SAMPLE. Computer Algebra System (Classpad 330 using OS 3 or above) Application selector. Icolns that access working zones. Icon panel (Master toolbar)

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1 A P P E N D I X B Computer Algebra System (Classpad 330 using OS 3 or above) B.1 Introduction For reference material on basic operations of the calculator, refer to the free downloadable documentation provided by Casio on the website: download.php. The Classpad calculator operates in many ways like your computer, with various drop down menus which are accessed using the stylus and tapping items on the Menu bar. Shortcuts in particular applications are found in the pictorial toolbar. You will find that Cut, Copy and Paste in the Edit menu very useful. Application selector Icolns that access working zones Icon panel (Master toolbar) The screen provides access to various in-built programs of the machine, selected by tapping the icons on the touch-sensitive screen using the stylus. To stop a process that is in action, tap. 693

2 694 Essential Mathematical Methods Units 1&2CAS The screen The various areas of the screen referred to in these notes are shown in this diagram. In some programs, such as the screen is split into two halves. If you wish one of these to fill the whole screen, tap to ensure it is selected (bold border) and tap on the Master toolbar.tapping it again will re-split the screen. Settings Menu bar Toolbar Work area input displayed on left, output displayed on the right. Status bar displays current mode settings The basic settings for the calculator are managed by tapping from any of the program screens. Quick changes can be made by tapping the appropriate item on the Status Bar. In they appear as shown at right. Tapping any of them toggles between the modes. Alg algebra mode automatically simplifies expressions. (Alternative Assist assistant mode does not automatically simplify expressions.) Standard results given in exact form (fraction, square root, etc.) (Alternative Decimal converts all results to decimal (approximate value).) Real real mode for real number calculations. (Alternate Cplx for complex number calculations.) Rad angles displayed in radians. (Alternative Deg angles displayed in degrees.) Executing in commands in Operator entries are shown on the left of the screen. Once entered, press and the answer will be displayed right-justified on the screen. Scroll arrows will appear if your entry, or the answer, is too wide for a single screen. Additional input options, such as square roots, alternative variables and trigonometric or calculus operations, are accessed by pressing the button on the calculator s keypad. If an answer is given in exact mode and you wish a decimal answer, highlight the answer with the stylus and tap.toconvert from decimal to exact, follow the same steps. Errors, deleting, inserting and clearing The button deletes the last character entered. You can use the central multi-directional arrow key, or the stylus, to return the cursor to the place where you wish to insert extra

3 Appendix B Computer Algebra System (Classpad 330 using OS 3 or above) 695 characters. If you have already executed a command, you can still use the stylus to go back and make a correction and then execute again. To clear the screen (in any program section of the calculator, select Edit Clear all.toclear values or functions which have been saved in memory locations, select Edit Clear all variables. Toclear individual variables, without clearing them all, use Variable manager.you can double-click on various subfolders to expand them and then select items to delete using the Edit submenu items. Note also that, like using a Windows computer program, you can highlight, copy, paste and drag-and-drop expressions, or parts of expressions, to create new expressions in the entry line. B.2 Using the Main menu In this section the Main menu is explored. Work through it to become acquainted with the menu. There are two ways to operate in this menu, but the Interactive menu is the simplest when used in conjunction with the stylus. In this section we shall to demonstrate some examples of how this is used. Solve This is used to solve equation and inequalities. The variables x, y and z are found on the hard keyboard. Other variables may be entered using the and selecting VAR.Variables are shown in bold italics. The abc keyboard allows you to type sentences, etc.; however, the letters are not always recognised as variables. If you choose to use the abc keyboard, you must type a x, for example, as ax will be treated as text. Enter ax + b = 0, highlight it with the stylus, tap Interactive Equation/inequality solve and ensure the variable selected is x. Solution returned is x = b a. Enter x 2 + x 1 = 0, and follow the same instructions as above. The answer is as shown. Note: x 2 + x 1 = 0isentered, but the calculator converts this to standard algebraic notation when EXE is pressed. Also note in this example that = 0 has been omitted. If the right-hand side of an equation is zero, it is unnecessary to enter it.

