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HSC Extension 2 Mathematics
HSC Extension 2 Mathematics
HSC Extension 2 Mathematics Topics Complex Numbers Curve Sketching Polynomials Integration Conics Volumes Mechanics Harder Extension 1
HSC Extension 2 Mathematics Topics     Complex Numbers      Curve Sketching      Polynomials      Integration      Conics ...
Maths Extension 2 Complex Numbers Complex Number Rules z x iy A complex number is represented in this form The conjugate z is x iy z x iy z rcis _____OR z r cos isin r x2 y 2 y tan 1 x z1z2 z1 z2 The Modulus Argument Mod Arg form of representing a complex number The Modulus of a complex number Note z r The Argument of a complex number Note arg z left Multiplication and Division rules of the Modulus and the Argument z1 z 1 z2 z2 arg z1z2 argz1 argz2 2 z arg 1 arg z1 arg z2 2 z2 z z x2 y 2 below More complex number rules z z z 2 z 2 x2 y2 z z 2x z1 z2 z1 z2 z z 2yi z z 1 1 z z 2 2 z1 z2 z1 z2 arg z arg z z 1 1 z 2 z z zn 1 z 1 zn 1 zn 2 z 1 _______________n is odd zn 1 z 1 zn 1 zn 2 z 1 _______________n is odd zn 1 z 1 z 1 zn 2 zn 4 z 1 _________ n is even arg zn n arg z http www geocities com fatmuscle HSC 1
Maths Extension 2 - Complex Numbers  Complex Number Rules   z   x   iy    A complex number is represented in this form   T...
Maths Extension 2 Complex Numbers General Ideas of Complex Numbers Complex Numbers are written in the form of a real part and an imaginary part Complex Number x iy Real x Imaginary y i i2 i3 i4 1 1 i 1 P x y Modulus z r r x2 y 2 Argument y arg z tan 1 x Modulus Argument form of a Complex Number z r cos isin z rcis x r x rcos z x iy rcis isin r cos isin rcis cos Euler s Formula ei cos isin DeMoivre s Theorem y r y rsin sin zn rncis n cos i sin n cos n i sin n http www geocities com fatmuscle HSC 2
Maths Extension 2 - Complex Numbers  General Ideas of Complex Numbers Complex Numbers are written in the form of a real pa...
Maths Extension 2 Complex Numbers Proof by Mathematical Induction cos i sin n cos n i sin n Let n 1 LHS 1 _ cos i sin cos i sin RHS cos 1 i sin 1 cos i sin true for n 1 Assume true for n k _ cos i sin k Let n k 1 _ cos i sin k 1 cos i sin 1 cos i sin k cos i sin cos k i sin k cos cos k sin sin k i cos sin k i sin cos cos k i sin k cos k 1 i sin k 1 RHS cos k i sin k cos k 1 i sin k 1 true for n k 1 True for n 1 True for n k True for n k 1 True for all positive integer values of n http www geocities com fatmuscle HSC 3
Maths Extension 2 - Complex Numbers  Proof by Mathematical Induction  cos     i sin      n   cos n     i sin n   Let n   1...
Maths Extension 2 Complex Numbers Expressing Complex Numbers in Mod Arg Form 3 3i Mod z x2 y 2 9 9 18 3 2 Arg y tan 1 x 3 3 tan 1 tan 1 1 135 3 4 z rcis 3 3 3 3i 3 2 cos i sin 4 4 Expressing Complex Numbers in x iy form 2cis 3 2 3 3 2 cos i sin 2 2 1 1 2 i 2 2 1 i Express in x iy form from a quadratic formula 1 1 4 x 1 3 2 i 2 2 1 3 1 2 1 3 i 1 3i 2 2 2 Conjugate pairs http www geocities com fatmuscle HSC 4
Maths Extension 2 - Complex Numbers  Expressing Complex Numbers in Mod-Arg Form    3   3i Mod z    x2   y 2  9 9   18  3 2...
Maths Extension 2 Complex Numbers Theory a ib c id a ib c id Example Addition 3 4i 2 5i a ib c id a c i b d a ib c id a c i b d Subtraction 3 4i 2 5i a ib c id Multiplication ac iad ibc bd 3 4i 2 5i ac bd i ad bc a ib c id a ib c id c id c id Division 3 4i 2 5i 3 4i 2 5i 5 9i 3 4i 2 5i 1 i 6 15i 8i 20 14 23i 3 4i 2 5i 2 5i 2 5i ac iad ibc bd c2 d 2 6 15i 8i 20 4 25 ac bd bc ad i 2 c2 d 2 c d2 26 7 i 29 29 http www geocities com fatmuscle HSC 5
Maths Extension 2 - Complex Numbers  Theory  a   ib     c   id    a   ib       c   id   Example Addition  3   4i     2   5...
