COURSE OUTLINES PART ONE

• • • • Maxima and Minima of Functions of a Single Independent Variable Tangents and Normals Differentiation Techniques of Differentiation

PART TWO

• Techniques of Integration: Indefinite Integrals, Integration by Parts, Definite Integrals, Improper Integrals • • Applications to Engineering Systems Introduction to Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE)

PART THREE

• • • Properties and Evaluation of Matrices Introduction to Symmetric and Skew-symmetric Matrices Simple simultaneous Linear Equations

1

PART I

DIFFERENTIATION 1.0

1.1

Maxima and Minima, Tangents and Normals

Gradients: The gradient of a straight line is the rate of change of y with x and is measured by =

the increase in the value of y divided by the corresponding increase in the value of x. Thus, if = the tangent at that point and when the equation of the curve is known differentiation provides a method of calculating the gradient exactly. Example: Calculate the gradient of the curve Solution: 1.2 =9 − 4 + 5. When x = 3, =3 −2 + 5 − 7 at the point (3, 71). + , = gradient of the line. The gradient of a curve at a particular point is the gradient of

= 74 = gradient of the curve. = ( ),

Tangents and Normals: For the curve given by

gives the gradient of the at that point,

tangent at any point. Given the co-ordinates of a point on the curve and the value of the equation of the tangent can be easily obtained. Example: Find the equation of the tangent to the curve Solution: = + 3 + 2; =3 = + 3 + 2 when x = – 2

Therefore the required equation is

= 15 + 18 or

+ 3. When x = – 2,

= 15. Also, when x = – 2, y = – 12.

− 15 − 18 = 0.

A normal is a line perpendicular to the tangent. Recall from geometry that if two lines are perpendicular then the product of their gradients is – 1. Example: Calculate the equation of the normal for the previous example at the