Wave Equation, part 3

Now we consider a boundary/initial value problem with non-zero initial velocity t.

Here the initial position function f is the same as before

Graph of f

The initial velocity function g has the graph shown below.

Graph of g

We know that, in this case, the solution y(x,t) of the initial/boundary value problem may be approximated by

Here, the half-range Fourier sine approximations to g are
Half-Range Sine Expansions for g

.
Thus, the coefficients d_k are given by
Formula for d[k]

Calculate the coefficients d_k, and check the half-range approximations to g.
Graph the approximation A2 to the solution of the initial/boundary value problem as a function of x for a range of values of t between 0 and 8. How does the solution of this problem differ from the one with zero initial velocity?