Maths Extension 2 Complex Numbers Finding Square Roots of Complex Numbers x iy x iy a ib a ib 2 a2 b2 2aib x a2 b2 y 2aib Eg Find the Square Root of 5 12i 5 12i 5 12i a ib a2 b2 2aib 5 a2 b2 5 12 2ab 6 ab 6 a b 36 b2 2 b Simultaneous Equations Find b 5b2 36 b4 0 b4 5b2 36 0 b2 4 b2 9 b 2 a 6 2 3 a 6 2 3 Because square roots of 3 2i complex numbers come in ____OR 3 2i conjugate pairs 5 12i To test these we just square them 3 2i 2 3 2i 3 2i 9 12i 4 5 12i 3 2i 2 3 2i 3 2i 9 12i 4 5 12i http www geocities com fatmuscle HSC 6
Maths Extension 2 - Complex Numbers  Finding Square Roots of Complex Numbers x   iy x   iy    a   ib    a   ib 2   a2     ...
Maths Extension 2 Complex Numbers Complex Numbers on the Argand Diagram Z2 Z1 Subtraction of Vectors Flip the tail and do a normal addition z1 zz z1 z2 Addition of Vectors Add tip to tail z1 z2 z1 z2 z1 zz z2 z1 Z3 Z4 Z2 Z1 Multiplication of vectors arg z3 arg z1 arg z2 z3 z1 z2 z4 iz2 Multiplication of i is a rotation through an angle of in the anti clockwise direction 2 Division of i is a rotation through an angle of in the clockwise direction 2 http www geocities com fatmuscle HSC 7
Maths Extension 2 - Complex Numbers  Complex Numbers on the Argand Diagram  Z2  Z1  Subtraction of Vectors Flip the tail, ...
Maths Extension 2 Complex Numbers LOCUS z a ib z x iy z 4 3i z 2 5i by first principles x iy 4 3i 4 x i 3 y 4 3i 4 x 2 3 y 2 2 x 2 5 y 2 x 2 y 2 8 x 6 y 25 x 2 y 2 4 x 10 y 29 4 x 10 y 4 8x 6y 25 4x 4 16 y x 1 y 4 4 2 5i z a ib k x iy 2 5i 2 x i 5 y k is the distance z 2 3 by first principles x iy 2 3 x 2 2 y2 x 2 y 2 2 3 2 3 __ 9 arg z a arg z b arg z 1 arg z 1 ________OR 4 arg z 1 arg z 1 4 A B Can either be 2 circles Check the angle Work out the locus by drawing a line and estimating the angle A arg z 1 B arg z 1 http www geocities com fatmuscle HSC 8
Maths Extension 2 - Complex Numbers  LOCUS  z      a   ib      z      x   iy    z         4   3i       z         2     5i ...
Maths Extension 2 Graphs Graphing y f x y f x y 10 2 x 2 3 x 2 2 3 5 x 10 5 5 10 5 x C 10 5 10 y f x C y C is positive the function is shifted up C is negative the function is shifted down 10 5 10 5 5 C 10 y f x C y 10 C 0 shift function right C units C 0 shift function left C units The function becomes y f x C 5 x 10 5 5 5 C 10 C 10 http www geocities com fatmuscle HSC 1
Maths Extension 2 - Graphs  Graphing y   f x  y   f  x  y 10  2   x-2  -3   x-2  2 -3  5 x 10  5  5  10  5  x  C 10  5  10...
Maths Extension 2 Graphs y C f x y C 1 the function is steeper 0 C 1 the function is shallower 10 5 x 10 5 5 10 5 10 y f Cx Multiply the x co ordinate by 1 C y 10 C 1 thinner steeper C 1 fatter shallower x 1 remains the same 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 2
Maths Extension 2 - Graphs  y   C. f  x  y    C   1, the function is    steeper      0   C   1, the function is    shallow...
Maths Extension 2 Graphs y f x y Flip about the x axis All positive become negative All negative become positive 10 5 x 10 5 5 10 5 10 y f x y 10 Flip about the y axis 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 3
Maths Extension 2 - Graphs  y       f  x  y    Flip about the x-axis   All positive become negative   All negative become ...
Maths Extension 2 Graphs y f x y 10 All negative y values become positive 5 x 10 5 5 10 5 10 y f x y 10 Positive x values are mirrored about the y axis 5 x 10 5 5 10 5 10 y f x y 10 Positive y values are mirrored about the x axis 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 4
Maths Extension 2 - Graphs  y   f  x  y 10    All negative y values become positive  5 x 10  5  5  10  5  10  y   f  x   y...
