As usual, we begin by wiping memory and loading what we need. restart; with(plots): with(orthopoly):

From the Supplement to Lab 5: cosine series routines There are two differences between this and what was in Lab 6. assume(n,integer);assume(m,integer): In the definition of the cosine I've change the period to 2L c := (x,n) -> cos((2*Pi*n*x)/(2*L)): and the integral below for A goes from -L to L, not from 0 to L: A:=proc(expr,var,n) evalf(simplify(int(expr*c(var,n),var=-L..L)/int(c(var,n)*c(var,n),var=-L..L))); end proc: cosineFP:=proc(expr,var,n) A(expr,var,0)+add(A(expr,var,m)*c(var,m),m=1..n); end proc: From the solutions to Lab 5: sine series rountines Again, we change the sine term so the period is 2L s := (x,n) -> sin((2*Pi*n*x)/(2*L)): and the integrals go from -L to L B:=proc(expr,var,n) evalf(simplify(int(expr*s(var,n),var=-L..L)/int(s(var,n)*s(var,n),var=-L..L))); end proc: sineFP:=proc(expr,var,n) add(B(expr,var,m)*s(var,m),m=1..n); end proc: Finally, we define a full Fourier series that is the sum of the sine and cosine series fullFP := proc(expr,var,n) cosineFP(expr,var,n)+sineFP(expr,var,n); end proc:

First a routine to calculate the coefficients, copied from the Lab 6 solutions. legcoeff := proc (f,x,n) local a; a := evalf(((2*n+1)/2)*int(f*P(n,x),x=-1..1)); return(a); end: Now the function that calculates the series expansion. Note that I've used add instead of sum. legseries := proc(f,x,nmax) local s; s := add(legcoeff(f,x,n)*P(n,x),n=0..nmax); return(collect(s,x)); end:

First we define the function of interest (replace this with the function given in the Homework) and define L: f := x-> (x^2-1)*sin(x): L:=1: Examples of using each of the series fitting routines above: Fourier series: fourier5:=fullFP(f(x),x,5);

LCwtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyomSSNQaUdGJiIiIkkieEdGKEYsJCErOyE0a24kISM1LUYkNiMsJEYqIiIjJCIrSnU8Nk0hIzYtRiQ2IywkRioiIiQkISs1dUoiZSohIzctRiQ2IywkRioiIiUkIitdJylwblJGPi1GJDYjLCRGKiIiJiQhK0lcNTk/Rj4=

Taylor series...NOTE THAT TO GET TERMS up to say, x^5, you need to tell Maple to go to the 6th term: ftaylor5:=unapply(convert(taylor(f(x),x,6),polynom),x);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCg5JCEiIiokRioiIiQjIiIoIiInKiRGKiIiJiMhIigiI1NGJUYlRiU=

And, finally, Legendre: fleg5 := legseries(f(x),x,5);

LChJInhHNiIkIStTIkdIKioqISM1KiRGIyIiJCQiKysrWGc2ISIqKiRGIyIiJiQhKytdUDk7Ric=

Below, a sample plot with a legend, reminding you how it is done...in the graph you make the four different functions will actually look different from each other.: plot([f(x),fourier5,ftaylor5(x),fleg5],x=-1..1,color=black,linestyle=[1,2,3,4],legend=["f","Fourier","Taylor","Legendre"]);

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