Chapter 7 Computer Algebra System and Tutorial Modes (ALGEBRA FX 2.
7-1-1 Using the CAS (Computer Algebra System) Mode 7-1 Using the CAS (Computer Algebra System) Mode On the Main Menu, select the CAS icon to enter the CAS Mode. The following table shows the keys that can be used in the CAS Mode. COPY H-COPY PASTE REPLAY i k Inputting and Displaying Data Input in the Algebra Mode is performed in the upper part of the display, which is called the “ input area.” You can input commands and expressions at the current cursor location.
7-1-2 Using the CAS (Computer Algebra System) Mode If all the result does not fit on the display, use the cursor keys to scroll it. k Inputting List Data List: {element, element, ..., element} • Elements should be separated by commas, and the entire set of elements should be enclosed within {curly braces}. • You can input numeric values and expressions, equations, and inequalities as list elements.
7-1-3 Using the CAS (Computer Algebra System) Mode k Inputting Vector Data Vector: [component, component, ..., component] • Components should be separated by commas, and the entire set of components should be enclosed within [square brackets]. • You can input numeric values and expressions as vector component entries. ○ ○ ○ ○ ○ Example To input Vector (1 2 3) !+( [ )b,c,d !-( ] )w k Performing an Algebra Mode Operation There are two methods that you can use for input in the Algebra Mode.
7-1-4 Using the CAS (Computer Algebra System) Mode k Manual Formula and Parameter Input You can use the function menus, K key, and J key in combination to input formulas and parameters as described below. • 3(EQUA)b(INEQUA) t}/{s s } ... {inequality} • {>}/{<}/{t •Kkey • {∞}/{Abs}/{ x!}/{sign} ... {infinity}/{absolute value}/{factorial}/{signum function* 1} • {HYP} ... {hyperbolic}/{inverse hyperbolic} functions • {sinh}/{cosh}/{tanh}/{sinh–1}/{cosh–1}/{tanh–1 } •Jkey • {Yn}/{rn}/{Xtn}/{Ytn}/{Xn} ...
7-1-5 Using the CAS (Computer Algebra System) Mode ○ ○ ○ ○ ○ Example To assign M to row 1 column 2 of variable A when the matrix is assigned to it 1 2 3 XYZ ah(M)aav(A) !+( [ )b,c!-( ] )w ○ ○ ○ ○ ○ Example To recall the value of variable A when the list {X, Y, Z} is assigned to it av(A)w ○ ○ ○ ○ ○ Example To recall the first component (A [1]) of variable A when vector (X Y Z) is assigned to it av(A)!+( [ )b !-( ] )w 20010102
-1-6 Using the CAS (Computer Algebra System) Mode k Function Memory and Graph Memory Function memory lets you store functions for later recall when you need them. With graph memory, you can store graphs in memory. Press the J key and then input the name of the graph.
7-1-7 Using the CAS (Computer Algebra System) Mode k Answer (Ans) Memory and Continuous Calculation Answer (Ans) memory and continuous calculation can be used just as with standard calculations. In the Algebra Mode, you can even store formulas in Ans memory. ○ ○ ○ ○ ○ Example To expand (X+1)2 and add the result to 2X 1(TRNS)b(expand) (v+b)x)w Continuing: +cvw k Replay Contents Replay memory can be used in the input area.
7-1-8 Using the CAS (Computer Algebra System) Mode SET UP Items u Angle ... Unit of angular measurement specification • {Deg}/{Rad} ... {degrees}/{radians} u Answer Type ... Result range specification • {Real}/{Cplx} ... {real number}/{complex number} u Display ... Display format specification (for approx only) • {Fix}/{Sci}/{Norm} ...
7-1-9 Using the CAS (Computer Algebra System) Mode u To save a calculation history to solution memory (Save) On the initial solution memory screen, press 1(SAVE). Press 1(YES) to save the calculation history to solution memory. Pressing i returns to the solution memory initial screen. • Pressing 6(NO) in place of 1(YES) returns to the solution memory initial screen without saving anything. u To clear solution memory contents (Clear Memory) On the initial solution memory screen, press 2(DEL • A).
7-1-10 Using the CAS (Computer Algebra System) Mode u To display solution memory contents (Display Memory) On the initial solution memory screen, press 6(DISP). This displays the oldest expression and result in solution memory. The bottom line shows the record number. • 6(DISP) is disabled when there is no data in Solution memory. • To display the next record Press 6(NEXT). • To display the previous record Press 1(BACK).
