The fundamental theorem of calculus

What we have seen is that to each function of the form $f:[0,\infty)\rightarrow[0,\infty),\;f(x) = kx^p$ where k > 0 and $p\in\{0,1,2,\dots\}$ we can assign to new functions, the slope function, we gives us a reasonable definition of the slope of the graph at at point, and the area function which gives us a reasonable formula for the area under the graph. The slope function of f is denoted by f' and the area function by the rather complicated notation:

$$\int_0^a f(u)\;du$$

With the usual convention that x⁰ = 1 even when x = 0 we have shown that for our limited set of functions,

$$\begin{array} {rcl} \int_0^a f(u)\;du & = & {\displaystyle k\frac{a^{p+1}}{p+1},}\\ f'(a) & = & pa^{p-1}.\end{array}$$

Here is the surprising part: Make a new function $F:[0,\infty)\rightarrow[0,\infty)$,

$$F(x) = \int_0^x f(u)\;du = k\frac{x^{p+1}}{p+1}.$$

Then

$$\begin{array} {rcl} F'(x) & = & (k/(p+1))(p+1)x^{p+1-1}\\ & = & kx^p\\ & = & f(x)\end{array}$$

and

$$\begin{array} {rcl} \int_0^a f'(u)\;du & = & \int_0^a kpu^{... ...u \\ & = & kpa^p/p\\ & = & ka^p\\ & = & f(a) - f(0).\end{array}$$

Together, these two observations are known as the fundamental theorem of calculus, and our object will be to see for what functions besides these simple power functions these results hold.