Application of the Maxima in the Teaching of Science Subjects at Different Levels of Knowledge

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Advances in Science and Technology Research Journal Volume 8, No. 24, Dec. 2014, pages 44–50 Original Article DOI: 10.12913/22998624/566

APPLICATION OF THE MAXIMA IN THE TEACHING OF SCIENCE SUBJECTS AT DIFFERENT LEVELS OF KNOWLEDGE

Anna Makarewicz1

1 Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38, 20-618 Lublin, Poland, e-mail: [email protected]

Received: 2014.10.18 ABSTRACT Accepted: 2014.11.12 In this paper I present a program Maxima, which is one of the best computer algebra Published: 2014.12.01 system (CAS). The program is easy to use and offers many possibilities. In the text the interface, basic commands, and various examples to facilitate complex calculations in almost every field of mathematics are preented. I prove that the program can be used in secondary schools, upper secondary schools and college. Program’s capabilities are presented, including, among others, differentiation and symbolic integration, symbolic solving equations (including differential), simplifying algebraic expressions, matrix operations, the possibility of drawing graphs of 2D / 3D and others.

Keywords: computer program, numerical computation, symbolic computation, symbolic differentiation, symbolic integration, graph, graph three-dimensional, function.

INTRODUCTION •• symbolic solving equations (including differential), Maxima is a computer program, performing •• symbolic solving systems of equations, mathematical calculations, both symbolic and nu- •• symbolic differentiation and integration, meric. The use of symbolic computation allows to •• matrix operations, solve many mathematical problems in an accurate •• drawing graphs of functions, manner. Maxima can work as a calculator, which •• perform calculations in the field of probability can be used in secondary schools. It is able to theory, create very complex parametric function graphs, •• define your own functions by the user, implicit function graphs and graphs of function •• export of the results to the following formats: of several variables, (with such problems faced HTML, LaTeX, and graphic formats PNG, students of almost all departments), these are nec- PostScript. essary basics that form the base for further de- Thanks to William Schelter program is free. velopment and specialization in a specific area, Since then, the Maxima has been developing quite for example mechanics, electrical engineering, intensively in versions for Windows and Linux. computer science, construction, aviation, broadly Since version 5.101 program already contains understood technologies, etc. an integrated overlay wxMaxima (graphi- This program stems from program macsyma cal user interface), which provides bet- developed at the Massachusetts Institute of Tech- ter formatting mathematical formulas and nology for the Department of Energy. The soft- makes them easier to enter and modify. ware features include: Maxima is a simple and intuitive software. Learn- •• performing numerical calculations with any ing the basics of the Maxima does not require a accuracy, large amount of time. Maxima facilitates better •• simplifying algebraic expressions and trigono- and faster understanding of mathematics. It al- metric, lows you to experiment, consider a lot of alter-

44 Advances in Science and Technology Research Journal vol. 8 (23) 2014 native solutions and create visualization. It also •• less than < helps to save time and to control and eliminate •• greater than or equal to >= errors in solving complex accounting tasks. •• less than or equal to <= •• to assign a value : •• defining a function := BASICS MAXIMA Constants: Maxima works at input %i and outputs %o. •• %e – base of natural logarithm Inputs and outputs are numbered that in combina- •• %i – the imaginary unit tion with the %i or %o creates a unique identifier. •• %pi – π Maxima distinguishes quantity of letter, so the •• inf – ∞ function f(x) and F(x) are not the same. Variable •• minf – −∞ %i stores the result of the last command. •• %gamma – Euler’s constant Each instruction must be completed: a semi- Example: colon (;) – displays the „result” or dollar ($) – do Let’s try to perform the first arithmetic opera- not display the result (but it is calculated). The tions (ie, Maxima will be used as a calculator) and program accepts standard arithmetic operators show the interface (Figure 1). that are intuitive, so that the program can be used Note that you can refer to the previous result as a better class of the calculator in secondary with % (percent), eg: schools: (%i4) % – 20; (33–20) • • adding + (%o4) 13; •• subtraction – •• multiplication * You can also refer to any previous result, •• dividing / which we received in a given session, which •• exponentiation ^ or ** greatly facilitates the work and calculations. For •• matrix multiplication . this purpose we use the term %o or %i. Let us see how this works in practice: Comparison operators are also intuitive: (%i5) %o2 * 3; (-10*3) •• equality = (%o5) -30 •• inequality <> (%i6) %o3 – %o4; (33–13) •• greater than > (%o6) 20

Fig. 1. First arithmetic operations

45 Advances in Science and Technology Research Journal vol. 8 (23) 2014

To receive a number in decimal form, use the expression which is located before the comma. So function float, here is an example: far, Maxima was used as a better class of calcula- (%i1) float(2/7); tor. Now we will present what is beyond the reach (%o1) 0.28571428571429 of a calculator. Numerical calculations in Maxima can be performed with any precision, the command EXPRESSIONS fpprec: n will be performed with an accuracy of n-significant digits (standard setting fpprec: The program Maxima is possible to declare 16). We can also change the accuracy of the dis- algebraic expressions. You can save even the played results by typing fpprintprec: n, here is most complex expression-patterns. These ex- an example: pressions can be freely converted, and you can (%i1) float(23/112); find their values at certain, fixed parameter val- (%o1) 0.20535714285714 ues. Maxima has many features that allow modi- (%i2) fpprintprec:4$ fying, simplifying and developing expressions (%i3) float(23/1121); and these are the main problems of students (%o3) 0.021 from secondary schools. Thanks to Maxima, Maxima handles well with large numbers and they may check their thinking because Maxi- is better than the calculator. Let us try to calculate ma allows you to control the partial results and the 2128 and 30! Factorial operation is very dif- calculations and all the results are displayed in ficult to imagine, especially for high school stu- symbolic form which enables precise analysis of dents and ordinary calculators cannot handle such reasoning. Let us present selected functions on large numbers, Maxima by calculating “the facto- algebraic expressions: rial of any value” helps you to realize how much •• ev(expression,condition) – modifies the - ex the number is changing under its influence, which pression based on a condition, greatly helps to develop student’s imagination. To •• subst(b,a,expression) – after the command do this in Maxima, we write: variable a in the expression will be replaced by (%i1) 2 ^ 128; variable b, (%o1) 340282366920938463463374607431768 •• expand(expression) – develops algebraic ex- 211456 pressions, (%i2) 30!; •• ratsimp(expression) – simplifies algebraic ex- (%o2) 265252859812191058636308480000000 pressions, •• radcan(expression) – simplifies expression, Operator ‘ (apostrophe) •• factor(expression) – decomposes expression factors, Operator ‘ prohibits the calculation of the val- •• trigexpand(expression) – develops trigono- ue of its argument. metric expression, Example: •• trigreduce(expression), trigsimp(expression) – (%i1) diff(sin(x),x) simplifies trigonometric expression (%o1) cos(x) •• rectform – shows the expression in the form (%i2) ’diff(sin(x),x); A + B i. (%o2) (sin(x)) •• polarform – shows the complex expression in the polar form re iθ Symbolic calculations are very good and •• logcontact – simplifies the expression such as useful, as during subsequent operations the log x + b log y. computer does not make the mistake associ- ated with an approximation of the number. Example: Nested mathematical operations are often so (%i1) expand((b+c)^5); // expands the expression complicated that it is impossible to estimate in brackets 5 4 2 3 3 2 4 5 it’s value. Then you can ask the program to (%o1) c +5*b*c +10*b *c +10*b *c +5*b *c+b give the approximate result of the calculation. Now we will factorize the expression by enter To get the value of the expression we place the the command: comma after it and call the command number. It (%i2) factor(%); requires the return of the approximate value of the (%o2) (c+b)5

