built in command "TaylorApproximation" (shown below).
Maple worksheet: math2130-spring2011-quadratic_approximations.mw

Below the code are the solutions for (most of) the problems found 

in our "Quadratic Approximations" supplement. 

"Edit → Execute → Worksheet" to make Maple show the command outputs. 

>

restart; with(VectorCalculus): evalSF := proc(f,X,A)   # Computes f with X = A (a scalar valued function version of evalVF).   subs([seq(convert(X,list)[i]=convert(A,list)[i],i=1..nops(convert(X,list)))],f); end proc: LinearApproximation := proc(f,X,A)   # Computes the linear approximation (1st order Taylor polynomial)   # of f based at X = A.   #   # f is an expression (not a function) with variables X.   # X is a vector which lists the variables on which f depends.   # A is a vector.   local f_A, delf_A;   # computes f(A)   f_A := evalSF(f,X,A);     # the gradient of f evaluated at X = A.   delf_A := evalVF(Gradient(f,convert(X,list)),A);   # Return the Linear Approximation of f at X = A.   simplify(f_A + delf_A.(X-A)); end proc: QuadraticApproximation := proc(f,X,A)   # Computes the quadratic approximation (2nd order Taylor polynomial)   # of f based at X = A.   #   # f is an expression (not a function) with variables X.   # X is a vector which lists the variables on which f depends.   # A is a vector.   local f_A, delf_A, Hf_A;   # computes f(A)   f_A := evalSF(f,X,A);     # the gradient of f evaluated at X = A.   delf_A := evalVF(Gradient(f,convert(X,list)),A);   # the Hessian of f evaluated at X = A.   Hf_A := evalSF(Hessian(f,convert(X,list)),X,A);   # Return the Quadratic Approximation of f at X = A.   simplify(f_A + delf_A.(X-A)+((1/2)*LinearAlgebra[Transpose](convert(X-A,Matrix)).Hf_A.convert(X-A,Matrix))[1,1]); end proc:

 

Exercise #1:   at  

>	X := <x,y>; A := <0,0>; f := exp(x+2*y); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(1)

 

Exercise #2:   at  

>	X := <x,y>; A := <2,3>; f := x^2+x*y^2; L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(2)

 

Exercise #3:   at  

>	X := <x,y>; A := <-1,0>; f := x^4*y^2+x*y-2*y; L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(3)

 

Exercise #4:   at  

>	X := <x,y>; A := <2,0>; f := 2*x^2+3*y^4+exp(x*y^2); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(4)

 

Exercise #5:   at  

>	X := <x,y>; A := <3,Pi>; f := sin(x*y); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(5)

 

Exercise #6:   at  

>	X := <x,y>; A := <1,0>; f := ln(x^2+y); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(6)

 

Exercise #7:   at  

>	X := <x,y>; A := <sqrt(Pi),sqrt(Pi)>; f := cos(x*y); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(7)

 

Exercise #8:   at  

>	X := <x,y,z>; A := <1,2,3>; f := x*y*z; L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(8)

 

Exercise #9:   at  

>	X := <x,y,z>; A := <-1,0,1>; f := x^4+y^4+z^4; L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(9)

 

Exercise #10:   at  

>	X := <x,y,z>; A := <2,1,-2>; f := x^2*z+x*y^2+y^3*z^2+exp(x*y+z); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(10)

 

Exercise #11:   at  

>	X := <w,x,y,z>; A := <0,0,0,0>; f := w+x^2+y^3+z^4+w^2*cos(x*y^2); L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(11)

 

Exercise #12: Let   have linear and quadratic approximations  

                     (based at ) called and . Show that in 

                     this case . What can be said when  

                      ? 

A linear approximation of a linear function is just the function itself and since the  

linear approximation is already perfect, the quadratic approximation will be no 

different. Now if is a quadratic function, we can expect (and in fact prove) 

that although it probably won't match its linear approximation 

anymore. 

>	X := <x,y>; A := <a,b>; # I'm avoiding the name "A" which is already in use. f := cA*x+cB*y+cC; L := LinearApproximation(f,X,A); Q := QuadraticApproximation(f,X,A); H := Hessian(f,convert(X,list));

 

 

 

 

 

 

	(12)

 

Exercise #13: Write down the -order terms for the Taylor polynomial expansion  

                     of at . 

Exercise #14: Write down the 4-order terms for the Taylor polynomial expansion  

                     of at . 

I'm not sure why, but... 

Here is exercise #5 reworked without using any fancy commands (minus printing 

our the Hessian). Each partial is computed and then evaluated at the point (just like  

we'd do by hand). 

>	f := (x,y) -> sin(x*y); 'f(x,y)' = f(x,y); a := 3; b := Pi;

 

 

 

 

	(13)

 

>

fx := diff(f(x,y),x); fy := diff(f(x,y),y); fxx := diff(f(x,y),x,x); fxy := diff(f(x,y),x,y); fyy := diff(f(x,y),y,y); fxS := simplify(subs(x=a,y=b,fx)); fyS := simplify(subs(x=a,y=b,fy)); fxxS := simplify(subs(x=a,y=b,fxx)); fxyS := simplify(subs(x=a,y=b,fxy)); fyyS := simplify(subs(x=a,y=b,fyy)); L := simplify(f(a,b) + fxS*(x-a) + fyS*(y-b)); Q := simplify(L + (1/2)*(fxxS*(x-a)^2+2*fxyS*(x-a)*(y-b)+fyyS*(y-b)^2));

 

 

 

 

 

 

 

 

 

 

 

 

	(14)

 

Maple has a nice built in command to compute multivariate Taylor polynomials.
So my "LinearApproximation" and "QuadraticApproximation" functions aren't 

needed. We can just compute the Taylor polynomials of orders 1 and 2. 

Let's redo exercise #5 (yet again and again no Hessian) using the  

"TaylorApproximation" command. 

>	with(Student[MultivariateCalculus]):

 

>	simplify(TaylorApproximation(sin(x*y),[x,y]=[3,Pi],1)); simplify(TaylorApproximation(sin(x*y),[x,y]=[3,Pi],2));

 

 

	(15)

 

Now we can also compute 3rd, 4th, ... order Taylor polynomials as well. 

Here are the 3rd, 4th, and 5th order Taylor polynomials (continuing our previous 

example). 

>	simplify(TaylorApproximation(sin(x*y),[x,y]=[3,Pi],3)); simplify(TaylorApproximation(sin(x*y),[x,y]=[3,Pi],4)); simplify(TaylorApproximation(sin(x*y),[x,y]=[3,Pi],5));

 

 

 

	(16)

 

This command can also produce plots and animations. Although, you need to fight with them 

a bit to get them to show what you want. 

>	TaylorApproximation(sin(x*y),[x,y]=[3,Pi],2,x=2.2..3.8,y=2.3..3.9,output=plot);

 

>	TaylorApproximation(sin(x*y),[x,y]=[3,Pi],10,output=animation);