Maths Extension 2 Graphs y f x y If y value is 1 y value is smaller but still 1 If y value is 1 y value is the same If y value is 0 y 1 y value is larger but still 1 If y value is 0 y value doesn t exist 10 5 x 10 5 5 10 y 5 2 1 5 1 10 0 5 y 2 f x 5 0 5 0 x 0 5 1 y 10 sketch y f x and y f x Both Brown and Green are y 2 f x 5 x 10 5 5 10 5 10 y f x 2 y f x y C 20 If y value 0 y remains 0 1 y becomes steeper above 0 y 1 y becomes steeper below if C is even y f x 0 15 10 5 x C if C is odd y f x has the same sign as f x C 10 5 5 10 5 10 http www geocities com fatmuscle HSC 5
Maths Extension 2 - Graphs  y   f  x   y    If y value is   1, y value is smaller but still   1   If y value is   1, y val...
Maths Extension 2 Graphs y 1 f x y 5 or y f x 1 x Asymptotes where y 0 For f x y values 1 becomes small _________ y values 0 y 1 becomes big Same case for y 0 but negative Points along y 1 and y 1 stay the same 5 5 5 y f 1 x y or 10 x f y INVERSE FUNCTION Flip about the line y x 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 6
Maths Extension 2 - Graphs  y   1 f   x   y 5  or y     f   x        1  x    Asymptotes where y   0   For f x , y values  ...
Maths Extension 2 Graphs Graphing y f x y g x y g x y sin x y 10 5 x 10 5 5 10 5 10 y f x g x y y co ordinates are added 10 5 x 10 5 5 10 5 10 y f x g x y 10 g x co ordinate is subtracted from the f x y co ordinate 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 7
Maths Extension 2 - Graphs  Graphing y   f x , y   g x  y   g  x  y   sin x  y 10 5 x 10  5  5  10  5 10  y   f   x    g  ...
Maths Extension 2 Graphs y f x g x y y co ordinates are multiplied 10 y 5 5 x x 5 10 5 5 5 10 5 10 5 y f x g x y Reciprocal Multiplication rule 1 f x g x 5 x 5 5 5 y 10 5 x 10 5 5 10 5 10 http www geocities com fatmuscle HSC 8
Maths Extension 2 - Graphs  y   f   x .g   x  y    y co-ordinates are multiplied  10  y  5 5 x x 5  10  5  5  5  10  5 10 ...
Maths Extension 2 Graphs y f g x Composite functions y f u u g x y 10 Take f x e x g x sin x 5 x 10 5 5 10 5 10 y 10 10 5 y u 5 x u 10 5 5 10 x 10 5 5 5 5 10 10 10 http www geocities com fatmuscle HSC 9
Maths Extension 2 - Graphs  y   f  g  x     Composite functions y   f  u   , u   g  x   y 10  Take  f   x    e x g   x    ...
Maths Extension 2 Polynomials Polynomials Definitions and properties of polynomials Factors Roots Fields Q Rational R Real C Complex Finding zeros over the complex field Factorization Division of polynomials Remainder Factor Theorem Rational Roots Multiplicity Theorem Repeated Roots Relationship between the roots and coefficients of a polynomial equation http www geocities com fatmuscle HSC 1
Maths Extension 2 - Polynomials  Polynomials   Definitions and properties of polynomials   Factors   Roots   Fields   Q Ra...
Maths Extension 2 Polynomials Definitions and properties of polynomials Polynomial Expression P x p0xn p1xn 1 p2xn 2 pn 1x pn where p0 0 Coefficients p0 p1 p2 p3 Leading term pnxn Constant p0 If pn 1 It is a monic If p0 p2 p3 0 Then P x is a zero polynomial Example 1 P x 3x4 x3 7x2 2x 3 Coefficient of x4 x3 Leading term is 3x4 Constant is 3 Is ___3 Is ___ 1 More points _ai Are the coefficients of the polynomial _a0 Is the constant term _anxn Is the leading term _an Is the leading coefficient _P x Polynomial of degree n _an 1 The polynomial is a monic _P x 0 Null polynomial _P x Expression of polynomial _P x 0 Polynomial equation http www geocities com fatmuscle HSC 2
Maths Extension 2 - Polynomials  Definitions and properties of polynomials Polynomial Expression P x    p0xn   p1xn-1   p2...
