle between the two functions.

Example: Graph 1

( ) 3 5 and ( ) 12

f x x g x x= + = − . Use trace to move around the graph.

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The Zoom menu provides some options for viewing graphs.

ZOOM 1 draws a box around an area of a graph for a closer view.

Using the function entered earlier as Y1, draw a box around the low point in quadrant 4 for a closer view. Press ZOOM 1, ENTER. Then move the cursor above and to the left of the region to expand. Press ENTER. Use the right arrow key to move as far to the right as desired, then use the down arrow key to move down as far as desired.

Press ENTER. The calculator will use the indicated box as the window.

ZOOM 2 accomplishes the same thing, but the user does not need to set a box. Using TRACE , move the cursor to the desired location.

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Press enter, and the calculator will zoom in on that location.

ZOOM 3 will keep the cursor on the desired point, and will make the dimensions of the viewing window larger.

ZOOM 4 will change the x coordinates for the window to decimal values. Enter a graph and press TRACE. The x coordinates will show in the bottom left corner of the viewing window as 0.1, 0.2, 0.3, …. This generates a very small window, but the size can be increased by pressing WINDOW and multiplying the Xmin, Xmax, Ymin and Ymax values by the same number. ZOOM 5 generates a square viewing window. Since the display screen is wider that it is high, the standard viewing window is slightly skewed. This view adjusts the dimensions so that each tick on the x axis and each tick on the y axis has the same width. ZOOM 6 reverts to a standard [-10, 10, -10, 10] viewing window. ZOOM 7 selects a viewing window that is appropriate for viewing trigonometric functions. ZOOM FIT theoretically selects a viewing window that fits the function that is entered. Practically speaking, the view of the function is often less than desirable. It is often much more practical to select the viewing window using information from the table and setting the window by pressing WINDOW and entering appropriate values.

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The Calculate (CALC) MenuThe Calculate (CALC) MenuThe Calculate (CALC) MenuThe Calculate (CALC) Menu

The CALC menu provides access to many features that will be regularly used in the class. value returns a single y value when the user enters an x value. zero provides a means for finding the zeros or x intercepts of a function. minimum provides a means for finding the local or relative minima of a function. maximum provides a means for finding the local or relative maxima of a function. intersect provides a means for finding any points of intersection when two or more functions are graphed. value: Enter a function at Y=. Then press GRAPH. Once the graph is on the screen, press 2nd TRACE and select value. Enter the desired value next to X=. The calculator will return the associated y value. Note, the point in question must be displayed on the screen. Otherwise, the calculator will return an error message.

Example: Use the value feature to find (3)f if 3 22 3( ) 5 9

3 4f x x x x= − − + .

Now press 3 ENTER.

Note the outcome when X= 12 is entered.

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The highest value of x displayed is 10, so value will not return a y value for X = 12. zero: Enter a function at Y= and graph it. Press 2nd TRACE and select zero. The calculator asks, “Left Bound?” Use the left arrow key to move the cursor so that it is to the left of the zero. Press ENTER. The calculator asks, “Right Bound?” Use the right arrow key to move the cursor so that it is to the right of the zero. Press ENTER. The calculator asks, “Guess?” Press ENTER. The location of the zero will appear at the bottom of the screen. Repeat the process to find the other zeros (if any) of the function.

Example: Find the leftmost zero of the function 3 22 3( ) 5 9

3 4f x x x x= − − + .

Start by graphing the function and adjusting the viewing window, if needed.

Press 2nd TRACE and select zero.

Move the cursor so that it is to the left of the zero.

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Press ENTER.

Note the arrowhead pointing right near the top of the screen. Now use the right arrow key to move the cursor to the right of the zero. Press ENTER.

Note the arrowhead pointing left. The two arrowheads are setting the interval within which the calculator will search for the zero. Notice that the first point had a negative y value and the second point has a positive y value. This guarantees that there is a zero in between. Now press ENTER to find the zero.

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Example: Now find the zero that is near x = 1. Press 2nd TRACE and select zero. Move the cursor so that it is to the left of the zero. The location of the zero means the left bound needs to have a positive y value.

