Overview

Derivative can be described informally as a rate of change and as such it is extremely important in Science and applications. In this chapter we introduce derivatives as limits, establish their properties and use them in studying deeper properties of functions and their graphs. We also extend the Taylor Theorem from polynomials to power series and explore it for applications (within mathematics).

13.1 Introduction

Let us think of a body moving along a straight line. Let the distance of this body from a fixed point be a known function of time, say f. During the time interval [ t, t+ h] the body travels the distance f(t+h) ? f(t) and the average velocity will be

The velocity shown on the speedometer of a car or a plane is the instantaneous velocity and for our moving body this is the limit of the above expression as h ?0.

Consider another example, this time from geometry; see Figure 13.1. The secant line joining the points (x, f(x)) and (x+h, f(x+h)) has the slope

Fig. 13.1: Secant becoming a tangent

As h ?0 the points (x, f(x)) and (x+h, f(x+h)) coalesce and the secant becomes a tangent. This corresponds to moving the ruler aligned along the secant line carefully so that the point (x+h, f(x+h)) moves towards (x, f(x)) until it reaches its limit position, and then the secant becomes a tangent.

Definition 13.1

The number

(13.1)

is called...