(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] FUNCFIT :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] FUNCFIT is a fun classroom activity to familiarize students with the shapes of various functions. Groups write non-technical geometric descriptions of graphs and exchange them with other groups. Then students try to recreate graphs from the descriptions. The exercise is appropriate for pre-calculus or beginning calculus students. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect] FileName: FUNCFIT :[font = subsection; inactive; preserveAspect] Full title: Function Fitting to Descriptions of Curves :[font = subsection; inactive; preserveAspect] Last Update: 5/29/96 :[font = subsection; inactive; preserveAspect] Developer: Lynn Kiaer, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA :[font = subsection; inactive; preserveAspect] Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@rose-hulman.edu. FAX: 812-877-3198. :[font = subsection; inactive; preserveAspect; endGroup] Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] This is a cooperative learning exercise. :[font = subsection; inactive; preserveAspect] 1. Each student is given a function definition and asked to graph the function, and then to write a non-mathematical description of the appearance of the function. :[font = subsection; inactive; preserveAspect] 2. Students form pairs, with each partner having the same function. The partnership refines the written description, and signs it. :[font = subsection; inactive; preserveAspect; startGroup] 3. Pairs exchange written descriptions (but not functions) with a nearby pair (a pair which started with a different function). Pairs attempt to find the function that fits the description. This may be an iterative process. When a pair proposes a function, the description writers may respond :[font = subsubsection; inactive; preserveAspect] You've got it. :[font = subsubsection; inactive; preserveAspect] That can't be the function because it doesn't satisfy a particular aspect of the description. :[font = subsubsection; inactive; preserveAspect; endGroup] Ouch! We have to modify the description! :[font = subsection; inactive; preserveAspect; endGroup] Additional rounds give the students practice matching functions to descriptions. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Function, fitting functions. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites: :[font = subsubsection; inactive; preserveAspect] Prerequisites will vary depending on the functions used. I would envision using polynomials (up to cubic), sines, cosines, exponentials and logarithms. :[font = subsubsection; inactive; preserveAspect; endGroup] It would probably be worth doing this exercise as a part of a general review of functions, and then again after the students have studied derivatives. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management: :[font = subsubsection; inactive; preserveAspect] This will depend somewhat on the complexity of the functions being used, but 2-3 minutes for steps 1 and 2, 10-15 minutes for step 3, and 8-10 minutes for additional iterations seems about right. :[font = subsubsection; inactive; preserveAspect; endGroup] Students who are stuck looking for a function should be instructed to sketch a graph that matches the description. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Depending on the kinds of functions used, students could be asked to find the exact function or merely a generic version. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] The purpose of this exercise is to build the skills and intuition about functions that students need in order to, for example, choose an appropriate function form for a collection of data points. :[font = subsection; inactive; preserveAspect] Extensions :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect; startGroup] Example 1 :[font = input; preserveAspect] f[x_] = 2 x^3 - 3 x^2 + x + 4; :[font = input; preserveAspect] Plot[f[x],{x,-3,1.5}]; :[font = input; preserveAspect] Plot[f[x],{x,0,1}]; :[font = subsubsection; inactive; preserveAspect; endGroup] Description: This function has a value of 4 when x is 0. It decreases rapidly when x<0, becoming negative before x gets to -1. As x increases from 0, the function increases just a little bit (not more than 4.1), then decreases a little bit (not less than 3.9), all before x gets to 1. When x is 1/2 and when x is 1, the function value is 4 again, and then, for x>1, the function increases rapidly in value. :[font = subsection; inactive; preserveAspect; startGroup] Example 2 :[font = input; preserveAspect] g[x_] = 100 - 100 Exp[-x/2]; :[font = input; preserveAspect] Plot[g[x],{x,-1,10}]; :[font = subsubsection; inactive; preserveAspect] Description: This function is always increasing. Its value is 0 when x is 0, and decreases sharply as x gets more and more negative. The function has a horizontal asymptote at y=100. :[font = subsubsection; inactive; preserveAspect; endGroup] This description would probably have to be refined in order for the function fitters to obtain the coefficient of x in the exponent. :[font = subsection; inactive; preserveAspect; startGroup] Example 3 :[font = input; preserveAspect] h[x_] = 3 + 2 Sin[2x - Pi/4]; :[font = input; preserveAspect] Plot[h[x],{x,0,2Pi}]; :[font = input; preserveAspect] Plot[h[x],{x,-.01,.01}]; :[font = input; preserveAspect; startGroup] h[0] //N :[font = output; output; inactive; preserveAspect; endGroup] 1.585786437626905 ;[o] 1.58579 :[font = subsubsection; inactive; preserveAspect] Description: This function oscillates between 1 and 5. It repeats itself every Pi units. When x is 0, the value of the function is about 1.58579. :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] In this case, I would encourage students not to use the exact value of the function at this point (3 - Sqrt[2]), partly because data doesn't look like this, and partly because it gives away the `shape' of the function. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] The Fitters may have "fits" because the proposers have not given sufficient information to "nail down" a unique function in their mind. Indeed there is no unique function which satisfies these finite conditions, although there might be a unique function of a given form say one like this: A + B Sin( C x + D). ^*)