Maths Extension 2 Polynomials Factors and Roots Factor A polynomial that divides into another and has remainder 0 x 2 is a factor of x2 4 Root P 2 is a root of x2 4 _x 2 x 0 2 Fields Q Rational Integer numbers 1 2 3 R Real Irrational Numbers Surdic roots occur in conjugate pairs If a b is a root so too will a b C Complex Numbers over the complex field a ib Complex roots occur in conjugate pairs If a ib is a root so too will a ib Example Factorize x4 2x2 15 over Q R C Y2 2Y 15 Y 5 Y 3 Q x2 5 x2 3 2 R x 5 x 5 x 3 C x 5 x 5 x 3i x 3i http www geocities com fatmuscle HSC 3
Maths Extension 2 - Polynomials  Factors and Roots Factor A polynomial that divides into another and has remainder 0  x   ...
Maths Extension 2 Polynomials Finding zeros over the complex field 1 If one root is complex then one of the other roots is it s conjugate 2 If 1 i is a root x 1 i is the factor So x 1 i x 1 i x2 2x 2 How do we get this x z x z 2 x z z x z z Where x2 2x 2 _z z _z z zz 1 i 1 i 2 2 Example Find all the zeros of P x x4 x3 2x2 6x 4 over C 1 i is a zero If 1 i is a root then 1 i is also a root By multiplying out the factors x2 2x 2 2 _x 2x 2 4 _x _x4 3 x 2x3 _x3 _x3 _x2 2x2 2x2 4x2 2x2 2x2 2x2 x 6x 6x 2x 4x 4x 2 4 4 4 0 The factorized equation is x 1 i x 1 i x 2 x 1 The zeros are 1 i 1 i 2 1 http www geocities com fatmuscle HSC 4
Maths Extension 2 - Polynomials  Finding zeros over the complex field 1. If one root is complex, then one of the other roo...
Maths Extension 2 Polynomials Factorization and Division of Polynomials Factorizing polynomials 1 Simple factorizing Trinomials Grouping Difference of 2 squares etc 2 Quadratic Formula 3 Completing the Square Simple factorizing _x2 x 2 _x4 1 Quadratic formula _x2 2x 3 x 2 x 1 x2 1 x2 1 x 1 x 1 x2 1 x 1 x 1 x i x i _x 2 22 4 1 3 2 1 2 8 2 2 2 2 2 1 2i ___________ x a 2 _4x 3x 2 x 1 _x 2i x 1 2i 3 32 4 4 2 2 4 3 23 8 3 23i 8 3 3 x 8 23i x 8 23i 8 8 We can only find the factors of the polynomial not the constant outside Completing the Square _x2 2x 3 x2 2x 1 2 x 1 2 2 x 1 2 2i x 1 2i x 1 2i x 1 2i x 1 2i _4x2 3x 2 4 x2 3 1 4 2 4 x 2 3 4 3 2 9 64 Complete the square then add end term to satisfy the equation i 2 1 Difference of two squares 4 x 8 23i 64 3 4 x2 8 2 23i 8 x 23 64 2 3 8 23i 8 http www geocities com fatmuscle HSC 5
Maths Extension 2 - Polynomials  Factorization and Division of Polynomials Factorizing polynomials 1. Simple factorizing. ...
Maths Extension 2 Polynomials Division of polynomials P x Dividend A x Divisor Q x Quotient R x Remainder 3x4 x3 7x2 2x 3 x 2 3x3 5x2 17x 32 67 LONG DIVISION x 2 4 3x 3x4 3x3 x3 6x3 5x3 5x3 5x2 7x2 7x2 10x2 17x2 17x2 17x 2x 2x 34x 32x 32x 32 3 3 64 67 Example 2 Divide and find a such that R x 0 x 2 For R x 0 3 x x3 x2 ax2 2ax2 a 2 x2 a 2 x2 2a 2 2a a 2 x ax ax 2z 4 x 4 a x 4 a x 4 a 6 6 2 4 a 2a 2 0 2 a 1 http www geocities com fatmuscle HSC 6
Maths Extension 2 - Polynomials  Division of polynomials P x  Dividend    A x    Divisor     Q x     Quotient    R x    Re...
Maths Extension 2 Polynomials Factor and Remainder Theorems Remainder Theorem If a polynomial P x is divided by x a then the remainder is P a Example 1 x 2 a 2 P 2 3 2 4 2 3 7 2 2 2 2 3 48 8 28 4 3 67 x 2 4 3x 3x4 3x3 x3 6x3 5x3 5x3 5x2 7x2 7x2 10x2 17x2 17x2 17x 2x 2x 34x 32x 32x 32 3 3 64 67 Factor Theorem For any polynomial P x if P a 0 then x a is a factor of P x OR For any polynomial P x if x a is a factor of P x then P a 0 http www geocities com fatmuscle HSC 7
Maths Extension 2 - Polynomials  Factor and Remainder Theorems Remainder Theorem   If a polynomial P x  is divided by  x  ...
