Some commonly encountered sets and their notations

Transcription

1 NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their notations the empty set N natural numbers i.e. {0,,,3.. } 3 Z integers 4 R real numbers 5 C comple numbers A = {, 4, 9 } - tabular form A = { = n, n =,, 3, } - set builder form If is an element of A, we write A. B is a subset of a set A if every element of B is an element of A. Notation: B A A B is the collection of all elements which are in either A or B; i.e. A B = { A or B }. A B is the collection of those elements which are in both A and B; i.e. A B = { A and B }.

2 The number of elements in a finite set A is denoted by n(a) OR A Some results : If A and B are disjoint, n(a B) = 0 n(a B) = n(a) + n(b) In general, n(a B) = n(a) + n(b) n(a B) Important Elementary Functions. Trigonometric functions sin cos tan sec = cos cosec (or csc = cot = tan sin

3 . Eponential functions e Indices Laws (for any positive real number a) a a y = a +y a a y = a +y (a ) y = a y 3. Logarithmic functions ln Logarithm Laws (for any positive base) log y = log + log y log y = log log y log n = nlog 4. Inverse trigonometric functions sin - cos - 3

4 tan - 5. Absolute value functions y if 0 = if 0 e.g.. =., -3.7 = 3.7 4

5 Comple Numbers COMPLEX NUMBERS i = C = { a + bi : a, b R : and i = } Real part of z = a + bi, Re(z) = a Imaginary part of z = a + bi, Im(a + bi) = b Algebra of Comple Numbers Addition & Subtraction (a + bi) + (c + di) = (a + c) + ( b + d)i (a + bi) (c + di) = (a - c) + ( b - d)i Multiplication (a + bi) (c + di) = (ac - bd) + ( bc + ad)i Conjugate of z = a + bi, z* = a - bi zz* = (a + bi)(a bi) = a + b Division a + bi = Argand Diagram 5

6 The comple number z = + yi can be represented by the point P(, y) in the Cartesian plane. y Imaginary ais P(, y) + yi Real ais Argand diagram Modulus & Argument Let z = a + bi Modulus of z, z = a + b (= length of OP) Argument of z, arg z = the angle α in the interval (-π, π ] which OP makes with the positive -ais Eample Find the modulus and argument of these comple numbers: + i, - + i, - - 3i and 3+ i Polar/Trigonometric Form 6

7 y r P(, y) = r cosθ y = r sinθ θ r = + y z = + y i = r cosθ + i( r sinθ ) = r (cosθ + i sinθ ) --- trigonometric or polar form of z. Euler formula ( eponential form ) It can be shown that cosθ + i sinθ = (Euler Formula) From this, we can deduce that iθ e θ = iθ i θ ( e i e ) θ iθ i sin and cos θ = ( e + e ) DeMoivre s Theorem ( cosθ + i sinθ ) n = cos nθ + isin nθ n is a positive integer Applications of DeMoivre s Theorem 7

8 Eamples Evaluate ( + 3 i ) 9 Find sin θ + sinθ + sin3 θ + L + sinnθ 3. The compound angle formulae are: 8

9 NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE Limits DIFFERENTIATION lim f ( is the value of f( as approaches c c from the left. We call it the left limit of f( as tends to c. lim f ( is the value of f( as approaches c c+ from the right. We call it the right limit of f( as tends to c. Consider a function shown below: y f : R R whose graph is (,4) (,0) What is the value of

10 (i) lim f ( (ii) lim + f ( When the left and right limits coincide, we call the common value the limit of f( as approaches c: lim f ( = lim f ( = lim f ( c c c+ Otherwise, we say that Eample lim c f ( does not eist. The function f is defined on [-3, 3] by f 5 + k = + 3 ( Find the value of k for which Sketch the graph of f. < 3 lim f ( eists. Continuity A function f is said to be continuous at = c if (i) lim f ( = lim f ( = L ( L is finite) c c+ (ii) f ( c) = L Roughly speaking, a function f is continous at = c if there is no jump on the graph of y = f( at = c, and in this case, lim f ( = f ( c) c

11 Derivatives For a function f(, its derivative at = a is defined by Note that if the above limit does not eist, we say that f is not differentiable at =a. Eamples. f( = We have f '( = lim h 0 f ( + h) h f ( = lim( + h) = h 0 ( + h) = lim h 0 h f( = ( 0) 3

12 Techniques of Differentiation List Of Derivatives Function sin cos tan cosec sec cot e Derivative cos - sin sec - cosec cot sec tan - cosec e ln / sin - cos tan - 4

13 Rules of Differentiation d d Chain Rule ( f o g) ( = f '( g( ) g'( d d Product Rule f ( g( = f '( g( + g'( f ( d f ( = d g( f '( g( g'( f ( ( g( ) Quotient Rule Eamples Differentiate with respect to (i) cos 3 ( e ) (ii) (iii) ln tan + 3 e sin

14 Implicit Differentiation Eample If tan ( 3 y) + e -3y dy =, find d. Parametric Equations For eample, a circle, centre (0, 0) and radius 3 units, is given by the pair of parametric equations : = 3cost and y = 3sin t where 0 t < π is the parameter. dy dy dt dy d = = d dt d dt dt d y d dy d dy dt = = d d d dt d d Eample d y =. d e Let = e t and y = t - e t. Show that t Derivative of epressions of the form f ( g ( Eample Differentiate (sin e 3 w.r.t. 6

15 Evaluation Of Limits (Limits Revisited) How to deal with lim f ( when f is c discontinuous or undefined at = c?? L Hospital s rule if lim f (,lim g( c c f ( lim c g ( ) = f '( lim c g '( ) are both zero or both infinity. It is important to note that the above rule is only applicable to indeterminate cases of the form 0 0. or Eamples Evaluate (i) (ii) lim 4 lim (iii) tan tan 5 lim 0 sin 4 + 5sin 6 7

16 (iv) lim ( + 5)(3 + ) (Limits at infinity) (v) lim e 3 4 (vi) lim ln( ) 0+ 8

17 NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS LECTURE 3: INTEGRALS Standard Integrals ( a 0) n+ n ( a + b) ( +C ( n ) ) a a + b) d = ( n + d = ln a + b + C a + b a a + b a+ e d = e b a + C sin( a + b) d = cos( a a b) cos( a + b) d = sin( a a b) sec ( + b) d = tan( a a b) csc ( + b) d = cot( a a b) d = sin a a + C C + C a + C + a + C d = tan + C a a a + + a d ln + C a a = a

18 Eamples Find d 9 (i) 4 + (ii) d (iii) d 4 Trigonometric Identities cos A = ( + cos A) sin A = ( cos A) sin Acos A = sin A tan A = sec A cot A = csc A sin A cos B = sin( A B) + sin( A + [ B) ] A sin B = cos( A B) cos( A + cos A cos B = [ cos( A B) + cos( A + B) ] sin [ B) ]

19 Eamples Find (i) ( sin cos5 d (ii) sin ( sin ) d + d (iii) tan ( tan 3cos Integration By Substitutions Eamples f '( g' ( f ( ) d = g( f ( ) + C Find (i) (ii) e 4 + e (ln 3 d d (using the substitution u = e ) (using the substitution u = ln ) (iii) ( cos 4 + sin d (iv) sin 3 cos 6 d 3

20 Integration By Parts or u ( v'( d = u( v( u'( v( d udv = uv vdu Eamples Find (i) 0 e d (ii) ln d (iii) cos d (iv) e sin3d (These notes are based on the book Introductory Mathematics by Ng Wee Seng.) 4