to produce Runge–Kutta methods. A closely related derivation is to substitute the forward finite difference formula for the derivative, in the differential equation . Again, this yields the Euler method. A similar computation leads to the midpoint method and the backward Euler method. Finally, one can integrate the differential equation from to and apply the fundamental theorem of calculus to get: Now approximate the integral by the left-hand rectangle method (with only one rectangle): Combining both equations, one finds again the Euler method. This line of thought can be continued to arrive at various linear multistep methods. Local truncation error The local truncation error of the Euler method is the error made in a single step. It is the difference between the numerical solution after one step, , and the exact solution at time . The numerical solution is given by For the exact solution, we use the Taylor expansion mentioned in the section Derivation above: The local truncation error (LTE) introduced by the Euler method is given by the difference between these equations: This result is valid if has a bounded third derivative. This shows that for small , the local truncation error is approximately proportional to . This makes the Euler method less accurate (for small ) than other higher-order techniques such as Runge-Kutta methods and linear multistep methods, for which the local truncation error is proportional to a higher power of the step size. A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem. If has a continuous second derivative, then there exists a such that In the above expressions for the error, the second derivative of the unknown exact solution can be replaced by an expression involving the right-hand side of the differential equation. Indeed, it follows from the equation that Global truncation error The global truncation error is the error at a fixed time , after however many steps the method needs to take to reach that time from the initial time. The global truncation error is the cumulative effect of the local truncation errors committed in each step. The number of steps is easily determined to be , which is proportional to , and the error committed in each step is proportional to (see the previous section). Thus, it is to be expected that the global truncation error will be