Press ENTER.

Now move the cursor to that it is to the right of the zero. The point will need to have a negative y value.

Now press ENTER.

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Now press ENTER and read off the zero.

minimum: This feature will find the x and y coordinates of a relative (or local) minimum of the function. A relative or local minimum is a point on the graph that is lower than the points around it. Enter a function at Y= and press GRAPH. Move the cursor so that it is to the left of the low point. Press ENTER. Move the cursor so that it is to the right of the low point. Press ENTER. Then press ENTER and read off the coordinates of the relative (local) minimum. Example: Find the coordinates of the relative (local) minimum of the graph of

3 22 3( ) 5 9

3 4f x x x x= − − + .

Enter the function in the Y= menu and press GRAPH. Adjust the viewing window if needed.

The relative (local) minimum is located in the 4th Quadrant. Press 2nd TRACE and select minimum .

Press ENTER.

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Move the cursor so that it is to the left of the low point.

Press ENTER. Move the cursor so that it is to the right of the low point.

Press ENTER.

Note the two arrowheads at the top that are marking the interval within which the calculator will look for a low point. Press ENTER.

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The x and y coordinates of the relative (local) minimum are displayed at the bottom of the screen. maximum: The process of finding a relative (local) maximum is identical to the process for finding a relative (local) minimum. Example: Example: Find the coordinates of the relative (local) minimum of the graph of

3 22 3( ) 5 9

3 4f x x x x= − − + .

Enter the function in the Y= menu and press GRAPH. Adjust the viewing window, if needed.

Press 2nd TRACE and select maximum.

Press ENTER.

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Move the cursor so that it is to the left of the high point.

Press ENTER.

Move the cursor so that it is to the right of the high point.

Note the two arrowheads at the top of the screen define an interval within which the calculator will look for a maximum value. Press ENTER.

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Press ENTER and read off the coordinates of the relative (local) maximum.

intersect: This feature finds the point(s) at which two functions intersect. Enter the two functions on the Y= menu. Press GRAPH. Adjust the viewing window if needed to see the point(s) of intersection. Press 2nd TRACE and select intersect. The calculator asks “First curve?” and wants confirmation that the cursor is on one of the two functions whose intersection is sought. Look at the top of the screen to see which of the functions is selected. If this is correct, press ENTER. It will be necessary to move the cursor close to the point of intersection before beginning the process. (If there are several equations entered in the calculator, use the up and down arrow keys to toggle between the functions and press ENTER when the correct one is displayed at the top of the screen). Now the calculator asks “Second curve?” and wants confirmation that the cursor is on the other desired function. If this is correct, press ENTER. (If not, toggle through the functions until the correct one is displayed at the top of the screen and press ENTER). Press ENTER one more time. The point of intersection will be displayed at the bottom of the screen. If two functions have more than one intersection point, repeat the process to find the others.

Example: Find the point of intersection if 1

( ) 3 5 and ( ) 12

f x x g x x= + = − .

Enter the two functions on the Y= menu and press GRAPH.

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Press 2nd TRACE and select intersect.

The cursor is on one of the functions (see the display at the top), so press ENTER.

The cursor has move to the other function (see the display at the top), so press ENTER.

The calculator is waiting for verification. Press ENTER.

The point of intersection is displayed at the bottom of the screen.

Example: Find the points of intersection if 3 22 3( ) 5 9

3 4f x x x x= − − + and

1( ) 1

2g x x= − .

Enter the two functions on the Y= menu and press GRAPH. Use a standard viewing window.

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Note that there are three points of intersection. Start by finding the leftmost point of intersection. Press 2nd TRACE and select intersect.

Press ENTER.

Move the cursor so that it is close to the Quadrant 3 point of intersection.

Press ENTER. The cursor will move to the other function. Note the function displayed at the top of the window.

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Press ENTER. Then press ENTER again.

The third quadrant point of intersection is shown at the bottom of the screen. Next, find the intersection point that appears to be near x = 1. Press 2nd TRACE and select intersect.

Move the cursor close to x = 1.

Press ENTER. The cursor will move to the other function. (Note the function displayed at the top of the screen.)