4 696 Essential Mathematical Methods Units 1&2CAS Enter abt w + t = wt for w; follow the instructions above and select w as the variable. Solve x 3 x 2 x + 1 = 0 for x. Solve 2x + 2 < 3 for x. See the screen above for solutions returned. Note: is found in when is activated and < is in.ifthe answer is not in the form required, it is often possible to cut and paste it in the next entry line and use Interactive Transformation simplify as shown here. Solve simultaneous equations uses a dedicated entry style. Note carefully the screen at right. To enter, tap in the window and enter the equations and variables as shown. For more than two equations, tap until the number of equations required is given. Factor To factorise is to transform the expression to a different form. This command is found in Interactive Transformation factor. Examples: To factorise x 3 2x over the rational numbers, use factor. To factorise over the real numbers, select rfactor. Some further examples are provided here. See the screens for the results. factor a 2 b 2. factor a( 3 b 3. ) 2 factor x (x 1) factor ( 2x 4 x 2) over the rationals. factor ( 2x 4 x 2) over the reals. This command can also be used to give the prime decomposition (factors) of integers.

5 Appendix B Computer Algebra System (Classpad 330 using OS 3 or above) 697 Expand To expand expressions, use Interactive Transformation expand. expand (a + b) 3. ( expand a b) The expand command can also be used to form partial fractions. In this case, enter and highlight the expression, select Interactive Transformation expand, select the Partial Fraction option and set the variable as x. 1 Expand x 2 1. Expand x 3 + 2x + 1. x 2 1 Note: The top screen shows all the examples; the bottom screen shows how to enter for partial fractions. Zeros To find the zeros of an expression in the menu, use Interactive Equation/inequality solve and ensure you set the variable. The calculator assumes you are solving an equation for which one side is zero. Zeros of x 2 1 for x. Zeros of x 2 y 2 for y. Zeros of x 2 y 2 for x. Zeros of x 2 y for y. Zeros of x 2 4x + 8 for x.nosolutions. Zeros of x 2 4x + 1 for x.two solutions. Zeros of x 2 4x + 4 for x. One solution. Approximate Switch mode in the status bar to Decimal. If an answer is given in Standard (Exact) mode, it can be converted by highlighting the answer and tapping in the toolbar.

6 698 Essential Mathematical Methods Units 1&2CAS Combining fractions This command gives an expression with a common denominator and. The denominator is returned in factored form. Enter and highlight 1/(x 1) + 1/(x + 1). Then select Interactive Transformation combine. Enter and highlight y/(x y) + y/(x + y). Then select Interactive Transformation combine. Solve numerically There are several ways to find numerical solutions to equations. In each of these ways, only one solution is given. You can vary the guess (Value) or the bounds of the search (Lower and Upper) to find particular solutions. If an expression such as x 2 x 2.1isentered without an equals sign, the calculator will assume the expression is equal to zero and solve the equation. Enter and highlight x 2 x 2.1 = 0, select Interactive Equation/inequality solve and tap the solve numerically option. Note that the lower and upper bounds are set to and a guess of 1 has been entered to return the first solution: For the second solution shown, the first line has been copied and pasted (or dragged) to the next entry line and the guess x = 2 has been entered to return the solution x = Alternatively, use from the Main menu and enter Lower and Upper bounds. A guess may also be entered, but is not necessary. Note: The bounds selected arise from a quick sketch which indicated that the quadratic equation has one positive and one negative solution. A third method involves using the Graph program and finding the solution using the GSolve application.

7 Appendix B Computer Algebra System (Classpad 330 using OS 3 or above) 699 B.3 Graphing Graphing using the Casio Classpad is usually done in the menu. The screen is split with the equation entry screen and the graph screen both shown. The normal configuration is the y = form; however, by tapping the arrow in the menu bar, alternative forms of equation and inequality may be selected. After entering the equation, tap the select box to the left of the equation and then tap to produce the graph in the lower screen. More than one equation can be entered and you will note tabs allowing more than one set of equations to be entered and stored for other uses. Note: In the example, the upper box is selected shown by abold boundary. Tapping the lower half of the screen will select the graph section and the menu bar will change to the operations available on the graph. To alter the viewing window for the graph, various Zoom menus are available or the window can be manually set using. Tap on the icon panel to allow the selected window (graph or equation) to be re-sized to fill the screen. Tapping again returns both windows to the screen. To find features of the graph, ensure the graph window is selected (bold frame), then tap Analysis GSolve and select the menu item you wish. If seeking the root, the calculator will return the left-hand root shown in the graph window by default. To find the other root, use Zoom-Box to zoom into an area containing only the desired root. Analysis GSolve will not return this root. Use Zoom Previous to return to the previous view. Analysis Trace allows use of the round arrow cursor on the calculator to move along the curve and display grid points in the lower part of the screen.

