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 all positive integers n, where ()f n denotes the nth derivative of f. This result was an exercise in Tutorial 1, and the proof is basically done via Mathematical Induction.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 9: Calculus Page 3 of 21 Here, let us just interest ourselves in finding a formula for the nth derivative of certain functions. Exercise 2 Find a formula for the nth derivative of the following functions (i) 52e xyx ; (ii) e sinxyx . Let us end the discussion on derivatives by introducing two important results in real analysis involving derivatives. There is no need to know how to prove them, but you should be able to have an intuitive understanding of why they are true. Rolle’s Theorem Rolle’s Theorem states th at if f is a continuous function on an interval [ a, b] and differentiable on (a, b), given that f(a) = f(b) = 0, then there exists a c in the interval (a, b) such that f ( ) 0c  . Mean Value Theorem The Mean Value Theorem states that if f is a continuous function on an interval [ a, b] and differentiable on (a, b), there exists a c in the interval (a, b) such that f ( ) f ( )f ( ) bac ba    . We can understand this result geometrically in the following manner: Typically one draws the graph of a function y = f(x) which happens to make several turns (i.e. has inflection points) on an interval [a, b] and the straight line connecting the points ( a, f(a)), (b, f(b)) on the same set of axes. Then one observes that the slope of this line is equal to the slope of the tangent to the graph of the function at least at one value of x in (a, b).

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 9: Calculus Page 4 of 21 Applications of the Mean Value Theorem Example 2 The function f ( ) e xx  has derivative f ( ) e xx  for all real x. We will now show that e1x x for all real x. If x = 0, the inequality is an equality. If x > 0, applying the Mean Value Theorem, we know that there is a real number c such that 0 cx and 0e e e ( 0)xc x   . Since e1c  for 0x , 0e e e ( 0) e 1x c x x x x       . Repeat a similar argument for x < 0 and we are done. Exercise 3 If 0 ab , show the following: (i) 11 22 tan tan11 b a b a baba     ; (ii) 1 ln 1a b b b a a     ; (iii) 22 22 3 a ab bab  . 2 Integration What is commonly referred to as Integration in the A -level syllabus is actually two separate concepts, involving the computation of areas and the inverse of differentiation, i.e. finding which functions, when differentiated, will result in the given functi on. That there is an intimate connection between these two concepts is the statement of a seminal theorem, known appropriately as the Fundamental Theorem of Calculus. 2.1 Riemann Sums Let f :[ , ]ab  be a continuous function. One way to compute the area bounded by the curve y = f(x), the x-axis, the lines x = a and x = b is as the limit of Riemann sums. Definition 2 A partition of [a, b] is a finite set of points {}kx , k = 0, 1, …, n such that 01 ... na x x x b     . We will only use partitions in which the points {}kx are evenly spaced, that is 1 ba kk nxx   for all k = 1, 2, …, n. Thus ()k k nx a b a   . Such partitions are also known as regular partitions.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 9: Calculus Page 5 of 21 Definition 3 The Riemann sum associated to f and a regular partition with n subintervals, denoted by S(f, n) is given by 11 11 1(f , ) ( )f ( ) f ( ) nn k k k kk b a kS n x x x a b a nn          . S(f, n) can be taken to be an approximation to the area under the curve, with the case for n = 6 illustrated below: To get a better approximation to the area, it seems natural to increase the number of rectangles used, so we can attempt to let n → ∞ to com pute the exact area. Unfortu nately, it is not always the case that a limit will exist. In those cases in which the limit exists, it is defined to be the (Riemann) definite integral f ( ) d b a xx . In other words, 1 1f ( ) d lim f ( ) b n n ka b a kx x a b a nn       , provided the limit exists. For the above Riemann sums, we used the left end point of the rectangles to compute the height 1f ( )kx  . We could equally well have used the right end point, giving the following equivalent result 1 f ( ) d lim f ( ) b n n ka b a kx x a b a nn       . For the A -levels, you do not need to worry about whether the limit exists.