MAPLECALC John J. Avioli Christopher Newport University Newport News, VA 23606 javioli@pcs.cnu.edu During the Fall '94 semester, Christopher Newport University introduced the computer algebra system Maple into all sections of Calculus I. It is planned that by the end of the Fall '95 semester, Maple will be used in all sections of elementary calculus, linear algebra and ordinary differential equations courses. Christopher Newport University is an urban institution, primarily serving a local student body, many of whom work and attend CNU on a part time basis. In the Fall '94 semester, CNU offered four sections of Calculus I, three day sections and one evening section, with a total of 89 students. In all sections of the course, Maple projects make up 10% of the student's grade for the course. Maple is used to enhance the teaching of the calculus through illustrations of traditional concepts in the subject, to solve more "real" problems which involve hard, long, tedious calculations, to permit more time spent on concepts rather than on calculations and, generally, to illustrate to the student that computer software is a valuable tool in a mathematician's tool box of methods to solve problems. The students in the four sections had access to a Mathematics Computer Laboratory which contained five Macintoshs and a laser printer. Faculty teaching the course used a Macintosh connected to an N - View display system on a moveable table which was located on the floor where all four sections of the course were taught. The computer lab was open thirty hours per week, having day, evening and Saturday hours, and was supervised at all times by either faculty or student computer consultants. Student consultants, were given approximately eight hours of training in Maple before the semester started. In addition, these consultants were given copies of Maple projects before the projects were given to students and copies of prepared "Maple lessons" that instructors used in class lectures. Two text were required for the course : Calculus, fourth edition, by Anton and Discovering Calculus with Maple, by K. Harris. A detailed syllabus was prepared and distributed to each instructor in the course. The syllabus covered the same material as covered in previous semesters but, in addition, suggested when to incorporate Maple into the lecture. A manual containing the syllabus and copies of eleven (11) sample lectures was distributed to each Calculus I instructor. The sample lectures were stored on the hard disk of the computer that was used for classroom demonstrations so that instructors could use the actual lesson or modify the lesson to suit their needs. Not all instructors knew Maple. The lectures were written to teach (instructor and student) the syntax of Maple and ways to use it in a Calculus class. No other instructional materials are needed in order to use Maple in the course. A brief summary of each of the lectures follows. Lesson I : Introduction to Maple I. Topics covered include accessing Maple, initializing a disk, saving a file, the operators +, - , * , / , ^ , how to quit a Maple session, plotting two dimensional graphs, defining expressions, and how to use the commands Digit, evalf, fsolve, solve. Lesson II : Introduction to Maple II. This lesson includes defining a function as mapping and as a procedure, using Maple to find and simplify difference quotients, using the "for" loop statement to write a procedure to make a table of values for a given function, and the "if - then - else" conditional statement to define piece - wise defined functions as Maple procedures. Lessons I and II were used the first week of class to set the tempo for the course. Lesson III : Intuitive Limits. This lesson approaches limits from an intuitive point of view. Given a function f and a number a , one uses Maple to 1. construct a table of values for f near a, 2. to plot the graph y =3D f(x) near x =3D a, and 3. to simplify the function. With this information, one makes a guess at the limit of f(x) as x approaches a. One and two sided limits and infinite limits are discussed. Lesson IV : Limits. The Maple command "limit" is introduced in this lesson. To demonstrate the need for working the problem different ways to check for consistent results, an example is given whereby Maple gives an incorrect result. This lesson is given after the student works a sufficient number of limit problems using algebraic techniques without the use of the computer. Lesson V : Differentiation. This lesson begins with finding the derivative of a function using Maple's limit command to take the limit of the difference quotient as Dx approaches 0. Maple's "diff" command (take derivative) is then introduced. Using graphs, an example is given, demonstrating the approximating qualities of the tangent line at a point to values of the function near the point. The lesson concludes with using Maple to do implicit differentiation. The lesson is given after differentiation rules and implicit differentiation is covered without computer enhancement. Lessons VI & VII : Applications of the derivative. Related rate problems, graphing problems, asymptotes, and maximum- minimum problems are worked using Maple. It is recommended that this lesson be introduced after the student has worked problems of this type without the use of Maple. The problems in this lesson involve long tedious calculations that are done best by the computer. To introduce graphing type problems, an example is given where using Maple's plot command doesn't produce a very good graph because an inappropriate range for the graph was specified. In the example, critical points of the function and possible points of inflection are found using Maple. "Adjusting" the range according to this information, Maple produces a "good" graph of the function detailing important aspects of the graph. Lesson VIII : Newton's Method. Newton approximations to the zeros of f(x) are computed using a Maple procedure that is given in the lesson. In addition, graphs are displayed illustrating Newton's approximations as the the intercepts of tangent lines to the graph; an appropriate example is given illustrating that Newton's approximations are good estimates of solutions of f(x) =3D 0. Lesson IX : Indefinite integration. Maple's "changevar" and "int" commands are introduced. The changevar command is used to perform substitutions in indefinite integrals. The student is asked to evaluate