Taylor Polynomials of a Function of Two Variables

The unapply command can be used to define our function f(x,y)

> f := unapply(sqrt(16-3*x^2+3*y^2),(x,y));

The Taylor Polynomials of a function of two variables may be found by using the mtaylor command.

The parameters are analagous to those we use with the taylor command. The first parameter is an expression

for the value of a function of two variables, the second is of the form [x=a,y=b] and specifies the poiint

about which you want the expansion to be computed, and the third is an integer that is one more than the

desired degree of the polynomial. Thus to find the third degree Taylor polynomial of f(x,y) = sqrt(16-3*x^2+3*y^2)

about the point (1,2), we type

> mtaylor(f(x,y),[x=1,y=2],4);

Note that Maple unfortunately oversimplifies the first degree terms writing 16/5 - 3/5 x + 6/5 y when it

should write 5 - 3/5(x-1) + 6/5(y-1). Whenever you use Maple to find a Taylor polynomial, you should

correct this mistake that Maple makes.

The linearization or tangent plane at (1,2) of the same function is found as follows:

> mtaylor(f(x,y),[x=1,y=2],2);

To define the Tangent Function, i.e. a function that we can use to find linear approximations of our

given function, the unapply command can be used

> T := unapply(mtaylor(f(x,y),[x=1,y=2],2),(x,y));

The linear approximation to our given function at say (1.2,1.7) would then be given by

> T(1.2,1.7);

The graphs of the function and its linearization can be drawn together using the display command from the plots library.

> a := plot3d(f(x,y),x=-5..5,y=-5..5):

> b:= plot3d(T(x,y),x=-5..5,y=-5..5):

> with(plots):

Warning, the name changecoords has been redefined

> display(a,b,axes=boxed);

>