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EPIC_3D (Exploration Programs In Calculus)

James Burgmeier and Larry Kost University of Vermont

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EPIC was designed to be useful in several ways.

For instructors, EPIC provides an environment to explore concepts and examples for presentation in the classroom. Even if EPIC is not used directly during class, it still provides an excellent opportunity for instructors to easily and accurately prepare examples for class.
EPIC provides easy to follow classroom demonstrations. Instructors have a variety of options for in-class presentations.

Some are willing to be "experimental" during class, essentially letting the discussion drive the choices of examples. It is here that the robustness of EPIC (a significant design component to insure that it won't crash no matter what input is given) is very reassuring. Errors will be noted, but one can easily recover and continue with minimal interruption.
Other instructors prefer to prepare sample functions, which they can carry from their office to the classroom on floppy disks, or send over the network. These functions may be recalled quickly in class, providing illustrations of the concepts under discussion.
A third option is to prepare screens in one's office, save these screens on a disk, and then recall the screens in the classroom. This "slide" mode for EPIC provides a means to display a sequence of examples without the need to wait for the computer to do calculations.

EPIC was also designed to be useful to students. Most of the homework problems and all of the major topics in beginning Mathematics courses (College Algebra, Calculus, Numerical Methods) and many of the problems in differential equations can be checked and/or "illuminated" by using EPIC.
The name chosen for EPIC -- again, which stands for Exploration Programs In Calculus -- reflects the original goal of the software. By making the programs easy to use, and easy to enter functions, we hope that students would be encouraged to explore mathematics through EPIC.

Features

1. Function Input.

(a) Textbook-looking function input style

(b) Parameters allowed in the functions

(d) Explicit domain specification is allowed

(e) Functions may be entered as summations or products

(f) Recursively defined functions are allowed

(g) One or two functions may be entered

Items (a), (b), (c), (d) and (g) are shown in the sample "EPIC Function Input screen" shown below. Following that screen is a brief elaboration of these features.

(a)

There are no editing rules to remember. Just use the arrow keys to move the cursor to any position you wish, and type the characters. Characters may be entered in any order, changed anytime with no special key combinations or mouse clicks to make your intentions understood.
Exponents on trig functions such as sin²3x can be written just as in a textbook, or on the blackboard. Expressions such as sinx cosx, csc4x, coshax and sin cosx are properly understood as (sin(x))*(cos(x)), csc(4*x), cosh(a*x) and sin(cos(x)), respectively.

(b)

Parameters can be used anywhere; values are requested after the function is translated, before graphing begins. Values of parameters can be easily changed.

(c)

As shown above for the function G(x), functions defined by several sections may be entered. In this case, each section must be given an explicit domain, indicated by following the formula for the section with a comma, and then the domain specification. (Even though only 8 rows are shown for the functions, each function actually can have 29 rows; the input box scrolls vertically and horizontally if the formula becomes too large for the box given. Also, in one function mode, there are 18 rows showing. In one or two function mode, the width of the input boxes can be extended by removing the help box at the right (F1).

(d)

Even for a one section function, explicitly giving the domain is also allowed. For example,

F(x) = 2x - x², x <= 1

can be used to illustrate the branch of the parabola whose inverse function is 1 - sqrt(1-x).

(e)

Pressing the F10 key changes the input screen into one where the function is defined by a summation. Pressing it again changes the sum to a product. Values of the upper and lower limits are entered according to your wishes (except that infinity is not allowed as a limit!).

(f)

The definition of F may involve G, and vice versa. Furthermore, the definition of F (or G) can involve itself. Here is one of our favorite examples:

F(x) =

x², x <= 1
=========
F(x-1), x > 1

2. 2D Graphs.

The "EPIC screen" below shows the graphs of the two functions defined on the "EPIC screen" shown above. In the transition from entering the functions to getting their graphs, requests were made for the graphing window coordinates (the defaults of -5,5 by -5,5 were taken), and for the parameters a and b in the definition of F(x). As shown in the graph below, a = -5 and b = 1 were entered.

Notice that the graphs of discontinuous functions are drawn discontinuously - a feature frequently missing with many graphing software packages!

The menu at the right of the graphs gives some idea of the options available. However, many of these choices have submenu choices. For example, the next "EPIC screen" shows the choices available under the Roots choice.

The next screen shows the results of making the choice Intersection of F & G. An input box appears asking for a start value for finding an intersection point. (Newton's method is used, with the derivatives computed symbolically before evaluation.) The value shown in the "x = " box was obtained by clicking the mouse (notice the mouse in the main portion of the graph). Accepting this value finds the result

x = 2.795904, F(x) = .2752889

as the point of intersection near the start value.

Additional 2D Graphing Features
Functions Defined by Tables	Symbolic Derivatives	Zoom
Graphs of Derivatives	Graphs of Second Derivatives	Taylor Polynomials
Tangent Lines	Secant Lines	Three Forms for General Lines
Movement of Lines	Roots of Functions	Roots of First Derivatives
Roots of Second Derivatives	Intersection Points	Area under Functions
Area between Functions	Integrals versus Area	Left, Right, Middle Riemann sums
Trapezoidal Rule	Simpson's Rule	Gauss Quadrature Orders 2 - 8
Function Combinations + - * /	Function Translations and Reflections	F(G(x)) and G(F(x))
Function Inverses	Newton's Method, Graphically and Numerically	Save and Recall Screens
Built-in Mathematical Calculator	Keyboard and Mouse Friendly	Menu-driven

The next "EPIC" screen shows the results of using the Trapezoidal Rule on the piecewise defined function G(x) defined above to estimate the value of the integeral from -4 to 4 of G(x), using 100 subintervals.

3. 3D Graphs.

EPIC supports graphs of functions of two variables (say x,y) defined on a region R in the x-y plane. Default choices for R are (i) a rectangle, and (ii) a circle. In addition, R may be specified by two functions: y = F(x) and y = G(x). The EPIC screen below shows the graph of

Z(x,y) = 5 + 3sin(x - 2y)

defined on the rectangle [-4,4] by [-4,4].

The next screen shows the same function drawn on the domain R defined by 0 <= x <= 2, 1 + cos(x) <= y <= e^sin(x).

These two functions were entered on a function input screen identical to that described above for 2D functions. While running EPIC, they may be viewed anytime by selecting choice `Region' from the menu. A small graph of the region R is shown at the top of the EPIC screen, to the right of the formula for Z(x,y). Also, note that according to the information in the upper right corner of the screen, the viewing angles for the viewer looking at the surface have been changed: theta = -45 degrees is the angle in the x-y plane measured from the positive x-axis; looking down on the plane, positive is counterclockwise. The angle phi = 15 degrees is the polar angle, measured from the positive z-axis.

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