46 Examples. Examples. Examples. 1. Let f(x) = xx be the function which derivative we try to calculate, then we obtain: 1. Let f(x) = xx be the function which derivative we try to calculate, then we obtain: 1. Let f(x) = xx be the function which derivative(%i1) we diff(x^x,x); try to calculate , then we obtain: (%i1) diff(x^x,x); (%i1) diff(x^x,x); (%o1) xx*(log(x)+1) x x (%o1) x *(log(x)+1) (%o1) x *(log(x)+1) 2. Let the function be given by f(x) = . Calculation derivative of the second degree in Maxima looks2. Le ast the follows: function be given by f(x) = sin (𝑥𝑥) . Calculation derivative of the second degree in Maxima 2. Let the function be given by f(x) = . Calculation derivative of the second degree𝑥𝑥 in Maxima • subst(b,a,expression) – after the command variable a in the expression will be replaced bysin (𝑥𝑥) sin (𝑥𝑥) looks as follows: • subst(b,a,expression)looks as follows: – after the command variable(%i1) a in the diff(sin(x)/x,x,2); expression will be replaced by 𝑥𝑥 variable b, 𝑥𝑥 variable• subst(b,a,expression) b, – after the command variable (%i1)a in the diff(sin(x)/x,x,2); expression will be replaced by variable• expand(expression)subst(b,a,expression) (%i1)b, diff(sin(x)/x,x,2); – –develops after the algebraic command expressions variable(%o1) a ,in the expression will be replaced by • subst(b,a,expression) – after the command variable a in the expression will be replaced by •variable expand(expression) b, – develops algebraic expressions, sin (𝑥𝑥) 2sin (𝑥𝑥) 2cos (𝑥𝑥) variable• subst(b,a,expression) b, – after the command variable (%o1)a in the expression will3 be replaced2 by •variable expand(expression)ratsimp(expression) (%o1)b, –– developssimplifies algebraic algebraic expressions expressions, , 𝑥𝑥 𝑥𝑥 𝑥𝑥 variable b, − sin (𝑥𝑥)+ 2sin (𝑥𝑥)− 2cos (𝑥𝑥) • expand(expression)ratsimp(expression)sin (𝑥𝑥 –)– developssimplifies2sin (𝑥𝑥) algebraic algebraic2cos (𝑥𝑥) expressions expressions3. T, ,he function be given3 by f(x,2 y) = . Let calculate its partial derivative: • expand(expression) – develops3 algebraic2 expressions, − 𝑥𝑥 + 𝑥𝑥 − 𝑥𝑥 2 2 • expand(expression)ratsimp(expression)radcan(expression)𝑥𝑥 – –– simplifies developssimplifies𝑥𝑥 algebraicexpression algebraic𝑥𝑥 expressions expressions, , , 𝑥𝑥 𝜕𝜕 𝑓𝑓 • expand(expression)− – +develops− algebraic expressions3., The function be given by f(x, y) = 3 . Let calculate its partial derivative: • radcan(expression)ratsimp(expression)3. The function – – simplifies simplifiesbe given byexpression algebraic f(x, y) = expressions, ,. Let calculate its partial derivative: 𝑦𝑦 2 𝜕𝜕𝜕𝜕𝜕𝜕2𝜕𝜕 • ratsimp(expression) – simplifies algebraic expressions(%i1)2, diff((x*y)+(x^2)/(y^3),x, 1, y, 𝑥𝑥1);∗ 𝑦𝑦2+ 𝑥𝑥 𝜕𝜕 𝑓𝑓 • radcan(expression)factor(expression)ratsimp(expression) – – –decomposes simplifies simplifies expression algebraic expression expressions, factors, 𝑥𝑥 , 𝜕𝜕 𝑓𝑓 3 • ratsimp(expression) – simplifies algebraic expressions3, 𝑥𝑥 ∗ 𝑦𝑦 + 𝑦𝑦 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 • factor(expression)radcan(expression) – – decomposes simplifies expression expression, 𝑥𝑥 factors,∗ 𝑦𝑦 +(%i1) 𝑦𝑦 diff((x*y)+(x^2)/(y^3),x, 1, y, 1);𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕 • factor(expression)trigexpand(expression)radcan(expression)(%i1) diff((x*y)+(x^2)/(y^3),x, – – decomposes simplifies – develops expression expression trigonometric 1, y,, 1);factors, expression(%o1) , • radcan(expression) – simplifies expression, 6𝑥𝑥 • trigexpand(expression)factor(expression) – decomposes – develops expression trigonometric factors, expression(%o1) , 4 • factor(expression)(%o1) – decomposes expression factors, 𝑦𝑦 • trigreduce(expression),trigexpand(expression) –trigsimp(expression) develops trigonometric – simplifies expressionTo calculate trigonometric1 −, the6𝑥𝑥 integrals expression both definite and indefinite we use integrate() command. • trigreduce(expression),trigexpand(expression)factor(expression) –6 𝑥𝑥decomposes –trigsimp(expression) develops expression trigonometric factors,– simplifies expression trigonometric, 4 expression 4 1 − 𝑦𝑦 • trigreduce(expression),trigexpand(expression)rectform – shows1 − the 𝑦𝑦 expression –trigsimp(expression) develops in trigonometricthe form A– simplifies+ Bexpression Toi. calculate trigonometric, the integrals expression both definite and indefinite we use integrate() command. • trigreduce(expression),trigexpand(expression)rectformTo – calculate shows the the expression integrals–trigsimp(expression) develops bothin trigonometricthe definite form A– and simplifies+ Bexpression iExamplen i.definite trigonometric 1.,we We use calculate integrate() expression command. • trigreduce(expression), trigsimp(expression) – simplifies trigonometriciθ expression7− 2𝑥𝑥 • polarformrectform – –shows shows the the expression complex expression in the form in A the + B pola Examplei. r form 1.re We calculate 2 • trigreduce(expression),Example 1. We calculate trigsimp(expression) – simplifies trigonometriciθ expression𝑥𝑥 +4 𝑥𝑥+13 • polarformrectform – –shows shows the the expression complex expression in the form in A the + B pola(%i1) i. r form integrate((7 re -2*x)/(x^2+4*x+13),x);∫ 7−2𝑥𝑥 𝑑𝑑𝑑𝑑 • logcontactpolarformrectform – ––shows showssimplifies the the expression complexthe expression expression in the7 −such form2𝑥𝑥 asin A thelog + B polax i.+ br formlog y. re iθ 2 2 ∫ 𝑥𝑥 +4𝑥𝑥+13 𝑑𝑑𝑑𝑑 • logcontactpolarformrectform – ––shows showssimplifies the the expression complexthe expression expression ∫in𝑥𝑥 the+ 4such form𝑥𝑥+13 asin 𝑑𝑑A𝑑𝑑 thelog + B polax (%i1) i.+ br formlog integrate((7 y. re iθ -2*x)/(x^2+4*x+13),x); • polarform(%i1) – shows integrate((7 the complex-2*x)/(x^2+4*x+13),x); expression in the polar form re iθ •Example: logcontactpolarform –– showssimplifies the complexthe expression expression such asin thelog polax(%o1) + br formlog y. re iθ •Example: logcontactpolarform –– showssimplifies the complexthe expression expression such asin thelog polax + br formlog11 y. re 2𝑥𝑥+4 • logcontact – simplifies the expression such as log x(%o1) + b log y. 2 •Example:(%i1) logcontact expand((b+c)^5);(%o1) – simplifies //the expands expression the expression such as log in x brackets + b log3 y.atan ( 6 ) − log (𝑥𝑥 + 4𝑥𝑥 + 13) • logcontact – simplifies the expression such as log xEach + b logstudent11 y. can 2easily𝑥𝑥+4 check the2 result by differentiating it and writing the following commands: (%i1)Example: expand((b+c)^5); 11 //2𝑥𝑥 expands+4 the expression2 in brackets Example:5 4 2 3 3 2 4 5 3 atan ( 6 ) − log (𝑥𝑥 + 4𝑥𝑥 + 13) (%i1)(%o1)Example: expand((b+c)^5);c +5*b*c 3+10*batan ( *c// 6expands+10*b) − log *cthe (𝑥𝑥+5*b expression+ 4*c+b𝑥𝑥 + 13 in) Eachbrackets student can easily check the result by differentiating it and writing the following commands: (%o1)(%i1)Example: expand((b+c)^5);c5Each+5*b*c student4+10*b can2 *c// easily expands3+10*b check3 *cthe2+5*b theexpression result4*c+b by5 in differentiating(%i2) brackets diff(%,x); it and writing the following commands: Now(%o1)(%i1) we expand((b+c)^5);c5 +5*b*cwill factorize4+10*b the2 *c// expandsexpression3+10*b3 *cthe2 +5*bby expression enter4*c+b the5 incommand (%i2)brackets :diff(%,x); (%i1) expand((b+c)^5);5(%i2) diff(%,x);4 2 // expands3 3 the2 expression4 5 in(%o2) brackets Now(%o1) we c5 +5*b*cwill factorize4+10*b the2*c expression3+10*b3*c2 +5*bby enter4*c+b the5 command: (%i2)(%o1) cfactor(%);5+5*b*c4+10*b 2*c3+10*b3*c2+5*b4*c+b5 11 2𝑥𝑥+4 Now we5 will factorize4 the2 expression3 3 2 by enter4 the5 command(%o2) : (2𝑥𝑥+4)2 2 (%o1) c(%o2)+5*b*c +10*b *c +10*b *c +5*b *c+b 𝑥𝑥 +4𝑥𝑥+13 Now(%i2) we factor(%); will factorize the expression by enter the command9:∗ ( 3611 +1) − 2𝑥𝑥+4 Now(%o2)(%i2) we (c+b)factor(%); will5 factorize 11 the expression2𝑥𝑥+4 by enter the command: (2𝑥𝑥+4)2 2 (2𝑥𝑥+4)2 2 (%i3) ratsimp(%); //− by𝑥𝑥 +this4𝑥𝑥+ command13 we get a simplified result Now we will5 factorize the expression+4𝑥𝑥+ 13byAdvances enter the in Sciencecommand and:9 ∗Technology( 36 +1) Research Journal vol. 8 (23) 2014 (%o2)(%i2) (c+b)factor(%); 9∗ ( 36 +1) − 𝑥𝑥 (%i2) factor(%);5 (%i3) ratsimp(%); // by this command we get a simplified result 4.(%o2)(%i2) Differentiation (c+b)factor(%); (%i3) ratsimp(%); and integration // by this command we (%o3)get a simplified result 4.(%o2)(%i2) Differentiation (c+b)factor(%);5 and integration DIFFERENTIATION(%o2) (c+b)5 AND INTEGRATION 2𝑥𝑥−7 4.The(%o2) Differentiation most (c+b) important and command integration to derive is diff (f (x), (%o3)x);(%o3) it r equires – 2 as the second argument the (%o3)5 𝑥𝑥 +4𝑥𝑥+13 4.The(%o2) Differentiation most (c+b) important and command integration to derive is diff (f (x), x); it r−equires2𝑥𝑥 −as7 the second argument the ∞ variable at which the2 𝑥𝑥function−7 f(x) is differentiated andExample can be 2. used 2Let tocalculate calculate the the definite partial integral derivatives. by writng command integrate(expression, 4. DifferentiationThe most important and2 commandintegration to derive is diff Example𝑥𝑥 2.+4 𝑥𝑥 Let+13 calculate the definite integral∞ 2 variableThe most at important which the command function f(x)to derive is differentiated is diff (f (x), and x); can it r equiresbe− used asto thecalculate second the argument partial derivatives. the − 𝑥𝑥 (fDiffThe4. (x),Differentiation mostcommand x); itimportant requires can− 𝑥𝑥 andhave +thecommand4𝑥𝑥 +integrationvariable a13 third to argument as derive the second is, definingdiff ar(f- (x), thevariable, Examplex); degree∞ it requires lower 2.of Letderivative aslimit, calculate the uppersecond. This the limit): argumentaredefinite s e lected integral the features by writng command integrate(expression, variableDiffThe mostcommand Exampleat important which can 2.the have Letcommand function calculatea third f(x)to argument derivethe is differentiateddefinite is, definingdiff integral (f (x), andthe x); degreecan it r equiresbe ofused derivativeby by as towriting writ thecalculate ngsecond. commandcommandThis the argumentare partial s integrate(expres-e integrate(expression,lected derivatives. the features ∫0 𝑒𝑒 2𝑑𝑑𝑑𝑑 gumentvariableTheof the most Maxima at at importantwhich which for the the the functioncommand function differential f(x) f(x)to isderive andis differentiated differentiated integral is diff calculus:(f (x), and variable,x); can it r equiresbe2 lowerused as tolimit, thecalculate secondupper the limit): argument partial derivatives. the −𝑥𝑥 variableDiffofThe the mostcommand Maxima variable,at important which canfor lowerthe thehave command function differential limit, a third upper f(x)to argument derive andis limit): differentiated integral is, definingdiff calculus:(f (x), andthe(%i1) x);sion, degreecan it −r variable,integrate(exp(equiresbe𝑥𝑥 ofused derivative as tolower thecalculate -second x*x),x,0,inf);limit,. This the upper argumentare partial se limit):lected derivatives. the features ∫0 𝑒𝑒 𝑑𝑑𝑑𝑑 andvariableDiff(n) cancommand atbe whichused can tothe have calculate function a third thef(x) argument partialis differentiated , derivadefining- andthe(%i1)∫ 0 degreecan𝑒𝑒 beintegrate(exp(-x*x),x,0,inf); 𝑑𝑑ofused𝑑𝑑 derivative to calculate. This the are partial selected derivatives. features variableDiffoff (x)the command -Maxima diff(f(x),x,n)at which canfor the thehave function differential a third f(x) argument andis differentiated integral, defining calculus: andthe(%i1) degreecan be integrate(exp( ofused derivative to calculate-.x*x),x,0,inf); This the are partial selected derivatives. features tives.Diff(n) command Diff(%i1) command canintegrate(exp( have can a havethird-x*x),x,0,inf); aargument third argument,, defining the degree of derivative. This are selected features Diffoff (x)the command -Maxima diff(f(x),x,n) canfor thehave differential a third argument and integral, defining calculus: the(%o1)(%o1) degree of derivative. This are selected features of(n) the Maxima for the differential and integral calculus: definingf (x) - diff(f(x),x,n) - theintegrate(f(x),x) degree of derivative. These are se- √𝜋𝜋 off(n) (x)the -Maxima diff(f(x),x,n) for the differential and integral calculus:(%o1) lected(n) features(%o1)- integrate(f(x),x) of the Maxima for the differential Expressions2 can be calculated numerically by f(n)(x) - diff(f(x),x,n) Expressions√𝜋𝜋 can be calculated numerically by using the float(expression) command, continuing the and(n) f( integral𝑥𝑥)𝑑𝑑𝑑𝑑 - i ntegrate(f(x),x)calculus:√𝜋𝜋 using the float(expression) command, continuing ∫f (x) - diff(f(x),x,n)integrate(f(x),x,a,b) or romberg(f(x),x,a,b) (calculate2 the integral numerically) ∫(n) f(𝑥𝑥)𝑑𝑑𝑑𝑑 - integrate(f(x),x)2 previousExpressionsthe previous example, can example, be we calculated get: we get: numerically by using the float(expression) command, continuing the f 𝑏𝑏(x) – diff(f(x),x,n)Expressions- integrate(f(x),x)integrate(f(x),x,a,b) can be calculated or romberg(f(x),x,a,b) numerically by using (calculate the float the(expression) integral numerically command) , continuing the ∫ f(𝑥𝑥)𝑑𝑑𝑑𝑑 - integrate(f(x),x) previous example, we get: 𝑏𝑏 -– i ntegrate(f(x),x)integrate(f(x),x) (%i2) float(%); ∫ 𝑎𝑎 ff( (𝑥𝑥𝑥𝑥))𝑑𝑑𝑑𝑑𝑑𝑑previous𝑑𝑑 − integrate(f(x),x,a,b) example, we get: or romberg(f(x),x,a,b)(%i2) (calculate float(%); the integral numerically) 𝑎𝑎 𝑏𝑏 f-( 𝑥𝑥diff(f(x,y),x))𝑑𝑑𝑑𝑑 integrate(f(x),x,a,b) or romberg(f(x),x,a,b)(%o2) (calculate 0.88622692545276 the integral numerically) ∫ ff( (𝑥𝑥𝑥𝑥))𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 −– integrate(f(x),x,a,b)integrate(f(x),x,a,b) or or romberg(f(x), romberg(f(x),x,a,b)(%i2) (calculate float(%); the integral numerically) ∫𝜕𝜕𝜕𝜕𝑎𝑎 𝑏𝑏 f-f( (𝑥𝑥diff(f(x,y),x)𝑥𝑥))𝑑𝑑𝑑𝑑𝑑𝑑(%i2)𝑑𝑑 − integrate(f(x),x,a,b) float(%); or romberg(f(x),x,a,b)(%o2) (calculate 0.88622692545276 the integral numerically ) 𝜕𝜕𝜕𝜕𝑏𝑏 - diff(f(x,y),x)integrate(f(x),x,a,b) or romberg(f(x),x,a,b) (calculateThe function the integral that performs numerically the integration) in ∫𝑎𝑎 f (𝑥𝑥)𝑑𝑑x,a,b)𝑑𝑑 − (calculate the integral numerically) (%o2) 0.88622692545276 ∫𝜕𝜕𝜕𝜕𝑎𝑎𝑏𝑏 -f (diff(f(x,y),x)diff(f(x,y),y)𝑥𝑥)𝑑𝑑𝑑𝑑 − a numerical way is romberg. The Romberg () and ∫ -f (diff(f(x,y),x)𝑥𝑥)𝑑𝑑(%o2)𝑑𝑑 − 0.88622692545276 ∫𝜕𝜕𝜕𝜕𝑎𝑎 -–f ( diff(f(x,y),x)diff(f(x,y),y) diff(f(x,y),x)𝑥𝑥)𝑑𝑑𝑑𝑑 − integrate () handles well with multiple integrals 𝜕𝜕𝜕𝜕 - diff(f(x,y),x) 𝜕𝜕𝜕𝜕 - diff(f(x,y),y) (not only with the constant limits), which will 𝜕𝜕𝜕𝜕 –- - diff(f(x,y),y) diff(f(x,y),y)diff(f(x,y),x,2) 2 - diff(f(x,y),y) surely be useful for students during calculating Examples.𝜕𝜕𝜕𝜕𝜕𝜕 𝑓𝑓 - -diff(f(x,y),y) diff(f(x,y),x,2) The function that performs the integration in a numerical way is romberg. The Romberg () and 𝜕𝜕𝜕𝜕2 2- diff(f(x,y),y) 𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥𝑓𝑓 -– diff(f(x,y),x,2) diff(f(x,y),x,2) fields, the volumes of solids and their surfaces. In 1.𝜕𝜕𝜕𝜕2 Let2 f(x) = xx be the function which derivative we try to calculate, thenintegrate we obtain: () handles well with multiple integrals (not only with the constant limits), which will surely 1.Examples.𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥 Let𝑓𝑓Examples. - diff(f(x,y),x,2)diff(f(x,y),y,2)f(x) = x be the function which derivative we tryaddition, to calculate anyone, then can we do obtain: two or three-dimensional 𝜕𝜕𝜕𝜕2 2 - diff(f(x,y),x,2) 𝜕𝜕2𝑓𝑓 -– diff(f(x,y),x,2) diff(f(x,y),y,2) The function that beperforms useful forthe studentsintegration during in a calculatingnumerical wayfields, is rtheomberg. volume Thes of Romberg solids and () theirand surfaces. In addition, 𝜕𝜕Examples.2𝑥𝑥 - diff(f(x,y),y,2) graphs, (we will explain it in chapter 9), this in (%i1)𝜕𝜕 𝑓𝑓2 - diff(f(x,y),x,2) diff(x^x,x);x x 1.𝜕𝜕2𝑦𝑦𝑥𝑥 Let𝑓𝑓1.2 - Letdiff(f(x,y),y,2) f(x) f(x) = x = bex bethe the function function which which derivative derivative we we tryThe try to calculatetofunction calculate ,that then, thenanyone performs we we obtain: canobtain: the do int twoegration or three in- dimensionala numerical waygraphs, is r omberg.(we will Theexplain Romberg it in c hapter() and 9), this in combination 𝜕𝜕𝑥𝑥2𝑓𝑓2Examples. x combinationintegrate () handles with symbolic well with calculations multiple integrals made in (not only with the constant limits), which will surely 𝜕𝜕1.𝑦𝑦𝑥𝑥 𝑓𝑓2Let -Examples. diff(f(x,y),y,2) f(x) -– diff(f(x,y),x,k,y,n) diff(f(x,y),x,k,y,n) = x be the function which derivative we tryThe to functioncalculate that, then performs we obtain: the integration in a numerical way is romberg. The Romberg () and 2𝑘𝑘2+𝑛𝑛 x integrate () handleswith well symbolic with multiple calculations integrals made (not in onlyMaxim with actively the constant growing limits), and developingwhich will spatialsurely imagination. We (%o1)(%i1)𝜕𝜕𝑥𝑥2𝑓𝑓 - diff(f(x,y),y,2) xdiff(x^x,x);*(log(x)+1)x Maximbe useful actively for students growing during and calculating developing fields, spatial the volumes of solids and their surfaces. In addition, 𝜕𝜕𝜕𝜕𝑦𝑦2(%i1)1. 𝑓𝑓Let - diff(f(x,y),x,k,y,n)f(x)diff(x^x,x); = x be the function which derivative weintegrateThe try tofunction calculate () handles that, then performs well we with obtain: the multiple int egration integrals in a numerical (not only waywith isthe romberg. constant The limits), Romberg which () will and surely 𝜕𝜕 𝑘𝑘𝑓𝑓𝑘𝑘2+ 𝑛𝑛- diff(f(x,y),y,2)𝑛𝑛 x be useful for students during calculating fields, the volumes of solids and their surfaces. In addition, Examples:𝜕𝜕(%i1)𝜕𝜕𝑦𝑦𝑥𝑥2 2𝜕𝜕𝑦𝑦1.𝑓𝑓 Letdiff(x^x,x);- diff(f(x,y),x,k,y,n) f(x) = x be the function which derivative weimagination.anyone try to can calculate do We two calculate:calculate, thenor three we: -obtain:dimensional graphs, (we will explain it in chapter 9), this in combination 𝜕𝜕𝑦𝑦𝑘𝑘𝑓𝑓𝑘𝑘2+𝑛𝑛 𝑛𝑛 Thex x function that performs the integration in a beintegratenumerical useful ()for way handles students is romberg. well during with The calculating multiple Romberg integrals fields, () and the(not volume only withs of thesolids constant and their limits), surfaces. which In will addition, surely 2.(%o1)𝜕𝜕𝜕𝜕𝑦𝑦𝑥𝑥 L(%o1)2𝜕𝜕e𝑦𝑦t𝑓𝑓 the x - *(log(x)+1) diff(f(x,y),x,k,y,n) xfunction*(log(x)+1)x be given by f(x) = . Calculationanyone derivative can do of two the orsecond three -degreedimensional2 6 𝑥𝑥 in Maxima graphs, (we will explain it in chapter 9), this in combination 1. 𝑘𝑘 (%i1)+Let𝑛𝑛 f(x)x diff(x^x,x); = x be the function whose derivative (%i1)with symbolic g(x,y):=x/(y^2); calculations made in2 Maxim actively growing and developing spatial imagination. We 𝜕𝜕(%o1)𝑦𝑦𝑘𝑘 𝑛𝑛 integratex- diff(f(x,y),x,k,y,n)*(log(x)+1) () handles well with multiplesin (𝑥𝑥) integralsbe (not useful only for with students the constant during1 limits),calculating4 which fields, will thesurely volume s of solids and their surfaces. In addition, 𝜕𝜕𝜕𝜕𝑥𝑥𝑘𝑘+𝜕𝜕𝑛𝑛𝑦𝑦(%i1)𝑓𝑓 - diff(f(x,y),x,k,y,n) diff(x^x,x); anyonewith symbolic can do calculationstwo or three -∫madedimensional∫ 𝑦𝑦in 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑Maxim graphs, actively (we growing will explain and itdeveloping in chapter spatial 9), this imagination. in combination We looks𝜕𝜕 𝑘𝑘we 𝑓𝑓as𝑛𝑛 try - follows: diff(f(x,y),x,k,y,n) tox calculate, then we obtain: (%i2) romberg(romberg(g(x,y),y,4,6),x,1,2);(%i1) g(x,y):=x/(y^2); 2.𝜕𝜕𝜕𝜕𝑥𝑥 𝑘𝑘L𝑘𝑘(%o1)+𝜕𝜕e𝑛𝑛𝑦𝑦t𝑓𝑓 𝑛𝑛thebe xfunctionuseful*(log(x)+1) for be students given by during f(x) = calculating 𝑥𝑥 . Calculation fields,calculateanyone the derivative volume can: dos of two thesolids orsecond three and -theirdegreedimensional surfaces. in Maxima graphs, In addition, (we will explain it in chapter 9), this in combination 2.