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Press ENTER.

Press ENTER.

The coordinates of the second point of intersection are shown at the bottom of the screen. Now repeat the process to find the third and final point of intersection. Press 2nd TRACE and select intersect.

Move the cursor so that it is near the third intersection point.

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Press ENTER.

Press ENTER.

Press ENTER.

The third point of intersection is given at the bottom of the screen. Note: There are two other features on the CALC menu, dy/dx and intf(x)dx. Other features do the same operations and are easier to use.

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Using Lists and Scatter PlotsUsing Lists and Scatter PlotsUsing Lists and Scatter PlotsUsing Lists and Scatter Plots

Clear any existing lists by pressing STAT, 4, 2nd Y, (comma), 2nd Z, ENTER. Press STAT, EDIT . Put the cursor in the L1 column. Enter the x coordinates of the data. Put the cursor at the top of the L2 column. Enter the y coordinates of the data. There must be the same number of entries in both columns. Example: Enter this information in LISTS. Then use STAT PLOT to graph. x 1 2 4 6 8 9 y 5 3 7 5 9 11 Start by clearing out any existing lists. Press STAT, 4, 2nd 1, (comma), 2nd 2, ENTER.

Press STAT and select 4.

Now state which lists to delete.

Press ENTER.

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Then press STAT EDIT.

The lists are now empty. Put the cursor under the heading L1. Enter the x values from the table of values by typing 1, ENTER, 2, ENTER, 4, ENTER, 6, ENTER, 8, ENTER, 9, ENTER. Now move the cursor so that it is under the heading L2. Enter the y values from the table of values.

Before graphing, verify that there are the same number of entries in L1 and L2. Ordered pairs are graphed using STAT PLOT. To access this, press 2nd Y=.

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All of the plots are turned off in this screen shot. We need to enable the first one. Since the 1: is highlighted, just press ENTER.

First, turn Plot1 on by pressing ENTER while the cursor is on the word ON. The type of graph selected is a scatterplot, which will just graph the ordered pairs. There are other options (line graph, bar graph, box and whiskers graphs) which are not appropriate for the applications in this class. The data for the ordered pairs come from L1 and L2. If the data are stored in different lists, use the 2nd of the keys numbered 3, 4, 5, and/or 6 to select the appropriate list(s). There are three choices for the mark, a square, a plus sign or a dot. The square is usually more visible. Most of the time, the only keystrokes needed to access the STAT PLOT are 2nd Y=, ENTER, ENTER. Now press GRAPH.

This is what happens when multiple functions are active in the Y= screen. Deactivate Y1, Y2 and Y3 on the Y= screen, and only the scatter plot will display.

Notice that one of the points is missing from the graph. The viewing window is too small. Change the viewing window to [-1, 15, -1, 15].

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Much better. It is also possible to activate a STAT PLOT from the Y= screen. Press Y=.

Notice the references to Plot1, Plot2 and Plot3 at the very top of the screen. To turn a STAT PLOT on or off, use the arrow keys to move the cursor to the desired STAT PLOT . Press ENTER. In the screenshot above, Plot1 is activated. In the screenshot below, it is disabled.

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Finding a Linear Regression ModelFinding a Linear Regression ModelFinding a Linear Regression ModelFinding a Linear Regression Model Enter the data from the table of values into L1 and L2. Press STAT, CALC , LinReg, ENTER. The top line gives the form of the equation. The next two lines give the values for a and b. When writing down the linear regression model, round to four decimal places. (The last line ( 2r ) gives a measure of the fit of the model to the data. The closer the number is to 1, the better the model fits the data.) On models of TI graphing calculators other than the TI-84, it may be necessary to press STAT, CALC , LinReg, ENTER, ENTER to find a linear regression. Note, if 2r is not displayed, enable the Diagnostic feature of the calculator by pressing 2nd 0 to access the CATALOG . Then press 1x− to enter the letter D. (The calculator is expecting a letter entry while in the catalog, so ALPHA is not needed before pressing 1x− .) Arrow down so that DiagnosticON is on the display. Press ENTER, ENTER. The display screen should read Done. Example: Using the data given in the example above under Using Lists and Scatter Plots, press STAT, CALC, LinReg, ENTER .

