1(a)

The local error at step j is the discretization error introduced in the j'th step of a numerical method for solving a differential equation. It would be the error in y_jif y_j-1 were correct. Thus it is $\psi_{j-1}(x_j) - y_j$, where $\psi_{j-1}(x)$ is the solution of the differential equation with initial condition y(x_j-1) = y_j-1.