𝑘𝑘 L𝑛𝑛et the functionx be given by f(x) = . Calculationwith symbolic derivative calculations of the second made degree in Maxim in Maxima actively growing and developing spatial imagination. We 𝜕𝜕𝜕𝜕𝑥𝑥 (%i1)𝜕𝜕𝑦𝑦(%o1)𝑓𝑓 diff(x^x,x); x *(log(x)+1) sin (𝑥𝑥) (%o2)calculate 0.12500002874223: 2 6 𝑥𝑥 (%i1)2. 𝑘𝑘Le t 𝑛𝑛 diff(sin(x)/x,x,2);theanyone functionx can dobe twogiven or by three f(x)-dimensional = sin (.𝑥𝑥 Calculation) graphs,with (we derivative symbolic will explain ofcalculations the2 it secondin chapter made degree 9) in, this inMaxim Maxima in combination actively growing and developing spatial imagination. We looks 𝜕𝜕𝑥𝑥looks(%o1)𝜕𝜕𝑦𝑦 as asfollows: x follows:*(log(x)+1) calculate: 12 46 𝑦𝑦𝑥𝑥(%i2) romberg(romberg(g(x,y),y,4,6),x,1,2); sin𝑥𝑥 (𝑥𝑥𝑥𝑥) ∫ ∫ 2 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 looks2. L asewitht follows:the symbolic function becalculations given by f(x)made = in Maxim. Calculation actively(%i1) growingderivative g(x,y):=x/(y^2);1 4 and of developingthe second degreespatial imagination.in Maxima We 2. Let2. Ltheet thefunction function be begiven given by by f(x) f(x) ==𝑥𝑥 .. Calculationcalculate derivative: ∫2 ∫6 𝑦𝑦𝑥𝑥 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑of the second degree in Maxima (%o1)(%i1)(%i1) diff(sin(x)/x,x,2); diff(sin(x)/x,x,2); sin (𝑥𝑥) (%i1) g(x,y):=x/(y^2);(%o2)2 0.12500002874223 looks as follows: sin (𝑥𝑥) SYSTEMS ∫1 2OF∫46 EQUATIONS𝑦𝑦𝑥𝑥 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Calculationcalculatesin (𝑥𝑥) derivative:2sin (𝑥𝑥) of2cos the (𝑥𝑥 )second degree𝑥𝑥 in (%i2)(%i1) romberg(romberg(g(x,y),y,4,6),x,1,2); g(x,y):=x/(y^2);2 (%i1)looks diff(sin(x)/x,x,2);sin as (𝑥𝑥 )follows:2sin (𝑥𝑥 ) 2cos (𝑥𝑥) 1 4 Maxima looks as2 3follows:6 𝑥𝑥 2 𝑥𝑥 (%i2) romberg(romberg(g(x,y),y,4,6),x,1,2); ∫ ∫ 𝑦𝑦5.𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Systems of equations (%o1)(%o1)(%i1) − diff(sin(x)/x,x,2);𝑥𝑥 + 𝑥𝑥 −2 𝑥𝑥 (%i1) g(x,y):=x/(y^2); 3. The function be ∫ 1given∫4 𝑦𝑦 by𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 f(x, y) = . Let calculate(%o2)(%i2)Some romberg(romberg(g(x,y),y,4,6),x,1,2); 0.12500002874223 its useful partial functions derivative: for solving equations (%o1)(%i1)(%i1) (%i1) sindiff(sin(x)/x,x,2); (sindiff(sin(x)/x,x,2);𝑥𝑥 ) (g(x,y):=x/(y^2);𝑥𝑥) 2sin2 (sin𝑥𝑥) (𝑥𝑥) 2cos2 cos ( 𝑥𝑥) ( 𝑥𝑥) 2 2 3 3 2 2 𝑥𝑥 and(%o2)(%i2) systems romberg(romberg(g(x,y),y,4,6),x,1,2); 0.12500002874223 of equations:Some useful f𝜕𝜕unctions 𝑓𝑓 for solving equations and systems of equations: (%o1)− sin–𝑥𝑥 (𝑥𝑥𝑥𝑥)+ 2sin𝑥𝑥 (𝑥𝑥𝑥𝑥)− 2cos𝑥𝑥 (𝑥𝑥𝑥𝑥) 3 (%o1) − + 3 − 2 5.(%o2) Systems 0.12500002874223 of equations (%i1)3. The diff((x*y)+(x^2)/(y^3),x,function(%i2)𝑥𝑥 romberg(romberg(g(x,y),y,4,6),x,1,2); be given𝑥𝑥 by f(x,𝑥𝑥 y)1, y,= 𝑥𝑥1);∗ 𝑦𝑦 + 𝑦𝑦 . Let calculate•• solve its(equation, partial derivative: variable) 𝜕𝜕–𝜕𝜕 𝜕𝜕solves𝜕𝜕 the equa- (%i1)3. (%o1)T diff((x*y)+(x^2)/(y^3),x,he− function sin (+𝑥𝑥) be2 singiven (−𝑥𝑥) by2 cosf(x, 1, ( 𝑥𝑥y,y)) 1);= 2 2. Let calculate its partial derivative:2 2 3 2 𝑥𝑥 5.(%o2) Systems 0.12500002874223 of equations• solve (equation, 𝜕𝜕 𝑓𝑓 variable) - solves the equation in the set of complex numbers, 3.3. TThehe functionsin𝑥𝑥 be (𝑥𝑥) given2𝑥𝑥sin by (𝑥𝑥) f(x,2𝑥𝑥 cosy)y) = (=𝑥𝑥 ) 2𝑥𝑥. Let calculatetion in its the partial set of derivative:complex numbers, 2𝜕𝜕 𝑓𝑓 − + 3 − 2 3 3 Some5. Systems useful of functions equations• linsolve for solving ([equations], equations [variables]) and systems - solves of equations: a linear system of equations, (%i1)3.Let T diff((x*y)+(x^2)/(y^3),x,hecalculate(%o2) function 0.12500002874223 𝑥𝑥its be partial given𝑥𝑥 derivative: by f(x, 1,𝑥𝑥 y,y) 𝑥𝑥 1);=∗ 𝑥𝑥 𝑦𝑦∗+𝑦𝑦 +𝑦𝑦𝑥𝑥 𝑦𝑦 . Let• calculate its partial derivative:𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑓𝑓𝜕𝜕𝜕𝜕𝜕𝜕 (%o1)(%i1) diff((x*y)+(x^2)/(y^3),x,− + − 1, y, 1); 3 2 Some5.• linsolveSystems useful ([equations],of functions equations for [variables]) solving equations2 – solves and a systems of equations: 3. The function be given by f(x, y) = 𝑦𝑦 . Let calculate its partial• algsys derivative: ([equations],𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 [variables]) - solves algebraic system equations in the set of complex numbers, if (%i1)(%i1) diff((x*y)+(x^2)/(y^3),x, diff((x*y)+(x^2)/(y^3),x,6𝑥𝑥 1, 1,y, 𝑥𝑥y,1); ∗1); 𝑦𝑦 + 𝑥𝑥 2 •Some solve useful (equation, functions variable) for solving - solves𝜕𝜕 equations the𝑓𝑓2 equation and insystems the set of of equations: complex numbers, 4 3 linear system of equations, 5. Systems of equations 𝑦𝑦 𝑥𝑥 you want to receive𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 solution𝑓𝑓 in the real numbers , first you should write the following command: (%o1) (%o1)(%i1)(%o1) 1 − diff((x*y)+(x^2)/(y^3),x, 𝑦𝑦 1, y, 𝑥𝑥1);∗ 𝑦𝑦 + 3 •Some •linsolvesolvealgsys useful (equation, ([equations], functions variable) for[variables]) solving - solves equations –-the solvessolves equation a algeand linear -insystems thesystem set of of of equations: complex equations, numbers, To calculate the integrals both definite and𝑥𝑥 in∗definite𝑦𝑦 + 𝑦𝑦 we use integrate() command. 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 (%o1)(%i1) diff((x*y)+(x^2)/(y^3),x,6𝑥𝑥 6 𝑥𝑥 1, y, 1); • algsyssolvelinsolve (equation, ([equations], ([equations],realonly: variable) [variables]) [variables]) true, - solves - so -the solveslves equation algebraic a linear in thesystem set of equationsof complex equations, in numbers, the set of complex numbers, if Some 4useful4 functions for solving equations and systemsbraic system of equations: equations in the set of complex (%o1)To calculate1 − 1 −𝑦𝑦6𝑥𝑥 𝑦𝑦 the integrals both definite and in- • linsolvealgsyssolve (equation, ([equations], ([equations],• variable)find_root [variables]) [variables]) - (fsolves (x), - x,so -the solveslvesa, equationb) algebraic– a this linear commandin thesystem set of findofequations complex equations, (through in numbers, the the set approximation) of complex numbers, zeros of ifthe function f in ExampleToTo calculate calculate 1. We the4 thecalculate integrals integrals both both definite definite and and indefinite indefinite weyou we numbers,use want use integrate() integrate()to receiveif you command. want solution command. to receive in the solutionreal numbers in the , first you should write the following command: definite(%o1)•1 solvewe− use𝑦𝑦 (equation,6 integrate()𝑥𝑥 variable) command. - solves the equation• algsyslinsolve in the ([equations],set ([equations], of complexthe interval [variables]) [variables])numbers, [a, b ]- (so-notice solveslves algebraicthat a linear f (a), systemf (b) should equationsof equations, have in different the set ofsigns). complex numbers, if To calculate the 4integrals both7 −definite2𝑥𝑥 and indefinite realonly:youwereal usewant integrate()numbers true,to receive , firstcommand. solution you inshould the real write numbers the fol -, first you should write the following command: 𝑦𝑦 6𝑥𝑥 2 Example• linsolve1. 1.1 We −We calculatecalculate ([equations],4 𝑥𝑥 + [variables])4𝑥𝑥+13 𝑑𝑑𝑑𝑑 - solves a linearyourealonly:• algsys want system ([equations], true,to receiveof equations, solution [variables]) in the - realsolves numbers algebraic , first system you shouldequations write in thethe followingset of complex command: numbers, if (%i1)ExampleTo calculate integrate((7 1.1 −We the 𝑦𝑦 calculate- integrals2*x)/(x^2+4*x+13),x);∫ both definite𝑑𝑑𝑑𝑑 and indefinite• find_root welowing use integrate()command: (f (x), x, a, command.realonly: b) – this true,command find (through the approximation) zeros of the function f in ExampleTo• calculatealgsys 1. We ([equations], calculate the integrals [variables]) 7both−27𝑥𝑥− 2definite𝑥𝑥 - so andlves i nalgebraicdefinite•realonly:you •find_rootfind_root wantwe system use true,to (f integrate() receiveequations (x),(f (x),For x, solutiona, x,example, b)command.ina, –theb) this in–set this tocommandthe of find realcomplexcommand a numberssolution find numbers, find(through to, first the ifysystem theo u shouldapproximation) of equations write the :zerosfollowing of the command: functionwe write: f in (%i1) integrate((7-2*x)/(x^2+4*x+13),x);2 2 the interval [a, b] (notice that f (a), f (b) should have different signs). Exampleyou want 1. We to calculate receive∫ solution𝑥𝑥 +𝑥𝑥74−𝑥𝑥+2+4𝑥𝑥13𝑥𝑥 in+𝑑𝑑13 the𝑑𝑑𝑑𝑑𝑑𝑑 real numbers•therealonly: find_root , firstinterval y otrue,u (f [a,should (x), b] (x,notice writea, b) that–the this followingf (a)command, f (b) command: should find (through have different the approximation) signs). zeros of the function f in (%o1)(%i1)(%o1)(%i1) integrate((7 integrate((7-2*x)/(x^2+4*x+13),x);-2*x)/(x^2+4*x+13),x);∫2 (through the approximation) zeros of the func- 9𝑥𝑥 − 8𝑦𝑦 = 4 Examplerealonly: 1. Wetrue, calculate ∫ 𝑥𝑥 +4𝑥𝑥7+−132𝑥𝑥𝑑𝑑𝑑𝑑 the• find_root interval (f[a, (x), b] (x,notice a, b) that– this f (a)command, f (b) should find (through have different the approximation) signs). zeros{ of the function f in (%i1) 11 integrate((72𝑥𝑥+4-2*x)/(x^2+4*x+13),x);2 Fortion example, f in the to intervalfind(%i1) a solution r1:[a, b] (notice to the systemthat f (a), ;of equationsf : we7 write:𝑥𝑥 + 2 𝑦𝑦 = 3 Each student can easily ∫check2𝑥𝑥 +4 7𝑥𝑥the−+213𝑥𝑥 result𝑑𝑑𝑑𝑑 by the interval [a, b] (notice that f (a), f (b) should have different signs). (%o1)(%i1) •3 find_root atan integrate((7( 6 (f )(x),-−2*x)/(x^2+4*x+13),x); logx, a, (𝑥𝑥 b) +– 2this4𝑥𝑥 + command13) find For(through(b) example, should the approximation) tohave find different a solution signs). zeros to the of systemthe function of equations f in : we write: Each(%o1) (%i1)student integrate((7 can easily check-2*x)/(x^2+4*x+13),x); the∫ 𝑥𝑥 result+4𝑥𝑥+ 13by𝑑𝑑 differentiating𝑑𝑑 it and writing the following commands: 9𝑥𝑥 − 8𝑦𝑦 = 4 differentiating(%o1) the11 interval it2 and𝑥𝑥+ [a,4 writing b] (notice the that following f (a), f (b)com should- For have example, different to signs).find(%o1) a solution 9 ∗ 𝑥𝑥 to− the8 ∗ system𝑦𝑦 = 4 of equations : { we write: 11 2𝑥𝑥+4 2 2 (%i1) r1: ; 79𝑥𝑥 +− 28𝑦𝑦 = 34 mands: 311atan ( 2𝑥𝑥6+4) − log (𝑥𝑥 + 4𝑥𝑥 + 13) For Forexample, example, to find to finda solution a solution to the to system the system of equations : { we write: (%i2)EachEach(%o1) student diff(%,x); student 3 atan can can easily( easily6 )check− checklog the (2𝑥𝑥 the result+ result4𝑥𝑥 by+ by13differentiating differentiating) (%i1) it r1:and it and writing writing the the following following; commands: commands: 97𝑥𝑥 −+ 82𝑦𝑦 = 43 (%i2)(%o1) diff(%,x);3 11atan ( 62𝑥𝑥)+4− log (𝑥𝑥 + 4𝑥𝑥 + 13) 9𝑥𝑥 − 8𝑦𝑦 = 4 { Each studentFor example, can easily to find check a solution the2 result to bythe differentiating system ofof(%i1) equations equations it r1: and 9 writing∗: 𝑥𝑥 −(%i2)8 the∗ 𝑦𝑦 following =r2:4 ;we we write: write: commands: ; 79𝑥𝑥 +− 28𝑦𝑦 = 34 11 2𝑥𝑥+4 (%o1) { (%o2)(%i2)(%o2)(%i2)Each diff(%,x); student diff(%,x);3 atan can ( easily6 ) − checklog (𝑥𝑥 the+2 result4𝑥𝑥 + by13 differentiating) (%i1)(%o1)(%i1) r1: r1: it and9 ∗ 𝑥𝑥writing9−𝑥𝑥 −8 ∗8 𝑦𝑦𝑦𝑦the== following44; commands: 7𝑥𝑥 + 2𝑦𝑦 = 3 (%i2)Each diff(%,x); student113 atan can( easily6 2𝑥𝑥)+−4 checklog (𝑥𝑥 the+ result4𝑥𝑥 + by13 differentiating) it and{ writing(%o2) the 7following ∗ 𝑥𝑥 + 2 ∗ commands:𝑦𝑦 = 3 (%i1)(2𝑥𝑥+ 4r1:)2 2 ; (%o1)(%i2) r2:9𝑥𝑥 9−∗8𝑥𝑥7𝑦𝑦−𝑥𝑥=+84∗2 𝑦𝑦𝑦𝑦 == 34; − 𝑥𝑥 +4𝑥𝑥+13 (%o2)(%i3)(%o2)(%i2) ratsimp(%); 9∗ ( diff(%,x);36 +1) //− by this command we get a (%i2)(%o1) r2: 9𝑥𝑥 9−∗8𝑥𝑥𝑦𝑦−=84∗ 𝑦𝑦 = 4;; (%i3)(%o2)simplified(%i2) ratsimp(%); 11diff(%,x); result11 // by2 this𝑥𝑥+24𝑥𝑥 +command4 we get a simplified result (%i3) 7s:solve([r1,r2],[x,y]);𝑥𝑥 + 2𝑦𝑦 = 3 (2𝑥𝑥+4)2 92∗ 𝑥𝑥 −2 8 ∗ 𝑦𝑦 = 4 (%i2) r2:9𝑥𝑥 7−∗8𝑥𝑥𝑦𝑦+=24∗ 𝑦𝑦 = 3; (%o1)(2 𝑥𝑥+4) +42𝑥𝑥 +13 (%o2) 9∗( 3611 +1) − 𝑥𝑥 𝑥𝑥2𝑥𝑥+44𝑥𝑥+13 (%o2) 9(∗2(𝑥𝑥+436)2 +1) −2 (%o2)(%i2) r2:9𝑥𝑥 7−∗8𝑥𝑥𝑦𝑦+=24∗ 𝑦𝑦 = 3; 𝑥𝑥 +4𝑥𝑥+13 (%o3) [[x=16/37,y=-1/74]] (%o3) (%i3) (%i3)(%o2) 9ratsimp(%); ∗ ( ratsimp(%); 36 11+1 ) −// by // bythis2 this𝑥𝑥 +command4 command we we get get a simplified a simplified result result (%i2) (r2:29𝑥𝑥𝑥𝑥+4−)2 8𝑦𝑦 = 42 ; (%o2)(%i3) 7s:solve([r1,r2],[x,y]);𝑥𝑥 7+∗2𝑥𝑥𝑦𝑦+=23∗ 𝑦𝑦 = 3 (%i3) ratsimp(%);2𝑥𝑥−7 11 // by𝑥𝑥 this+42𝑥𝑥 𝑥𝑥command++134 we get a simplified result 47 92∗( (362𝑥𝑥+4+)21) − 2 7 ∗ 𝑥𝑥 + 2 ∗ 𝑦𝑦 = 3 2 𝑥𝑥 +4𝑥𝑥+13 (%o2)(%i3) 7s:solve([r1,r2],[x,y]);𝑥𝑥 + 2𝑦𝑦 =Checking3 the solution. (%o3) (%i3) − 𝑥𝑥 ratsimp(%);+94∗𝑥𝑥(+1336 +1 )//− by this command we ∞get a simplified result Example(%o3)(%o2) 2. Let calculate 7 ∗ 𝑥𝑥 + 2the∗ 𝑦𝑦definite= 3 integral (%o3) (%i3) by writ[[x=16/37,y=7s:solve([r1,r2],[x,y]);𝑥𝑥 +ng2 command𝑦𝑦 =The3-1/74]] following integrate(expression, command substitute the result into to both equations. The subst() command put the (%o3) (%i3) 2 𝑥𝑥ratsimp(%);−27𝑥𝑥−7 // by this command we get 2a simplified result 2 (%o3) [[x=16/37,y=7𝑥𝑥 + 2𝑦𝑦 = 3-1/74]] 2 −𝑥𝑥 (%i3) s:solve([r1,r2],[x,y]);expression in the first argument into the expression the second argument variable,(%o3)− lower 𝑥𝑥 +𝑥𝑥24𝑥𝑥𝑥𝑥+ −+limit,4713𝑥𝑥+13 upper limit): ∞ ∞ Checking the solution. Example 2.− 2Let calculate the definite integral ∫0 𝑒𝑒 𝑑𝑑(%o3)𝑑𝑑 by [[x=16/37,y=writng command-1/74]] integrate(expression, Example(%o3) (%i3)𝑥𝑥 2.+ 4 7s:solve([r1,r2],[x,y]);Let𝑥𝑥𝑥𝑥+13+ calculate2𝑦𝑦 = 3 the definite integral ∞ by writng command integrate(expression, − 2𝑥𝑥−7 2 TheChecking(%o3)2 following [[x=16/37,y= the solution. comma-1/74]]nd substitute the result into to both equations. The subst() command put the (%i1)Example integrate(exp( 2. Let2 calculate2𝑥𝑥−7-x*x),x,0,inf); the definite integral −𝑥𝑥 −𝑥𝑥 by writng command (%i4) [subst(s,r1),subst(s,r2)]; integrate(expression, variable,variable, lower− lower𝑥𝑥 +limit,4𝑥𝑥 limit,+13 upper upper limit): limit): ∞ 2Checking the solution. (%o3) [[x=16/37,y=2 -1/74]] ∫0 𝑒𝑒0 𝑑𝑑expressionThe𝑑𝑑 following in the comma first argumentnd substitute into the the result expression into to the both second equations. argument The subst() command put the variable,Example lower −2. 𝑥𝑥Let limit,+4 calculate𝑥𝑥+ 13upper limit):the definite integral ∫ −𝑒𝑒∞𝑥𝑥 Checking𝑑𝑑𝑑𝑑 by writthe solution.ng command integrate(expression, (%i1)Example integrate(exp( 2. Let calculate-x*x),x,0,inf); the definite integral∫0 𝑒𝑒 Theexpression𝑑𝑑2𝑑𝑑 following by iwritn the commang (%o4)first command ndargument [4=4,3=3] substitute integrate(expression, in to the the result expression into to the both second equations. argument The subst() command put the (%o1)(%i1)variable, integrate(exp( lower limit, -upperx*x),x,0,inf); limit): −𝑥𝑥 2 Checking the solution. 0 expressionThe(%i4)−𝑥𝑥 following [subst(s,r1),subst(s,r2)]; in the comma first argumentnd substitute into the the result expression into to the both second equations. argument The subst() command put the (%i1)variable, integrate(exp(√𝜋𝜋 lower limit,-x*x),x,0,inf); upper limit): ∫ 𝑒𝑒 𝑑𝑑𝑑𝑑 4 The following command substitute the result∫0 into𝑒𝑒 expression(%i4) to𝑑𝑑 both𝑑𝑑 [subst(s,r1),subst(s,r2)]; equations.in the Solutionsfirst Theargument subst() do not in commandhaveto the to expression be put a real. the Forthe secondequation argument x -1 = 0, we obtain: (%o1)(%o1)(%i1) 2 integrate(exp( -x*x),x,0,inf); (%o4) [4=4,3=3] Expressions(%i1)expression integrate(exp(can be i ncalculated the first-x*x),x,0,inf); argument numerically in to by the using expression the (%i4) float the (expression)[subst(s,r1),subst(s,r2)]; second argument command , continuing the (%o1) √ 𝜋𝜋 √ 𝜋𝜋 (%o4) [4=4,3=3] (%i1) solve(x^4-1); previous example, we get: (%i4) [subst(s,r1),subst(s,r2)]; 4 (%o1)√ 2 𝜋𝜋 2 Solutions(%o4) [4=4,3=3] do not have to be a real. For equation x -1 = 0, we obtain: ExpressionsExpressions (%i4) can [subst(s,r1),subst(s,r2)]; can be becalculated calculated numerically numerically by byusing using the the float float(expression)(expression) command command, continuing, continuing the the 4 (%i2)(%o1) float(%);2 √ 𝜋𝜋 Solutions(%o4) [4=4,3=3] do not have (%o1) to [x=%i,x=be a real. -For1,x= equation-%i,x=1] x -1 = 0, we obtain: previousExpressionsprevious example, example, can be we calculated we get: get: numerically by using the(%i1) float solve(x^4(expression)-1); command , continuing the 4 (%o4)2√ [4=4,3=3]𝜋𝜋 Solutions do not have to be a real. For equation x -1 = 0, we obtain: previousExpressions example, can webe calculatedget: numerically by using(%i1) the float solve(x^4(expression)-Functions1); command realroots, continuing (W), realroots the 4 (W, accuracy) find by bisection method all real roots of a (%o2)(%i2)Expressions float(%); 0.886226925452762 can be calculated numerically by usingSolutions the float do(expression) not have to commandbe a real. ,For continuing equation the x -1 = 0, we obtain: (%i2)previous float(%); example, we get: (%o1)(%i1)4 solve(x^4[x=%i,x=--1);1,x= -%i,x=1] (%i2) float(%);Solutions do not have to be a real. For equation x -1 = 0, we obtain:polynomial. Polynomial coefficients must be rational numbers. The second parameter defines the previous example, we get: (%o1)(%i1) solve(x^4[x=%i,x=--1);1,x= -%i,x=1] -7 (%o2)(%o2)(%i2) 0.88622692545276 float(%); 0.88622692545276 Functions realrootsaccuracy (W), realroots with which (W, theaccuracy) root is searchedfind by bisection- default methodis 10 . Inall addition, real roots each of a student can calculate roots (%i1) solve(x^4-1); (%o1) [x=%i,x=-1,x=-%i,x=1] (%o2)(%i2) 0.88622692545276 float(%); polynomial.Functions(%o1) [x=%i,x= realroots Polynomial-1,x= (W),-%i,x=1] coefficientsrealroots (W, must accuracy) be rational find numbers.by bisection The method second all parameter real roots defines of a the (%o2) 0.88622692545276 Functionspolynomial. realroots Polynomial (W), coefficientsrealroots (W, must accuracy) be rational find numbers.by-7 bisection The method second all parameter real roots defines of a the (%o2)(%o1) 0.88622692545276 [x=%i,x=-1,x=-%i,x=1] accuracy with which the root is searched- default is 10 . In addition, each student can calculate roots polynomial.accuracyFunctions with realroots Polynomial which (W), the coefficientsrootrealroots is searched (W, must accuracy)- defaultbe rational findis 10 numbers.by-7. Inbisection addition, The method second each studentall parameter real canroots definescalcula of a te the roots Functions realroots (W), realroots (W, accuracy)accuracypolynomial. find by with bisection Polynomial which method the coefficientsroot all is realsearched roots must- of defaultbe a rational is 10 numbers.-7. In addition, The second each student parameter can calculadefineste the roots polynomial. Polynomial coefficients must be rationalaccuracy numbers. with which The thesecond root parameteris searched defines- default the is 10-7. In addition, each student can calculate roots accuracy with which the root is searched- default is 10-7. In addition, each student can calculate roots eg. quadratic equation (with parameters a, b, c: ax2 + bx + c = 0), and thus verify that the formulas learned in school are rely true, and they are zeros above quadratic function. Additionally, you can also check the correctness of Viete'a formulas, which improves memorization and provides impetus for deepening self-knowledge. 6. Differential Equations The function that solves ordinary differential equations of the first and second order is ode2 (eqn, zvar, nvar), where zvar is the dependent variable and nvar is the independent variable. To solve the differential equation we will use the operator: 'diff – which introduces a symbol of the derivative and does not perform differentiation, but prepare derivative which later is part of the differential equation.