7-1-11 Using the CAS (Computer Algebra System) Mode Algebra Command Reference The following are the abbreviations used in this section. • Exp ... Expression (value, formula, variable, etc.) • Eq ... Equation • Ineq ... Inequality • List ... List • Mat ... Matrix • Vect ... Vector Anything enclosed within square brackets can be omitted. u expand Function: Expands an expression.
7-1-12 Using the CAS (Computer Algebra System) Mode u solve Function: Solves an equation. Syntax: solve( Eq [,variable] [ ) ] solve( {Eq-1,..., Eq-n}, {variable-1,...,variable-n} [ ) ] ○ ○ ○ ○ ○ Example To solve AX + B = 0 for X 1(TRNS)e(solve)av(A)v+ X=–B A al(B)!.(=)aw ○ ○ ○ ○ ○ Example To solve simultaneous linear equation 3X + 4Y = 5, 2X – 3Y = – 8 1(TRNS)e(solve)!*( { ) da+(X)+ea-(Y)!.(=)f, ca+(X)-da-(Y)!.
7-1-13 Using the CAS (Computer Algebra System) Mode u trigToExp (trigToE) Function: Transforms a trigonometric or hyperbolic function to an exponential function. Syntax: trigToExp( {Exp/List/Mat/Vect} [ ) ] ○ ○ ○ ○ ○ Example To convert cos(iX) to an exponential function 1(TRNS)f(TRIG)d(trigToE)c!a(i)vw ex+ e—x 2 u expToTrig (expToT) Function: Converts an exponential function to a trigonometric or hyperbolic function.
7-1-14 Using the CAS (Computer Algebra System) Mode u combine (combin) Function: Adds and reduces rational expressions. Syntax: combine( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ] ○ ○ ○ ○ ○ Example To reduce the fraction (X + 1) / (X + 2) + X (X + 3) 1(TRNS)h(combin)(v+b)/ (v+c)+v(v+dw X3 + 5X2 + 7X + 1 X+2 u collect (collct) Function: Rearranges an expression, focusing on a particular variable.
7-1-15 Using the CAS (Computer Algebra System) Mode u cExpand (cExpnd) Function: Expands xth root of imaginary number. Syntax: cExpand( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ] ○ ○ ○ ○ ○ Example To expand 2i 1(TRNS)v(cExpnd)!x( )c!a(i)w 1 +i u approx Function: Produces a numerical approximation for an expression. Syntax: approx( {Exp/Eq/Ineq/List/Mat/Vect} [ ) ] ○ ○ ○ ○ ○ Example To obtain a numerical value for 1(TRNS)l(approx)!x( ○ ○ ○ ○ ○ Example 2 )cw 1.
7-1-16 Using the CAS (Computer Algebra System) Mode u diff Function: Differentiates an expression. Syntax: diff( {Exp/List} [, variable, order, derivative] [ ) ] diff( {Exp/List}, variable [, order, derivative] [ ) ] diff( {Exp/List}, variable, order [, derivative] [ ) ] ○ ○ ○ ○ ○ Example To differentiate X6 with respect to X 2(CALC)b(diff)vMgw 6X5 • X is the default when no variable is specified. • 1 is the default when no order is specified. u∫ Function: Integrates an expression.
7-1-17 Using the CAS (Computer Algebra System) Mode uΣ Function: Calculates a sum. Syntax: Σ( {Exp/List}, variable, start value, end value [ ) ] ○ ○ ○ ○ ○ Example To calculate the sum as the value of X in X2 changes from X = 1 through X = 10 2(CALC)e(Σ)vx,v,b,baw 385 uΠ Function: Calculates a product.
7-1-18 Using the CAS (Computer Algebra System) Mode u tanLine (tanLin) Function: Returns the expression for a tangent line. Syntax: tanLine( {Exp/List}, variable, variable value at point of tangency [ ) ] ○ ○ ○ ○ ○ Example To determine the expression for a line tangent with X3 when X = 2 2(CALC)i(tanLin)vMd,v,cw 12X – 16 u denominator (den) Function: Extracts the denominator of a fraction.
7-1-19 Using the CAS (Computer Algebra System) Mode u lcm Function: Obtains the least common multiple of two expressions Syntax: lcm( {Exp/List}, {Exp/List} [ ) ] ○ ○ ○ ○ ○ Example To obtain the least common multiple of X2 – 1 and X2 + 2X – 3 2(CALC)l(lcm)vx-b, vx+cv-dw X3 + 3X2 – X – 3 u rclEqn Function: Recalls multiple eqn memory contents. Syntax: rclEqn( memory number [, ...
