Differenciation and Integration of piecewise functions

Hello, I'm using matlab R2016a, and I cannot find a way to differentiate or integrate piecewise functions like the following:

f(x) = x if x<=1 and 2x if x>1

Can anyone help me? I need to do this both numerically and improperly Thanks Joao

Answer 1

[Walter Roberson]

You mention you are using R2016a. For MATLAB R2008b to R2016a, the only way to do that symbolically is to dodge into MuPAD in order to create a piecewise expression inside the symbolic engine. This situation changed with R2016b, which added a piecewise() function call.

G = evalin(symengine, 'piecewise([x < 1, x], [x > 1, 2*x])');

You can then test with a few different values:

>> subs(G,x,[-1 0 1 2])
ans =
[ -1, 0, NaN, 4]

Notice the NaN: that is there because your expression does not define the result for x of exactly 1. Boundary conditions are important for integration!

You can then

>> int(G)
ans =
piecewise(1 < x, x^2, x < 1, x^2/2)

For numeric work, create a function:

function y = f(x)
  y = nan(size(x));
  mask = x < 1;
  y(mask) = x(mask);
  mask = x > 1;
  y(mask) = 2 * x(mask);

You can then do

integral(@f,-5,5,'waypoints',1)

You need the 'waypoints' option because of the discontinuity.

You can cross-check with a numeric approximation:

x = linspace(-5,5,101);
y = f(x);
cumtrapz(x, y)

Why does this end in NaN? It's that pesky boundary condition again! If only some definite value had been assigned for the point at x = 1 ...