MA1102 Calculus Tutorial 5

Topics Covered:

Differentiation. The definition. Derivative of sum of functions. The product and quotient rule. Chain rule and application. Derivative of piecewisely defined functions. Derivative of trigonometric functions. Implicit differentiation.

Textbook: Chapter 5

Function can be thought of as a record of change. Continuity at a point says something about the change of the function locally, no abrupt behaviour, no sudden jump, small change in x produces small change in f (x). This idea is extended to the whole of the closed and bounded interval to give us some global information. Because small change in x produces small change in f (x), we may perhaps expect the continuous function to have some measure of the rate of change, the derivative at least at some point in the domain of f . This turns out to be false. Example of Bolzano, a continuous function which is not differentiable everywhere drives home the message that geometric intuition is not always reliable. The definition of the derivative as the limit of the difference quotient is a subtle one. Information at a point does not necessarily give global information. While you hone your skill at differentiation in this tutorial, bear in mind the application of derivative. While pathological examples like those of Bolzano, Weierstrass hardly ever present themselves in most application outside of mathematics, conceptually derivative is in some sense a rather deep concept. The definition rules here. You should expect to keep going back to the definition and examine the reason why you expect the limit of the difference quotient to exist. Ask question of this type, particularly when the mechanical rules for sum, product and quotient and the chain rule do not seem applicable. You may often have to examine the left and right limits too.