# Proof of Fundamental Theorem of Calculus

*The quadrature of all figures follow from the inverse method of tangents, and thus the whole science of sums and quadratures can be reduced to analysis, a thing nobody even had any hopes of before.*(Leibniz)*Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method.*(Leibniz

Let us now study the effect of the time step in solution of the basic IVP

- ,

by Forward Euler time stepping

We compare taking one step with time step with two steps of time step , for a given :

- ,

where is computed with time step , and we assume that the same intial value for is used so that . Assuming that is Lipschitz continuous with Lipschitz constant , we then find that

- .

Summing now the contributions from all time steps with , where is a final time, we get using that ,

- ,

where thus is computed with time step and with time step .

Repeating the argument with successively refined times step , we get

for the difference between computed with time step and computes with vanishingly small time step, since

- .

We have now proved the Fundamental Theorem of Calculus:

**Theorem** If is Lipschitz continuous, then the function defined by Forward Euler time-stepping with vanishing time step, solves the IVP: for , .

# Understanding the Fundamental Theorem

The proof shows what it means *to understand* the Fundamental Theorem of Calculus: This is to realize that (letting denote a finite time step and a vanishingly small step)

- ,

as a consequence of

- ,

where the sum is referred to as a *Riemann sum,* with the following bound for the difference

- ,

assuming is Lipschitz continuous with Lipschitz constant .

In other words, *understanding* the integral of a function , means to understand that:

- is determined by Riemann sums with vanishingly small step size, as the solution to the IVP , ,
- the difference between two Riemann sums with mesh size and , is bounded by (or more precisely by ).

# Even Better Understanding

As a serious student, you now probably ask: In precisely what sense the differential equation is satisfied by an Euler Forward solution with time step ? It certainly is so constructed, but can we get a direct verification? One way to do this is to associate a continuous piecewise linear function determined by the values at the discrete time levels ,again denoted by . We then have on each interval , by the definition of :

- ,

from which we conclude that

- .

We can thus say that satisfies the differential equation for all with a precision of . In other words, the* residual* is smaller than .

We have now understood the Fundamental Theorem even better, right?

We shall see below that extending a function defined on a discrete set of points to a continuous piecewise linear function, is a central aspect of approximation in general and of the *Finite Element Method FEM *in particular.

# To Think About

- What is fundamental about the Fundamental Theorem?
- Why is ? (compare with last argument)
- What is the effect of finite precision computation according to Constructive Calculus in Finite Precision? (see below)
- What is the Riemann sum error using the Trapezoidal Rule ?

Hint:

- .

Babbage’s Difference Engine No. 2 1847.

# Fundamental Theorem in Finite Precision

Lipschitz continuity in the presence of finite precision can be defined as follows: A real-valued function of a real variable is *Lipschitz continuous* *with Lipschitz constant in finite precision *, if for all and

- .

We see that here will effectively be bounded below by . With this extension of the concept of Lipschitz continuity to finite precision, the first step of the above proof takes the form

- .

In the second step, the repetition with successively refined times step , is performed until for some natural number , which gives

for the difference between computed with time step and computed with time step . Here is the 2-logarithm of and thus is a constant of moderate size (not large). In essence, we thus obtain the previous estimate with replaced by and appears as a lower bound of the time step

Next: Rules of Integration Previous: Rules of Differentiation

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