Page 295 - DMTH201_Basic Mathematics-1

Basic Mathematics – I Richa Nandra, Lovely Professional University

Notes Unit 11: Parametric Differentiation

CONTENTS
Objectives
Introduction

11.1 The Parametric Definition of A Curve
11.2 Differentiation of A Function Defined Parametrically
11.3 Second Derivatives

11.4 Parametric Functions
11.5 Summary
11.6 Keywords
11.7 Self Assessment
11.8 Review Questions

11.9 Further Readings

Objectives

After studying this unit, you will be able to:
Differentiate a function defined parametrically

Find the second derivative of such a function
Introduction

Some relationships between two quantities or variables are so complicated that we sometimes
introduce a third quantity or variable in order to make things easier to handle. In mathematics
this third quantity is called a parameter. Instead of one equation relating say, x and y, we have
two equations, one relating x with the parameter, and one relating y with the parameter. In this
unit we will give examples of curves which are defined in this way, and explain how their rates
of change can be found using parametric differentiation.

Instead of a function y(x) being defined explicitly in terms of the independent variable x, it is
sometimes useful to define both x and y in terms of a third variable, t say, known as a parameter.
In this unit we explain how such functions can be differentiated using a process known as
parametric differentiation.

11.1 The Parametric Definition of A Curve

In the first example below we shall show how the x and y coordinates of points on a curve can be
defined in terms of a third variable, t, the parameter.

Example: Consider the parametric equations
x = cos t y = sin t for 0 t 2 (1)
Note how both x and y are given in terms of the third variable t.

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