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The way the Taylor polynomials of a function of one variable progressively converge to the graph of the function like y = cos x is really quite impressive and is inherently interesting. We can extend this topic into three dimensions using CalcPlot3D.
As an exercise, I require my students to generate the linear and quadratic Taylor polynomials of a function of two variables using the partial derivatives of the function evaluated at a particular point.
\( \begin{eqnarray} f(x,y) &\approx L(x,y) = f(a,b) &+ f_x(a,b)(x-a) + f_y(a,b)(y-b) \qquad (1^{st}\text{-deg. Taylor poly or tangent plane})\\ f(x,y) &\approx Q(x,y) = f(a,b) &+ f_x(a,b)(x-a) + f_y(a,b)(y-b) \\ &+\frac{f_{xx}(a,b)}{2}(x-a)^2 &+ f_{xy}(a,b)(x-a)(y-b) + \frac{f_{yy}(a,b)}{2}(y-b)^2 \qquad (2^{nd}\text{-deg. Taylor poly})\end{eqnarray}\)
Exercise: | Determine the 1st and 2nd degree Taylor polynomials in two variables for the given function. Simplify both polynomials. Show all work including all partial derivatives and using the formula clearly with functional notation in the first step. Please also provide a printout of the given surface along with each of the Taylor polynomials. (That’s 2 printouts all together.) Include the point on the surface where the polynomial is tangent to the surface. Use the Format Surfaces option on the View Settings menu so that the Taylor polynomial is reverse color and transparent so it’s possible to tell the two surfaces apart. If necessary, zoom out and the rotate to a view that shows the surfaces clearly. Then use the Print Graph option on the File menu of the applet to print the graph. \(f(x,y) = \sin(2x) + \cos y\) for x,y near (0,0) |
Answers: | 1st-degree Taylor Polynomial of f: | 2nd-degree Taylor Polynomial of f: | |
\(L(x,y)=1+2x\) | \(Q(x,y)=1+2x-\frac{1}{2}y^2\) | ||
There is also a feature of the applet that will allow you to demonstrate higher-degree Taylor polynomials for a function of two variables.
Click here to open the CalcPlot3D applet in a new window.
Click here to open a pdf file which contains the instructions for the activity.
Paul Seeburger (Monroe Community College), "CalcPlot3D, an Exploration Environment for Multivariable Calculus - Taylor Polynomials of a Function of Two Variables (1st and 2nd degree)," Loci (November 2011), DOI:10.4169/loci003781