For example, we solve the differential equation 𝑑𝑑𝑑𝑑 2 2 2 (%i1) rr:'diff(y,t,1)=3*(t*y)^2-y^2*sin(t); 𝑑𝑑𝑑𝑑 = 3𝑡𝑡 𝑦𝑦 − sin (𝑡𝑡)𝑦𝑦

(%o1) 𝑑𝑑𝑑𝑑 2 2 2 (%i2) ro:ode2(rr,y,t);𝑑𝑑𝑑𝑑 = 3𝑡𝑡 𝑦𝑦 − sin// solution (𝑡𝑡)𝑦𝑦 of the differential equation Advances in Science and Technology Research Journal vol. 8 (23) 2014 (%o2) 1/y=-cos(t)-t3+%c //notice that %c is constant value (%i3) s:solve([r1,r2], [x,y]); (%i3)equation rs:ic1(ro,t=0,y=1); and we get: // initial condition y(0)=1 (%o3) [[x=16/37, y=-1/74]] (%o4) separable // variable-separated (%o3) 1/y=-cos(t)-t3+2 If Maxima cannot solve a differential equa- Checking the solution (%i4)tion, as method; a result // returns We can false. ask about the type of the equation and we get: The following command substitute the result (%o4) separable // variable-separated into to both equations. The subst() command put MATRICES the expression in the first argument into the ex- If Maxima can not solve a differential equation, as a result returns false. pression the second argument The Maxima has many functions related to (%i4) [subst(s,r1),subst(s,r2)]; 8. Matrices matrices. Matrices are introduced to the program (%o4) [4=4,3=3] Theby using Maxima the hasfunction many matrix, functions to performrelated to the matrices. ma- Matrices are introduced to the program by using Solutions do not have to be a real. For equa- thetrix function product matrix,we use ato dot perform (not *). the The matrix use ofproduct other we use a dot (not *). The use of other features we 4 tion x -1 = 0, we obtain: showfeatures on anwe example: show on an example: (%i1) solve(x^4-1); If we use the function genmatrix of undefined (%o1) [x=%i,x=-1,x=-%i,x=1] Ifparameter, we use the we function obtain genmatrixa general ofmatrix undefined with eleparameter- , we obtain a general matrix with elements aij.