8 700 Essential Mathematical Methods Units 1&2CAS B.4 Defining functions Expressions can be stored in memory location. For this the is used from and the variable A is entered using. Forexample x 2 3x A. The operations discussed previously can now be applied to A. B.5 Probability and counting Basic probability functions are found using the menu. To find n C r and n P r tap the key and then enter the values of n and r as shown. To find n!, type the value of n and tap the! key. Generate a random number between 0 and 1 by typing rand(. Continue to tap to produce more numbers. The function randlist(20) will create a set of 20 random numbers between 0 and 1. To generate a random integer between 25 and 50, type rand(2,20) as shown. To create a random list of numbers between 0 and 1, the Stat editor can also be used. In the line for the required list type randlist(20) to create a list of 20 random numbers. Note: The mode should be set to Decimal in the status bar at the bottom of the screen. The functions referred to here can be found in the catalogue. Go to and scroll to and select the function required.

9 Appendix B Computer Algebra System (Classpad 330 using OS 3 or above) 701 To find the value of a term in a binomial distribution, go to the Stat screen and enter Calc Distribution and scroll to Binomial PD. Enter the values for n, p and x and tap.tofind more values, use and enter the required values. Alternatively, define a function bi(n, p, x) = ncr(n, x) p x (1 p) (n x) This can be used for any binomial distribution problem. B.6 Trigonometric functions The choice of radian or degree measure is made by tapping the appropriate section of the Status bar at the bottom of the screen. Trigonometric equations are solved using the Interactive functions in. Initially, we shall consider the solution of the equation sin x = 0.5. The exact solution is given as x = 2k or x = 2k +. The parameter k is shown as 6 constn(1) and constn(2) on the screen. This indicates that these are the first and second parameters to be used. The screen shows the solution when the status bar is set to Standard or to Decimal. Tap on to scroll across for the rest of the solution. In cases where the domain is limited, enter the equation with the domain sin(x) = x 2, highlight the equation part only and then tap Interactive Equation/inequality solve and ensure the variable is set to x. After tapping EXE the B.7 Using the Calculus menu Four of the operations of the calculus menu are considered here. These functions appear on the keyboard screen and variables and expressions can be entered directly. (It is also possible to use the options in the Interactive Calculation drop-down menu.)

10 702 Essential Mathematical Methods Units 1&2CAS Differentiate This operation is use to differentiate expressions. Several examples are shown in the screen here. In tap, enter x in the darkened entry box and the expression x 2 + 2x in the empty box. Change the variable in the darkened box when differentiating with respect to a variable other than x. If a second derivative is required, tap set the order to 2 in the small raised entry box beside d. Integrate For indefinite integrals, in tap and enter ax 2 + bx + c in the space provided for the expression, then enter x as the variable and tap EXE. For definite integrals, the same process is followed except that the upper and lower bounds are entered. Limit The limits considered in this section will be those that are considered in the senior years of school mathematics. Right and left limits are shown in the sample screen. To enter limits use, tap and enter the required information in the entry boxes provided. Remember to use the VAR screen to enter variables. x 3 3x 2.

11 Appendix B Computer Algebra System (Classpad 330 using OS 3 or above) 703 Minimum (fmin) and Maximum (fmax) These operations return the value for which the maximum or minimum value (or the least upper bound or greatest lower bound of a function occurs). fmax returns the value for which a local maximum occurs only if this is the actual maximum for the interval being considered. Similarly for fmin. Use Interactive Calculate fmax or fmin A number of examples are show in the screen on the right. The entries for the example seeking the maximum value of x 3 3x 2 for x < 3 are show below.

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