RI Chapter 9 Calculus

RAFFLES INSTITUTION H3 Mathematics 9820 _____________________________________________________________________________________________ _________________________ Chapter 9: Calculus Page 1 of 21 Chapter 9 Calculus The topics for Calcul us, namely Differentiation, Integration, Power Series and Differential Equations, should be familiar from your study of H2 Mathematics. In this chapter we collect together some additional tips and techniques that might be useful in H3 Mathematics. SYLLABUS INCLUDES  Knowledge of the following topics (differentiation, Maclaurin series, integration techniques and differential equations) and suitable extensions  Proving statements involving derivatives and integrals CONTENT 1 Derivatives 2 Integration 2.1 Riemann Sums 2.2 Integration Techniques 2.2.1 Reduction Formula 2.2.2 Useful Substitutions 3 Maclaurin Series Techniques to find Series Expansions 4 Differential Equations Variable Separable Equations and Integrating Factor

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 9: Calculus Page 2 of 21 1 Derivatives It is important to understand that the derivative is defined via a limiting process. Definition 1 Let I be an open interval and 0x be a point in I. We say that a function defined on I is differentiable at I if the limit 0 0 0 f ( ) f ( )lim xx xx xx   exists. In this case, the limit is called the derivative of f at the point 0x , and is written as 0f ( )x . If this limit exists for every point in I, we say that f is differentiable on I and its derivative is written as f ( )x . Example 1 Let us use the definition above to find the derivative of 2f ( )xx . For any real number x, we have 22 0 0 0 f ( ) f ( ) ( )f '( ) lim lim lim(2 ) 2 . h h h x h x x h xx x h x hh           This derivation is also known as differentiation from first principles. Exercise 1 Find from first principles the derivatives of the following: (i) f ( ) nxx , where n is a positive integer; (ii) f ( ) sinxx . In similar fashion, we can work out the derivatives of simple functions and then use familiar results like the product, quotient and chain rules to work out derivatives of more complicated functions. Such techniques will be assumed to have been mastered and will not be pursued here. Higher Order Derivatives If a function is such that f ( )x is still differentiable, we can form the second derivative f ( )x and similarly for higher order derivatives. When it exists, th e nth derivative of y = f(x) with respect to x is denoted by ()f ( )n x or d d n n y x . Using the product rule and induction, we can prove the following generalized product rule: If f and g are n times differentiable, then ( ) ( ) ( ) 0 (fg) f g n n k n k k n k     for a