indefinite integrals using Maple's changevar command. With this exercise, it is hoped that the student will sharpen his/her "guessing" skills in trying to find substitutions for indefinite integrals that "reduce" the integral to a simple form. The int command (evaluate the integral) is defined and the student is encouraged to use the "diff" command on Maple's indefinite integral result to verify the correctness of Maple's result. Lesson X : Defining Area. This lesson considers the problem of finding the area bounded by x =3D a, x =3D b, y =3D 0, and y =3D f(x), where= f is continuous and nonnegative on [a,b]. Approximations of the area using "right rectangles" are first considered. Using Maple, graphs of the region and the approximating rectangles are displayed. Again using Maple, a formula in terms of sums of the rectangles is developed and then summed by Maple to produce the desired approximation of the area. The area of the region is defined to be the limit of the Nth approximation as N -> =83. Examples are given : find the Nth approximation as a sum of the approximating rectangles; use Maple to calculate the sum the Nth approximation; use the Maple's limit command to take the limit of the Nth approximation as N -> =83 and thus obtain the area of the region. Lesson XI : Definite Integration. The "changevar" command is used to make substitutions in definite integrals to reduce the integral to a simple form that can be evaluated easily. Integration is done directly using the "int" command and Maple's numerical approximation to a definite integral is demonstrated for an integral that Maple cannot evaluate exactly. Also, in this lesson, examples are given illustrating that the derivative with respect to x of the integral of f(t) dt from t =3D a to t =3D g(x) is f'(g(= x) * g'(x). Generally, the computer is not used as a substitute for manipulations or drill. Students are expected to preform all manipulations done in a "traditional " non computer enhanced course except those manipulations involving long, tedious computations. Another component of the course stressed using learned mathematical concepts to either verify Maple's results or to argue against a Maple result. Students were encouraged to have Maple solve a problem several ways, algebraically, numerically, and graphically, and look for results consistent with Maple's result. This was especially nice in the case of limits. The students were given periodic Maple assignments out of Harris' text and a Maple computer test given in parts throughout the semester. The projects (see Appendix A) required students to use Maple commands to solve calculus problems and in many cases had to verify or argue against Maple's result. Instructors gave three or four in class tests in addition to the Maple computer test and a comprehensive final exam. The in class tests and final exam didn't involve anything regarding the Maple component of the course. The experiment, incorporating Maple into the Calculus sections, is working quite well. In addition to the "regular" material covered in a non - technology enhanced calculus course, students learn a powerful computer software package. Hopefully, with the Maple component, they have a deeper understanding of calculus concepts. In my class of 29 students, one student told me that she didn't have the time to do computer laboratory work and therefore would drop the course for this reason. Thirteen percent (13%) of those students who took the first test decided to stay in the course but not do the Maple project component and therefore, would take a zero for the 10% Maple component of their grade for the course. Fifty - five percent of the students doing the Maple component, earned a grade on the first Maple project greater than or equal to the grade they earned on the first test. The average grade on the first Maple project was 83.5 and on the first test it was 75.0. Appendix A : Parts I and II of the Maple test used in Calculus I. The Maple test made up 10% of the student's grade for the course. ---------------------------------------------------------------------------- MAPLE PROJECT - PART I : LIMITS MATH 140 September , 1994 DUE : (Class time, one week after distribution of this project) =46or each of the following, use Maple's limit command to find the limit of "f(x)" as x approaches "a". Verify or argue against Maple's result using graphic, algebraic (simplification), and/or numerical (tables) means. Use the following format for each problem : define the function, use Maple's limit command, attempt to verify Maple's result. Use Maple's text mode to type any comments and your conclusion. At the top of your worksheet, type your name, the date, your course section and instructor. 1. f(x) =3D x^2 sin(1/x^2) , a =3D 0 2. f(x) =3D (x^(1/5) - 4) / (x - 1024) , a =3D 1024 3. f(x) =3D (1 + 2 x)^(1/x) , a =3D 0 4. f(x) =3D (x - 2) / ( (x^2 + 4)^(1/3) - 2) , a =3D 2- 5. f(x) =3D (6 - x) / (5 + 6 x^2)^(1/2) , a =3D -=83 6. f(x) =3D x^2 / ( (4 - x) (1 + x) ) , a =3D 4+ 7. If x =BE 0 then f(x) =3D x sin(x) - x^2 , else f(x) =3D 2 - x . a =3D 0 8. f(x) =3D (x + 3 x^2)^(1/2) - x , a =3D =83 ---------------------------------------------------------------------------- MAPLE PROJECT - PART II : DERIVATIVES Math 140 October 1994 DUE : Class time, November 7, 1994 I. Just for review : Let 2 x^3 + 7 x^2 - 20 x + 12 - x^4 f(x) =3D ------------------------------------------------ 5 x - (4/5) - (26/5) x^2 + x^3 Using the limit command find the limit of f(x) as x approaches 1 and then verify Maple's result algebraically, numerically, and graphically. II. If y =3D 5 sin(6 x^2) , find d^5y / d x^5. III. Find the x -coordinate of points on the graph of y =3D -58 x^5 - 90 x^4 + 53 x^3 - x^2 + 94 x + 83 where the tangent line is horizontal. Verify your result by plotting the graph of y =3D f(x) in appropriate ranges which include the (real) x coordinates you found above. IV. Let f(x) =3D 1 / (1 + x). Using Maple, find the equation (in slop= e intercept form) of the tangent line to the graph of y =3D f(x) at x =3D 0. Plot the graphs of y =3D f(x) and the tangent line on the same axis over an appropriate range to illustrate the approximating qualities of the tangent line for values of x near 0. V. Find the equation of the tangent line( in slope intercept form) to the graph of x y =3D sin(x + 4 y^4) at the point ( 0 , sqrt(Pi) / 2). Note : Type your name and comments or conclusions on your Maple worksheet.