7-1-20 Using the CAS (Computer Algebra System) Mode u exchange (exchng) Function: Exchanges the right-side and left-side expressions. Syntax: exchange( {Eq/Ineq/List} [ ) ] ○ ○ ○ ○ ○ Example To exchange the left-side and right-side expressions of 3 > 5X – 2Y 3(EQUA)f(exchng)d3(EQUA)b(INEQUA)b(>) fa+(X)-ca-(Y)w 5X – 2Y < 3 u eliminate (elim) Function: Assigns an expression to a variable.
7-1-21 Using the CAS (Computer Algebra System) Mode u absExpand (absExp) Function: Divides an expression that contains an absolute value into two expressions. Syntax: absExpand( {Eq/Ineq} [ ) ] ○ ○ ○ ○ ○ Example To strip the absolute value from | 2X – 3 | = 9 3(EQUA)j(absExp)K5(Abs)( 2X – 3 = 9 cv-d)!.(=)jw or 2X – 3 = – 9 2 1 u andConnect (andCon) Function: Connects two inequalities into a single expression.
7-1-22 Using the CAS (Computer Algebra System) Mode u clear (clrVar) Function: Clears the contents of specific equation (A to Z, r, θ ).* 1 Syntax: clear( variable [ ) ] clear( {variable list} [ ) ] ○ ○ ○ ○ ○ Example To clear the contents of variable A 6(g)1(CLR)b(clrVar)av(A)w ○ ○ ○ ○ ○ Example { } To clear the contents of variables X, Y, and Z 6(g)1(CLR)b(clrVar)!*( { )a+(X), a-(Y),aa(Z)!/( } )w { } u clearVarAll (VarAll) Function: Clears the contents of all 28 variables (A to Z, r, θ ).
7-1-23 Using the CAS (Computer Algebra System) Mode k List Calculation Commands [OPTN]-[LIST] u Dim Function: Returns the dimension of a list. Syntax: Dim List ○ ○ ○ ○ ○ Example To determine the dimension of list {1, 2, 3} K1(LIST)b(CALC)b(Dim)!*( { )b,c,d !/( } )w 3 u Min Function: Returns the minimum value of an expression or the elements in a list.
7-1-24 Using the CAS (Computer Algebra System) Mode u Max Function: Returns the maximum value of an expression or the elements of a list.
7-1-25 Using the CAS (Computer Algebra System) Mode ○ ○ ○ ○ ○ Example To determine the mean of the elements in list {1, 2, 3} when their frequencies are {3, 2, 1} K1(LIST)b(CALC)e(Mean)!*( { )b,c,d !/( } ),!*( { )d,c,b!/( } )w 5 3 u Median Function: Returns the median of the elements in a list. Syntax: Median( List [ ) ] Median( List, List [ ) ] The list must contain values or mathematical expressions only. Equations and inequalities are not allowed.
7-1-26 Using the CAS (Computer Algebra System) Mode u Prod Function: Returns the product of the elements in a list. Syntax: Prod List The list must contain values or mathematical expressions only. Equations and inequalities are not allowed. ○ ○ ○ ○ ○ Example To determine the product of the elements in list {2, 3, 4} K1(LIST)b(CALC)h(Prod)!*( { )c,d,e !/( } )w 24 u Cuml Function: Returns the cumulative frequency of the elements in a list.
7-1-27 Using the CAS (Computer Algebra System) Mode u A List Function: Returns a list whose elements are the differences between the elements of another list. Syntax: A List List The list must contain values or mathematical expressions only. Equations and inequalities are not allowed.
7-1-28 Using the CAS (Computer Algebra System) Mode u Seq Function: Generates a list in accordance with a numeric sequence expression. Syntax: Seq( Exp, variable, start value, end value, [increment] [ ) ] If you do not specify an increment, an increment of 1 is used.
7-1-29 Using the CAS (Computer Algebra System) Mode u SortA Function: Sorts the elements of a list into ascending order. Syntax: SortA( List [ ) ] The list must contain values or mathematical expressions only. Equations and inequalities are not allowed. ○ ○ ○ ○ ○ Example To sort the elements of list {1, 5, 3} into ascending order K1(LIST)c(CREATE)e(SortA)!*( { )b,f,d !/( } )w { 1, 3, 5 } u SortD Function: Sorts the elements of a list into descending order.
