Note: If you are already familiar with Matlab, you may use it to do this exercise instead of Maple. Please be sure to show the Matlab commands you use to complete the exercises.
Use Maple to plot the Bessel function J1(x) for a range sufficient to show the first three positive roots (not counting the one at x = 0). In the answer file Mylab06.txt, give an estimate of the zeros, based on reading the plot. Just to be clear, J1(x) is known to Maple as BesselJ(1,x) and in Matlab as besselj(1,x)
When you are satisfied with your result, copy your answer from the Maple window to the file Mylab06.txt. Please provide both input and output lines for the results you want to hand in.
The simplest way to do this is to use your mouse to copy from the Maple window and paste it into your emacs window.
Hint: For this exercise you need to evaluate the derivative of the
Bessel function. Maple insists on being much more precise about this
procedure than most scientists are used to. First you have to find
the function that is the derivative of the Bessel function. Then you
have to evaluate the derivative function at the desired point. There
are two ways to do this in Maple, namely, with diff or
D for the derivative.
f := x -> BesselJ(1,x); g := x -> evalf( D(f)(x) );
f := x -> BesselJ(1,x); g := x -> evalf( subs(y=x, diff(BesselJ(1,y),y) ) );
You don't need to write a Maple procedure for this. Use the same
technique you used for the fixed point method, i.e. write an
assignment statement that gives the new Newton Raphson estimate
of x based on the old estimate. Then repeat by hand until you
are satisfied that it has converged.
In the answer file, show the Maple commands you used and the result you found.
Note that with fsolve, you can specify a range of x values in which to search for the root. Check it out with ?fsolve.