Functions realroots (W), realroots (W, accu- (%i1)ments genmatrix(a,2,2); aij. racy) find by bisection method all real roots of a (%i1) genmatrix(a,2,2); polynomial. Polynomial coefficients must be ra- (%o1) tional numbers. The second parameter defines the 𝑎𝑎1,1 𝑎𝑎1,2 (%o1We) [ calculate2,1 2, 2the] determinant by writing the accuracy with which the root is searched- default We calculate𝑎𝑎 the𝑎𝑎 determinant by writing the following command: is 10-7. In addition, each student can calculate following command: roots e.g. quadratic equation (with parameters a, (%i2)(%i2) determinant(%); determinant(%); 2 b, c: ax + bx + c = 0), and thus verify that the (%o2) a1,1a2,2-a1,2a2,1 formulas learned in school are rely true, and they (%o2)We a can1,1a2,2 also-a1,2 a(symbolically)2,1 designate the in- are zeros above quadratic function. Additionally, Weverse can matrix: also (symbolically) designate the inverse matrix: you can also check the correctness of Viete’s for- (%i3) invert(%o1) ; // Simplified result which we mulas, which improve memorization and provide (%i3)obtain invert(%o1) by using the ; following // Simplified function: result which we obtain by using the following function: impetus for deepening self-knowledge. (%i4) ratsimp(%); //then we have: (%i4) ratsimp(%); //then we have: eg. quadratic equation (with parameters a, b, c: ax2 + bx + c = 0), and thus verify that the formulas (%o4) 𝑎𝑎2,2 −𝑎𝑎1,2 eg. quadratic equation (with parameters a, b,learned c: ax2 in + schoolbx + c =are 0 )rely, and true, thus and verify they that are the zeros formula aboves quadratic function. Additionally, you can also DIFFERENTIAL EQUATIONS a1,1a2,2−a1,2a2,1 a1,1a2,2−a1,2a2,1 learned in school are rely true, and they arecheck zeros theabove correctness quadratic of function. Viete'a formulas Additionally,, which you improves can also memorizatio [ −𝑎𝑎2,1n and provides𝑎𝑎 1impetus,1 ]for check the correctness of Viete'a formulas, whichdeepening improves self-knowledge. memorizatio n and provides impetus for(%o4) These a1 ,1symbolica2,𝑎𝑎22−,2a1,2a 2calculations,1 a1,1a2−,2𝑎𝑎− 1,a2also1,2a 2,make1 re- The function that solves ordinary differential These symbolic calculations also make remembering easier and preserve the fundamental rights that deepening self-knowledge. equations of the first and second order is ode2 memberinga1,1a 2easier,2−a1 ,2anda2,1 preservea1,1a2,2 −thea1, 2fundamentala2,1 6. Differential Equations are used by the−𝑎𝑎 2students,1 in all science𝑎𝑎1,1 , they also inspire to continue learning mathematics. (eqn, zvar, nvar), where zvar is the dependent rights that[ are used by the students in all] science, 6. Differential Equations a1,1a2,2−a1,2a2,1 a1,1a2,2−a1,2a2,1 Thevariable function and thatnvar solves is the ordinary independent differential variable. equations ThesetheyAlthough of alsothe symbolic first inspire the and Maxima calculationsto second continue is ordera symbolic learning also is ode2make mathematics.algebra ( rememberingeqn, system also easier allow and us preserve for numerical the fundamental operations, rights also thatwith The function that solves ordinary differentialzvar,To equations solve nvar) the, where differentialof the zvar first is andequation the second dependent we order will variable isuse ode2 the and (eqn, nvarareelements used Althoughis the by of independent thelinear the students Maximaalgebr variable. ina. allWeis a science cansymbolic calculate, they algebra also for example:inspire to continue learning mathematics. system also allow us for numerical operations, zvar, nvar), where zvar is the dependent variableTooperator: solve and the ‘diff nvar differential – is which the independent introduces equation we avariable. symbol will use of the the operator: 'diff – which introduces a symbol of the Althoughalso •with the theelements eigenvalues Maxima of linearis ofa symbolic the algebra. matrix algebra WeA and can systemtheir cal- multiplicities also allow us - eigefor numericalnvalues(A), operations, also with To solve the differential equation we will usederivativederivative the operator: andand does does'diff not –not which perform perform introduces differentiation, differentiation, a symbol but o preparef the derivative which later is part of the elementsculate, for of example: linear algebra. We can calculate for example: derivative and does not perform differentiation,differentialbut prepare but prepare equation. derivative derivative which which later later is part is part of ofthe the •• the• eigenvaluesrank of a matrix of the A matrix - rank(A), A and their mul- differential equation. differential equation. • the eigenvalues of the matrix A and their multiplicities - eigenvalues(A), tiplicities – eigenvalues(A), For exForample, example, we solve we solve the differential the differential equation equa - • eigenvalues of a square matrix A, with their multiples and their eigenvectors - 𝑑𝑑𝑑𝑑 •• rank• rankof a matrixof a matrix A – rank(A), A - rank(A), For example, we solve the differential equationtion 2 2 eigenvectors(A),2 (%i1)𝑑𝑑𝑑𝑑 rr:'diff(y,t,1)=3*(t*y)^2-y^2*sin(t); 𝑑𝑑𝑑𝑑 = 3•𝑡𝑡• 𝑦𝑦eigenvalues− sin (𝑡𝑡) 𝑦𝑦of a square matrix A, with their mul- (%i1) rr:’diff(y,t,1)=3*(t*y)^2-y^2*sin(t);2 2 2 𝑑𝑑𝑑𝑑 tiples• eigenvalues and their eigenvectors of a square – matrix eigenvectors(A), A, with their multiples and their eigenvectors - (%i1) rr:'diff(y,t,1)=3*(t*y)^2-y^2*sin(t); = 3𝑡𝑡 𝑦𝑦 − sin (𝑡𝑡)𝑦𝑦 • transposition of matrix A - transpose(A) (%o1)(%o1) •• transpositioneigenvectors(A), of matrix A – transpose(A) 𝑑𝑑𝑑𝑑 (%o1) (%i2) ro:ode2(rr,y,t);2 2 // solution2 of the differential 9. Graphs 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 2 2 2 (%i2) ro:ode2(rr,y,t);= 3𝑡𝑡 𝑦𝑦 − sin// solution (𝑡𝑡)𝑦𝑦 of the differential equation• transposition of matrix A - transpose(A) equation GRAPHS (%i2) ro:ode2(rr,y,t);𝑑𝑑𝑑𝑑 = 3𝑡𝑡 𝑦𝑦 − sin// solution (𝑡𝑡)𝑦𝑦 of the differential(%o2) 1/y=-cos(t)-t equation 3+%c //notice that %c is con- At the beginning of drawing graphs we will present and talk over some examples of functions and (%o2) 1/y=-cos(t)-t3+%c //notice that %c is constant value9. Graphs stant value their parameters. Let f1(x):= (%o2) 1/y=-cos(t)-t3+%c //notice that %c is constant value At the beginning of drawing graphs2 we will (%i3) rs:ic1(ro,t=0,y=1); // initial condition y(0)=1 (x −1) (%i3) rs:ic1(ro,t=0,y=1); // initial condition y(0)=1 At the beginning of drawing graphs2 we will present and talk over some examples of functions and 3 present and talk over some examples of functions (%i3) rs:ic1(ro,t=0,y=1); // initial condition(%o3) y(0)=1 1/y=-cos(t)-t +2 (%i1) f1(x):=(x^2-1)/(x^2+1)$ (x +1) 3 and theirtheir parameters.parameters. Let f1(x):= (%o3)(%i4) method; 1/y=-cos(t) // We-t +2can ask about the type of the 2 3 (x −1) (%o3) 1/y=-cos(t)-t +2 (%i2) plot2d(f1(x),[x,-8,8]); \\ Drawing2 the function y:=f1(x) in the range -8 <= x <=8 (%i4) method; // We can ask about the type of the equation(%i1) f1(x):=(x^2and we get:- 1)/(x^2+1)$ (x +1) (%i4) method; // We can ask about the type48 of the equation and we get: (%o4) separable // variable-separated (%i2) plot2d(f1(x),[x,-8,8]); \\ Drawing the function y:=f1(x) in the range -8 <= x <=8 (%o4) separable // variable-separated If Maxima can not solve a differential equation, as a result returns false. If Maxima can not solve a differential equation, as a result returns false. 8. Matrices 8. Matrices The Maxima has many functions related to matrices. Matrices are introduced to the program by using The Maxima has many functions related tothe matrices. function Matrices matrix, areto performintroduced the tomatrix the program product bywe using use a dot (not *). The use of other features we the function matrix, to perform the matrix productshow on we an useexample: a dot (not *). The use of other features we show on an example: If we use the function genmatrix of undefined parameter, we obtain a general matrix with elements aij. If we use the function genmatrix of undefined parameter, we obtain a general matrix with elements aij. (%i1) genmatrix(a,2,2); (%i1) genmatrix(a,2,2);

