Implicit Function Explorations

Transcription

1 Activities with Implicit Functions and Implicit Differentiation on the TI-89/Voyage 00 Dennis Pence Western Michigan University Kalamazoo, Michigan USA Abstract: Unfortunately the topic of implicit differentiation occurs early in most calculus texts (primarily so that there can be a more rigorous proof for the derivative power rule), when the student has no experience with or understanding of implicitly defined functions. Implicit function plotting on these calculators is embedded in 3D plotting, and it is generally not yet appropriate to study functions of several variables either. We present instead methods of working with implicit functions using the CAS solver, the numeric solver, tables, and function graphing all tools that are comfortable for a beginning calculus student. This work can naturally lead to a greater understanding of implicit differentiation, both when the implicit differentiation is done by hand and when carried out using a CAS process. Students in a first calculus course rarely have a very well developed concept of function. For most, the notion of a function is closely tied to a formula. If something cannot be typed as a simple formula into the Y= Editor in Function Graphing Mode, then it probably does not get included in the notion of function as they understand it. Even piecewise-defined functions (which can be typed into a function slot in the Y= Editor with difficulty using the when command) give students some trouble, with many considering the pieces as separate functions. Pre-calculus experiences virtually never deal with situations where a function cannot be written by an elementary formula. Thus it should be no surprise that these same students do not readily accept the notion of an implicitly-defined function. In fact, the vertical line test that they have been taught to use in order to decide whether a curve in the plane represents the graph of a function or not will lead they to reject most of the curves given to them now when we start talking about implicit functions. Most calculus texts spend very little time (if any) developing an understanding of the rather subtle notion of an implicitly-defined function. Instead, these textbooks rush immediately into the concept of implicit differentiation. Clearly some of the difficulties which students face in understanding implicit differentiation come from this lack of understanding for the very function they are trying to differentiate, the implicit function we like to assume they see in the given equation. I have found that spending separate time just working with implicit functions pays off later when we move on to implicit differentiation. Implicit Function Explorations 3 3 For definiteness, let s consider a folium of Descartes, namely x + y = 6xy near (3, 3). Given prior work with the graphs of conic sections, it is reasonable to expect the graph of this equation to give some curve in the xy-plane. It is also easy to check that the point (3, 3) satisfies the equation. Thus we seek to explore the claim that near x = 3 there is an implicitly defined function y(x). Our first exploration will use the Numeric Solver. While it is possible that the students have seen Newton s method by this time (and the solving could be done that way), we want the speed and simplicity of the built-in numeric solving process. 1

2 APPS Numeric Solver Enter equation Enter desired x-value and seed y = 3 Press F Solve Try other nearby x-values, same seed y = 3 A limited amount of this exploration will lead to the discovery that most nearby x-values result in an associated y-value, which is also near 3. However the example above where x = 3.5 (even with a seed of y = 3) requires more explanation. This is probably the appropriate time to explain the use of a seed value in the variable y for the Numeric Solver. An equation such as this one may have more than one solution for a certain x-value. We type a seed value in for y before we press F Solve in order to start the search for a solution near where we hope to find one. There is no guarantee that this will happen, but it tends to happen. The iterative process used by the calculator will also take much less time if we start with a seed near the final solution. [Students who have previously studied Newton s method should have enough practice with this to appreciate the worth of a good starting value.] You can also restrict the search for the solved variable with the bound, but we choose not to do that here. To demonstrate this, we go to x = 3 and use the seeds y = 3, 0, and 3. Before F Solve, type y = 3 y = 0 y = 3 Thus we can see that the curve representing the graph of this equation will not be the graph of a single function because it fails the vertical line test at x = 3. Attempts to use still further seeds will lead to the conclusion that you always end up with one of these three final answers, and it will re-enforce the idea that you end up with a solution near where you start (if that is possible). This further leads to the conclusion that there might be three different functions to consider near x = 3. Obviously this process of solving for the y-value associated with a specified x-value, one at a time, is tedious. Don t underestimate its pedagogical value, however. Now perhaps is the time to be more efficient. We would like to see these xy-pairs in a table and in a graph. The trick is to