7-1-30 Using the CAS (Computer Algebra System) Mode u List→Mat (L→Mat) Function: Converts lists into a matrix. Syntax: List→Mat( List [ , ... ,List ] [ ) ] ○ ○ ○ ○ ○ Example To convert list {3, 5} and list {2, 4} into a matrix K1(LIST)d(LIST→)b(L→Mat)!*( { )d,f 3 2 !/( } ),!*( { )c,e!/( } )w 5 4 u List→Vect (L→Vect) Function: Converts a list into a vector.
7-1-31 Using the CAS (Computer Algebra System) Mode k Matrix Calculation Commands [OPTN]-[MAT] u Dim Function: Returns the dimensions of a matrix. Syntax: Dim Mat ○ ○ ○ ○ ○ Example To determine the dimensions of the matrix below 1 2 3 4 5 6 K2(MAT)b(CALC)b(Dim)!+( [ )!+( [ ) b,c,d!-( ] )!+( [ )e,f,g !-( ] )!-( ] )w { 2, 3 } u Det Function: Returns the determinant of a matrix.
7-1-32 Using the CAS (Computer Algebra System) Mode u EigVc Function: Returns the eigenvector of a matrix. Syntax: EigVc Mat ○ ○ ○ ○ ○ Example To determine the eigenvector of the matrix below 3 4 1 3 K2(MAT)b(CALC)e(EigVc) !+( [ )!+( [ )d,e !-( ] )!+( [ ) [ 0.894427191 – 0.894427191 ] b,d!-( ] )!-( ] )w [ 0.4472135955 0.4472135955 ] Eigenvectors are stacked vertically on the display. In this example, (0.894427191 0.4472135955) are the eigenvectors that correspond to 5, while (–0.894427191 0.
7-1-33 Using the CAS (Computer Algebra System) Mode u Rref Function: Returns the reduced row echelon form of a matrix. Syntax: Rref Mat ○ ○ ○ ○ ○ Example To determine the reduced row echelon form of the matrix below –2 –2 0 –6 1 –1 9 –9 –5 2 4 –4 K2(MAT)b(CALC)g(Rref)!+( [ )!+( [ ) -c,-c,a,-g!-( ] )!+( [ ) b,-b,j,-j!-( ] ) 66 71 147 0 1 0 71 62 0 0 1– 71 1 0 0 !+( [ )-f,c,e,-e !-( ] )!-( ] )w u Ref Function: Returns the row echelon form of a matrix.
7-1-34 Using the CAS (Computer Algebra System) Mode u LU Function: Returns the LU resolution of a matrix. Syntax: LU( Mat, lower memory, upper memory) ○ ○ ○ ○ ○ Example To determine the LU resolution of the matrix below 6 12 18 5 14 31 3 8 18 The lower matrix is assigned to variable A, while the upper matrix is assigned to variable B.
7-1-35 Using the CAS (Computer Algebra System) Mode u Augment (Augmnt) Function: Combines two matrices.
7-1-36 Using the CAS (Computer Algebra System) Mode ○ ○ ○ ○ ○ Example To create a 2 × 3 matrix, all of whose entries are X K2(MAT)c(CREATE)e(Fill)v,c,dw X X X X X X u SubMat Function: Extracts a specific section of a matrix into a new matrix.
7-1-37 Using the CAS (Computer Algebra System) Mode u Diag Function: Extracts the diagonal elements of a matrix. Syntax: Diag Mat ○ ○ ○ ○ ○ Example To extract the diagonal elements of the matrix below 1 2 3 4 K2(MAT)c(CREATE)g(Diag)!+( [ )!+( [ ) b,c!-( ] )!+( [ )d,e !-( ] )!-( ] )w [ 1, 4 ] u Mat→List (M→List) Function: Converts a specific column of a matrix into a list.
7-1-38 Using the CAS (Computer Algebra System) Mode u Swap Function: Swaps two rows of a matrix. Syntax: Swap Mat, row number 1, row number 2 ○ ○ ○ ○ ○ Example To swap row 1 with row 2 of the following matrix 1 2 3 4 K2(MAT)e(ROW)b(Swap)!+( [ )!+( [ ) b,c!-( ] )!+( [ )d,e 3 4 !-( ] )!-( ] ),b,cw 1 2 u `Row Function: Returns the scalar product of a row of a matrix.
7-1-39 Using the CAS (Computer Algebra System) Mode u Row+ Function: Adds one row of a matrix and to another row.