𝑎𝑎1,1 𝑎𝑎1,2 We can also do three-dimensional graphs in Maxima, here's an example: 1,1 1,2 (%o1) [ 2,1 2,2] 𝑎𝑎 𝑎𝑎 We calculate𝑎𝑎 the𝑎𝑎 determinant by writing the following command: 2 2 (%o1) [ 2,1 2,2] Let fc(x,y):=2-x -y We calculate𝑎𝑎 the𝑎𝑎 determinant by writing the following command: (%i2) determinant(%); We can also do three-dimensional graphs in Maxima, here's an example: (%i1) fc(x,y):=2-x^2-y^2$ (%i2) determinant(%); Let fc(x,y):=2-x2-y2 (%o2) a1,1a2,2-a1,2a2,1 (%i1) plot3d( fc(x,y), [x,-8,8], [y,-8,8]); (%o2) a1,1a2,2-a1,2a2,1 We can also (symbolically) designate the inverse matrix(%i1): fc(x,y):=2-x^2-y^2$ We can also (symbolically) designate the inverse matrix: (%i3) invert(%o1) ; // Simplified result which we obtain(%i1) by plot3d(using the fc(x,y), following [x,-8,8], function [y,-8,8]);: (%i3) invert(%o1) ; // Simplified result which we obtain by using the following function: (%i4) ratsimp(%); //then we have: (%i4) ratsimp(%); //then we have: Advances in Science and Technology Research Journal vol. 8 (23) 2014

