8.2.1 Introduction

The derivative $f\text{'}(x_0)$ of a differentiable function $f$ describes how the function values change "in the vicinity of" a point $x_0$. If, for example, the derivative is positive, then the function $f$ is monotonically increasing. Geometrically, this means that the slope of a tangent line to the graph at the point $x_0$ is positive. The derivative provides a local view of the function at every point $x_0$. This way, a lot of detailed information can be collected.
Conversely, a "global characteristic" is obtained if a "summary" of the function is generated by summing up the weighted function values. In mathematics, this sum is called the integral or integral value of the function. Geometrically, this concept provides a way to calculate the area under the graph of a function. It was Bernhard Riemann who specified this approach and who gave his name to the so-called Riemann integral.