7-1-40 Using the CAS (Computer Algebra System) Mode k Vector Calculation Commands [OPTN]-[VECT] u Dim Function: Returns the dimension of a vector. Syntax: Dim Vect ○ ○ ○ ○ ○ Example To determine the dimension of the vector (1 2 3) K3(VECT)b(CALC)b(Dim)!+( [ )b,c,d !-( ] )w 3 u CrossP Function: Returns the cross product of two vectors.
7-1-41 Using the CAS (Computer Algebra System) Mode u UnitV Function: Normalizes a vector. Syntax: UnitV Vect ○ ○ ○ ○ ○ Example To normalize a vector (1 2 3) K3(VECT)b(CALC)f(UnitV) !+( [ )b,c,d 14 14 3 14 14 , 7 , 14 !-( ] )w u Angle Function: Returns the angle formed by two vectors.
7-1-42 Using the CAS (Computer Algebra System) Mode u Vect→List (V→List) Function: Converts a vector into a list. Syntax: Vect→List Vect ○ ○ ○ ○ ○ Example To convert vector (3 2) into a list K3(VECT)d(VECT→)b(V→List)!+( [ )d,c !-( ] )w { 3, 2 } u Vect→Mat (V→Mat) Function: Converts vectors into a matrix. Syntax: Vect→Mat( Vect [, ...
7-2-1 Algebra Mode 7-2 Algebra Mode The CAS Mode automatically provides you with the final result only. The Algebra Mode, on the other hand, lets you obtain intermediate results at a number of steps along the way. On the Main Menu, select the ALGEBRA icon to enter the Algebra Mode. The screens in this mode are the same as those in the CAS Mode. Operations in the Algebra Mode are identical to those in the CAS Mode, except for a number of limitations.
7-3-1 Tutorial Mode 7-3 Tutorial Mode On the Main Menu, select the TUTOR icon to enter the Tutorial Mode. k Tutorial Mode Flow 1. Specify the expression type. 2. Define the expression. 3. Specify the solve mode. k Specifying the Expression Type Entering the Tutorial Mode displays a menu of the following expression types. • Linear Equation • Linear Inequality • Quadratic Equation • Simul (Simultaneous) Equation Use the cursor keys to highlight the expression type you want to specify, and then press w.
7-3-2 Tutorial Mode The following shows the formulas available for each type of expression.
7-3-3 Tutorial Mode k Defining the Expression In this step, you specify coefficients and define the expression. You can select any of the three following methods for specifying coefficients. • {RAND} ... {random generation of coefficients} • {INPUT} ... {key input of coefficients} • {SMPL} ... {selection of coefficients from samples} • {SEED} ...
7-3-4 Tutorial Mode k Specifying the Solve Mode You can select one of the following three solve modes for the displayed expression. • {VRFY} ... {Verify Mode} In this mode, you input a solution for verification of whether or not it is correct. It provides a good way to check solutions you arrive at manually. • {MANU} ... {Manual Mode} In this mode, you manually input algebra commands, transform the expression, and calculate a result. • {AUTO} ...
7-3-5 Tutorial Mode You can press 4(MANU) to change to the Manual Mode or 5(AUTO) to change to the Auto Mode.
7-3-6 Tutorial Mode k Manual Mode Press 5(MANU) to enter the Manual Mode. As with the Algebra Mode, the screen is divided between an input area and a display area. This means you can select Algebra Mode commands from the function menu, transform the expression, and solve it. Operation is the same as that in the Algebra Mode. After you obtain a result, you can press 5(JUDG) to determine whether or not it is correct. • {DISP} ... Determines whether the expression in the display area is a correct solution.
7-3-7 Tutorial Mode ○ ○ ○ ○ ○ Example 4X2 = 16 True (X = 2, X = – 2) Besides “TRUE” the messages shown below can also appear as the result of verification. “CAN NOT JUDGE” appears in the Manual Mode, while the other messages appear in both the Verify Mode and Manual Mode.
7-3-8 Tutorial Mode k Auto Mode Press 6(AUTO) to enter the Auto Mode. In the Simultaneous Equation Mode, you must also select SBSTIT (Substitution Method) or ADD-SU (Addition/Subtraction Method). The Substitution Method first transforms the equation to the format Y = aX + b, and substitutes aX + b for Y*1 in the other equation. The Addition/Subtraction Method multiplies both sides of the expression by the same value to isolate the coefficient X (or Y).
7-4-1 Algebra System Precautions 7-4 Algebra System Precautions • If an algebraic operation cannot be performed for some reason, the original expression remains on the display. • It may take considerable time to perform an algebraic operation. Failure of a result to appear immediately does not indicate malfunction of the computer. • Any expression can be displayed in various different formats.
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