(%i1) f1(x):=(x^2-1)/(x^2+1)$ integral. This allows the student to realize what (%i2) plot2d(f1(x),[x,-8,8]); \\ Drawing the func- are the limits of integration especially if they are tion y:=f1(x) in the range -8 <= x <=8 (Figure 2). infinite and illustrate the use of multiple integrals We can also do three-dimensional graphs in in practice. It is invaluable help for pupils and it Maxima, here’s an example (Figure 3): proves that mathematics cannot learn “by heart”. Let fc(x,y):=2-x2-y2 To get the ability to rotate the graph (of course (%i1) fc(x,y):=2-x^2-y^2$ after running Maxima and write commands), we (%i1) plot3d( fc(x,y), [x,-8,8], [y,-8,8]); obtain: (%i2) plot3d(fc(x,y), [x,-8,8], [y,-8,8], [plot_for- Thanks to 3D graph, students can see, some- mat, openmath]) (Figure 4). times even read (from graph) for example maxi- mum and minimum value of functions in the range Draw.mac package through the command im- and extremums of function. The situation is very plicit () allows us to draw graphs of implicit func- complicated, especially when function is a functions both 2D and 3D, such as (Figure 5): tion of several variables, where, thanks to graph (%i1)load(draw); and visualize students can realy see the use of pat- (%i2)draw2d(implicit(sin(2*x)*cos(y)=0.2,x,- terns and certain solving schemas in practice. 3D 3,3,y,-3,3)); and 2D graphs also help in determining the lim- Using parametric() command allows us to its of integration, not only single but also multiple create graphs of parametric functions.