3 use the command line version of the numeric solver as the input in the Y= Editor. The syntax for the command is nsolve( EQUATION, VAR ) where we can (optionally) input a seed by typing y = 3, for example. The variable x will be set by the table or graphing process and the command will solve for a y-value near the seed. Command in CATALOG Seed is last argument as y=3 Table Setup Table Window (suggest xres = 3) Graph Both the table and the graph seem to suggest some implicitly-defined function y(x) exists for x- values near 3 (but this function seems to drop off between x = 3.1 and x = 3.). Note that since we took the default graphing style (Line), then we get a near vertical line where the drop off occurs. We will get a better understanding of this if we plot all three of the solution parts on the same graph using the style Dot so that there is no vertical line when we drop off to another function. Now you are really going to want a larger xres because function evaluation with this numeric solve command takes a long time! Note that we can scroll up and down in the table and we can trace these function graphs ZoomSqr It is interesting here to try to use the F5 Math commands to look at derivatives and tangent lines. While the Maximum command does work (slowly) for our first function, the error message for a derivative or a tangent line seems to indicate that these commands require a simple formula. 3

4 I should mention, before we go on to differentiation, that Implicit Plotting is implemented in the 3D graphing mode provided you are willing to discuss things in 3D. I prefer not to do so in a beginning calculus course, but it is nice to remind students about implicit functions when they are taking a later multivariate calculus course. The disadvantages here are you cannot trace along the curve (because it really is a 3D plot and the trace command moves on the surface) and you cannot look at tables. Select 3D graphing F1 Format Style z1=x^3+y^3 6x*y, 4.5 x 4.5, 3.5 y 3.5 Implicit Differentiation On the old TI-9, we used to do implicit differentiation by asking for the derivative with respect to x in an equation where we typed an (unknown) function y(x) in place of y in the equation. This part still works. However it is no longer easy to solve for the derivative. This trick no longer works. What you now need to do is to cursor up to highlight the output of the implicit differentiation. Press ENTER to have this equation pasted onto the command line. Then edit out all of the phrases d(y(x),x) and replace them with a variable name such as yp. Finally surround this edited equation with a solve command and solve for the variable name yp. 4

5 I still like the above process for demonstrating how they should operate by hand. But for more efficient later work, the Flash App Calculus Tools provides a nicer way to have the tool do the whole process of implicit differentiation and then solving for the derivative. However the student is expected to find the implicit derivative (by hand or using the tool), there needs to some graphical confirmation that this agrees with what we expect for a derivative of a function. We now add tangent lines to the graph we obtained above by entering the lines as additional functions in the Y= Editor. 3 3 dy x y x + y = 6 xy, = dx x y dy For the point ( x, y) = (3, 3), = 1, tangent line y = 3 ( x 3). dx dy For the point ( x, y) = (3, ), =.065, tangent line y = ( x 3). dx dy For the point ( x, y) = (3, ), = , tangent line y = ( x 3). dx y4= 3 ( x 3) y5 = ( x 3) y6 = ( x 3) 5

6 [Tip: Since the implicit functions plot so slowly, you will not want to have them redrawn again every time you change to a new tangent line above. Have only the three implicit functions stored in y1, y, and y3 plotted first and do F1 Save Copy As to a Picture file. Then graph the tangent lines one at a time, and Open the Picture file to have the implicit plot redrawn quickly.] The Implicit Differentiation in Calculus Tools also gave us the second derivative of the implicit function (and would have given higher orders as well). We do not usually do this in a beginning calculus course, but we easily could. Just implicitly differentiate the original equation twice and solve for the second derivative that results. Note that there are command line versions of the commands from Calculus Tools in the CATALOG. 8 y7= 3 ( x 3) ( x 3) 3 y8 = ( x 3) ( x 3) y9 = ( x 3) ( x 3) 6