Calculus Therom

A Numerical Insight for the Fundamental Theorem of Calculus

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The purpose of this article is not to provide some rigorous statement, neither a rigorous clever proof of the fundamental theorem of calculus. It is rather to give some sort of numerical insight on an intuitive simplified statement of the fundamental theorem of calculus.

Simplified intuitive statement of the fundamental theorem of calculus:
” Integration and differentiation are reverse operations on a function ##f(x)##. That means :

  1. if we take the integral of the derivative of a function, the result is the function itself (within some constant c): That is ##\int^x \frac{df}{dt}dt=f(x)+c##
  2. if we take the derivative of the integral of a function , the result is the function itself again. That is ##\frac{d\int^xf(t)dt}{dx}=f(x)##”

In the rest that will follow I am gonna give some non rigorous yet I believe intuitive and useful explanation for why 1 and 2 are true.

I shall use as starting points what I call the “loose” (or we can say numerical) definitions of the integration and differentiation operators.

The integration operator is defined as ##\int_{x_0}^{x_n} f(x)dx=\sum_{i=0}^{i=n-1}f(x_i)\Delta x_i## where ##\Delta x_i=x_{i+1}-x_i## and the ##\{x_i\}## is a partition of the interval ##[x_0,x_n]##, and in order for this definition to be close to the rigorous definition, ##n## must be large or equivalently the maximum of ##\Delta x_i, 0\leq i \leq n## must be small enough.

The differentiation operator is defined as ##\frac{df}{dx}=\frac{f(x+\Delta x)-f(x)}{\Delta x}## where again we need the ##\Delta x## to be small enough in order for this to be close to the real derivative of the function at point x.

We all know that the rigorous definitions involve the limit of the sum as ##n## goes to infinity (for the integration) , and the limit of the ratio as ##\Delta x## goes to zero (for the differentiation).

But if we start talking about limits , this will open the doorway for an analytical approach to this subject. For my purpose lets just stick to the numerical definitions, simply stating that the ##\Delta x_i## and ##\Delta x## should be small enough.

Furthermore lets assume that we choose the ##\{x_i\}## such that all the ##\Delta x_i=\Delta x## are equal and equal to the ##\Delta x## of the differentiation. Then lets see why 1. is true:

$$\int_{x_0}^{x_n=x}f'(t)dt=\sum_{i=0}^{n-1}f'(x_i)\Delta x_i=\sum_{i=0}^{n-1}\frac{f(x_i+\Delta x)-f(x_i)}{\Delta x}\Delta x_i=$$
$$=\sum_{i=0}^{n-1}[f(x_{i+1})-f(x_i)]=f(x_n)-f(x_0)=f(x)-f(x_0)$$

We see that because we have chosen ##\{x_i\}## such that all the ##\Delta x_i=\Delta x## many desired simplifications happen, and the integral ends up as a telescopic series, which gives the desired result. We know from the analytical thinking that the ##\{x_i\}## can be chosen in any way and 1 can still be true, but in the analytical approach  we are taking the limits for ##\Delta x_i \rightarrow 0## (for the integration) and ##\Delta x \rightarrow 0## (for the differentiation) , so we are making ##\Delta x_i## and ##\Delta x## both  infinitesimally small, so in a way they are sort of equal again.

Lets now see why 2. is true

$$\frac{d\int_{x_0}^{x_n=x}f(t)dt}{dx}=\frac{\int_{x_0}^{x_{n+1}=x+\Delta x}f(t)dt-\int_{x_0}^{x_n=x}f(t)dt}{\Delta x}=$$
$$=\frac{\sum_{i=0}^{n}f(x_i)\Delta x_i-\sum_{i=0}^{n-1}f(x_i)\Delta x_i}{\Delta x}=\frac{f(x_n)\Delta x_n}{\Delta x}=f(x_n)=f(x)$$

One perhaps oversimplified and naïve way of thinking about why 1 and 2 are true, is to say that integration involves summation and multiplication, while differentiation involves subtraction and division, therefore integration and differentiation are reverse operations because summation is the reverse of subtraction and multiplication is the reverse of division. Someone that has gone through the analytical and rigorous way of reading about the fundamental theorem of calculus, knows that there is more to that, however I believe this (perhaps naïve and oversimplified) thinking can be a (loose) but I believe intuitive basis for why integration and differentiation are reverse operations.

 

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