Fig. 2. Drawing the function

Fig. 3. Three-dimensional graphs

49 Advances in Science and Technology Research Journal vol. 8 (23) 2014

Fig. 4. Rotate the graph Fig. 5. Graphs of implicit functions

CONCLUSIONS The Maxima, thanks to its capabilities and ap- propriate selection tools, helps to solve common Very aesthetic visualization has the advantage problems in different fields, such us economic that by its form encourages learning by creating analysis (e.g. compound), physics (e.g. math- an attractive environment. The student can freely ematical pendulum), chemistry and engineering. rotate the graph to see the desired details. Manip- ulation of color allows, for example distinguish quickly negative number from positive number. REFERENCES The cursor allows reading values precisely. The 1. Lachowicz C.T.: Matlab, Scilab, Maxima. Opis i use of different kinds of visualization explains przykłady zastosowań, Wydawnictwo Politechniki thoroughly the course of the function, and is also Opolskiej, Opole 2005. invaluable help for student. 2. Staranowicz A., Duda P., Orłowski A.: Technolo- Computer graphs can approach the stu- gie informacyjne. Wydawnictwo SGGW, Warsza- dent to an unavailable reality, things which are wa 2007. outside his environment, phenomena and pro- 3. Portal projektu MAXIMA http://maxima.source- cesses taking place in a great distance, things forge.net/ and phenomena occurring us around. The fun- 4. Portal projektu Otwarta Nauka http://otwartanau- damental principles are more easily assimilated ka.pl/ by students thanks visual effects and symbolic 5. Portal matematyki obliczeniowej http://computer- calculations. basedmath.org/