L1 and L2 STAT cursor over to CALC

LinReg ENTER The linear regression model is 0.7692 2.8205y x= + with an 2r value of 0.71.

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Finding Other RegFinding Other RegFinding Other RegFinding Other Regression Modelsression Modelsression Modelsression Models There are several other regression models programmed into the calculator. To find a quadratic regression model, enter the data from the table of values into lists L1 and L2. Then press , STAT, CALC , QuadReg, ENTER.

The top line shows the form of the equation. The values for a, b and c are listed. The value for

2R is a measure of how well the quadratic regression model fits the data. The closer it is to 1, the better the fit. Rounding each of the values to four decimal places, the quadratic regression model is

20.1302 0.5325 4.9467y x x= − + with 2 0.7982R = . Use similar methods to find a cubic regression model:

So 3 20.0278 0.2865 1.1897 3.2801y x x x= − + + with 2 0.8136R = The process is similar for QuartReg (4th degree polynomial regression), ExpReg (exponential regression) and Logistic (logistic regression). Just select the appropriate regression from the STAT CALC menu.

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Automatically Entering a Regression Equation at Y=Automatically Entering a Regression Equation at Y=Automatically Entering a Regression Equation at Y=Automatically Entering a Regression Equation at Y= First, find the regression coefficients and make sure that that information is the current information on the main screen. Then press Y=, VARS, Statistics, EQ, RegEQ on the succeeding menus. Then press GRAPH to graph the regression equation. Example: Using the data given in the example above under Using Lists and Scatter Plots, graph the linear regression equation.

LinReg Y= VARS

Statistics Cursor over to EQ RegEQ

Graph

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If the STATPLOT is turned on, the calculator will display both the scatterplot and the linear regression model.

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Finding a Numeric DerivativeFinding a Numeric DerivativeFinding a Numeric DerivativeFinding a Numeric Derivative at x = A at x = A at x = A at x = A First, enter the function in Y= as Y1. Then return to the main screen. Press MATH, 8, VARS, Y-VARS, FUNCTION, Y1, (comma), X, A. Close parentheses and press ENTER. Example: Find '(2)f if 2( ) 3 7 4f x x x= − +

Enter function MATH 8

VARS Y-VARS FUNCTION ,

ENTER Enter , X,2) ENTER Note: If the function is stored in a different line than Y1, cursor down to the appropriate function line and press ENTER.

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Graphing the Derivative of a FunctionGraphing the Derivative of a FunctionGraphing the Derivative of a FunctionGraphing the Derivative of a Function

To view the graph of the derivative of a function, enter the function on the Y= menu. Then move to an empty line of the Y= menu and enter this string of commands MATH, 8, 2nd VARS, Y-VARS, Funtion, Y1, (comma) X, (comma) X). To display the graphs of both the function and the derivative, enable both on the Y= menu and press GRAPH. To display only the derivative, make sure it is the only function enabled on the Y= menu.

Example: Display the graphs of both 3 22 3( ) 5 9

3 4f x x x x= − − + and its derivative.

Enter the function at Y=. Enter Math, 8, VARS, y-VARS, Function, (comma) X, (comma), X) on an empty line of the Y= menu. Make sure both lines are enabled and any other entries in Y= are disabled. Then press GRAPH.

Adjust the viewing window to see them better, if needed.

Example: Display the graph of the derivative of 3 22 3( ) 5 9

3 4f x x x x= − − +

Using the display from the previous example, disable Y1 on the Y= screen and press GRAPH.

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FFFFinding a Riemann Suminding a Riemann Suminding a Riemann Suminding a Riemann Sum To calculate a Riemann Sum using left endpoints, follow these instructions: Enter the function at Y= as Y1. Enter the number or rectangles (n) as N, so press n, STO>, ALPHA, N. Enter the left endpoint of the interval (a) as A, so press a, STO>, ALPHA, A. Compute x∆ and store it as W. Press the number, then STO> ALPHA W . Press Y= and enter the following commands as Y5: sum(seq(Y1(A + K*W)*W, K, 0, N-1) Press VARS, Y-VARS, Function, 5, ENTER.

Example: Use a Riemann Sum to approximate the area under 21( ) 3

2f x x= + using 15

rectangles and left endpoints on the interval [1, 6].

Enter ( )f x as Y1 on 15 STO> ALPHA N 1 STO> ALPHA A the Y= menu.

Compute x∆ . ANS STO> W 2nd STAT (choose MATH ). Go to Y= and arrow down to Y5.

Result. Choose sum(. Result. Press 2nd STAT and Result. Press seq(.

choose OPS.

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Result. Result. Go to main screen. Press VARS, YVARS, Enter Y1(A + K*W)*W, K, 0, N-1) Function, Y5, ENTER Use Alpha keys for letters. To calculate a Riemann Sum using right endpoints follow these instructions: Enter the function at Y= as Y1. Enter the number or rectangles (n) as N, so press n, STO>, ALPHA, N. Enter the left endpoint of the interval (a) as A, so press a, STO>, ALPHA, A. Compute x∆ and store it as W. Press the number, then STO> ALPHA W . Press Y= and enter the following commands as Y6: sum(seq(Y1(A + K*W)*W, K, 1, N) Press VARS, Y-VARS, Function, 6, ENTER.

Enter ( )f x as Y1 on 15 STO> ALPHA N 1 STO> ALPHA A the Y= menu.

Compute x∆ . ANS STO> W 2nd STAT (choose MATH ). Go to Y= and Press Y6.

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Result. Choose sum(. Result. Press 2nd STAT and Result. Press seq(.

choose OPS.

Result. Result. Press VARS, Y-VARS, Enter Y1(A + K*W)*W, K, 1, N) Function, Y6, ENTER. Use Alpha keys for letters.

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Finding A Definite IntFinding A Definite IntFinding A Definite IntFinding A Definite Integralegralegralegral on [a, b] on [a, b] on [a, b] on [a, b] Enter the integrand at Y= as Y1. Then press MATH, 9, VARS, Y-VARS, Function, Y1, (comma), X, (comma), A, (comma), B).

Example: Evaluate 3

3 2

1

(3 5 7)x x dx−

− +∫

Enter 3 23 5 7x x− + at Y= as Y1. Press MATH, 9, VARS, Y-VARS, Function, Y1, (comma), X, (comma), -1, (comma) 3). Then press ENTER.

Enter the integrand MATH 9

VARS Y-VARS Function

Y1 (comma)X(comma)-1(comma)3) ENTER

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Entering ProgramsEntering ProgramsEntering ProgramsEntering Programs

It is unlikely that programming will be undertaken during the course. However, programs for drawing rectangles for Riemann sums and other calculus topics are easy to find on the internet, and using them can be fun. To enter a program, press PRGM and select NEW. Press ENTER. Using the Alpha keys on the calculator, enter a name for the program. (Note, the calculator is expecting letters, so there is no need to press ALPHA before each letter.) After entering the name, the display will show:

Copy the program into the calculator, using the Alpha keys for letters and operation keys for symbols. Press ENTER at the end of each line you type. Many of the commands needed in programming are accessible on the PRGM menu. Within the program, press PRGM and select CTL . There are 20 commands listed on this screen. Within the program, press PRGM and select I/O . Another 12 commands are listed on this screen. To use a command, select it and press ENTER. It will appear at the cursor site in the program. The display below show part of a program for drawing rectangles in Riemann sums problems. (Look in the unit 11 notes for some samples).

To run a program, press PRGM and with the cursor on EXEC, scroll down to select the program

to run. Before running this program, enter 21

4y x= as Y1.

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Press ENTER.

This screen verifies the request. Press ENTER. Now answer the prompts. In this Riemann sum problem, the calculator wants a value for A. Enter 2.

Now enter a value for B. Enter 10.

Next enter the number of rectangles. Enter 20.

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Next, decide between left endpoints, right endpoints and midpoints. Enter 0 for left endpoints. Press ENTER.