Maths Extension 2 Polynomials Rational Roots Let P x have degree n with integer coefficients p Suppose P x has a rational root of q _p q are prime integers Then p a0 and q an is divides into Example Given that P x 2x3 3x2 11x 6 has a rational root Find all the zeros of P x p Let the root be q _p 6_ _q 2 The possible rational roots are _p 6 3 2 1 _q 3 1 2 2 P 2 0 P 3 0 P 1 0 2 The zeros of P x are 2 3 1 2 The factorized equation is x 2 x 3 x 1 2 OR x 2 x 3 2x 1 Given that P x x3 4x2 x 6 has a rational root Find all the zeros of P x p Let the root be q _p 6_ _q 1 The possible rational roots are _p 6 3 2 1 _q 1 x 1 is a factor P 1 0 _x 1 _x3 _x3 _x2 4x2 x2 5x2 5x2 5x x 6 6 x 1 x 2 x 3 x 5x 6x 6x 6 6 0 The zeros of P x are 1 2 3 The factorized equation is x 2 x 3 x 1 http www geocities com fatmuscle HSC 8
Maths Extension 2 - Polynomials  Rational Roots Let P x  have degree n with integer coefficients. p Suppose P x  has a rat...
Maths Extension 2 Polynomials Multiplicity Repeated Roots _x a is a repeated root of P x or multiplicity r so x a Q x If P x x a r Q x If x a is of multiplicity r of P x then multiplicity r 1 or P x Proof Let P x x a r Q x P x x a r Q x Q x r x a r 1 1 x a r 1 x a Q x r Q x x a r 1 R x P x has a root at x a with multiplicity r 1 Product rule Example Find roots of P x x3 3x2 4 given that the polynomial has a double root P x x3 3x2 4 P x 3x2 6x P x has a single root so only need to differentiate once 0 3x x 2 _x 0 or 2______________Check both results by remainder theorem P 0 0 P 2 0 _x 2 is a double root To find the other root we can either use Long Division or Sum of the Roots Long Division x 2 2 x2 4x 4 x2 4x 4 _x3 _x3 3x2 4x2 x2 x2 Sum of the Roots Sum of the roots one at a time 2 2 _x 0 4x 4x 4x 1 4 4 4 0 b a 3 3 1 So the roots of P x are 2 2 1 http www geocities com fatmuscle HSC 9
Maths Extension 2 - Polynomials  Multiplicity     Repeated Roots _x   a is a repeated root of P x  or multiplicity r. so  ...
Maths Extension 2 Polynomials For polynomials which have Double roots Differentiate one time Triple roots Differentiate two times Quadruple roots Differentiate three times If number of multiple roots are not given keep differentiating until the P x P x etc can be factorized Example Factorize and find the zeros of P x x4 x3 3x2 5x 2 if it has multiple roots P x x4 x3 3x2 5x 2 P x 4x3 3x2 6x 5 P x 12x2 6x 6 6 2x 1 x 1 P 1 0 1 3 1 2 Example P x x3 3x2 9x C has a double root Find C P x 3x2 6x 9 3 x 3 x 1 _x 3 or 1 We can substitute both but our C will be different Both answers must be given P 3 27 27 27 C C 27 P 1 1 3 9 C C 5 http www geocities com fatmuscle HSC 10
Maths Extension 2 - Polynomials  For polynomials which have  Double roots Differentiate one time Triple roots Differentiat...
Maths Extension 2 Polynomials Relationship between the roots and coefficients of a polynomial equation Quadratic ax2 bx c b a c a Sum of roots 1 at a time Sum of roots 2 at a time product of roots Cubic ax3 bx2 cx d b a c a d a Sum of roots 1 at a time Sum of roots 2 at a time Sum of roots 3 at a time product of roots Quartic ax4 bx3 cx2 dx e b a c a d a e a Sum of roots 1 at a time Sum of roots 2 at a time Sum of roots 3 at a time Sum of roots 4 at a time product of roots If n is the number of roots 1 at a time to find out how many combinations there are we use n Cr For a quartic i i j 4 C1 4 C2 i j k 4 C3 i j k l 4 C4 http www geocities com fatmuscle HSC 11
Maths Extension 2 - Polynomials  Relationship between the roots and coefficients of a polynomial equation Quadratic   ax2 ...
Maths Extension 2 Polynomials Example Roots are for 2x3 4x2 3x 1 1 1 1 1 1 1 1 2 1 2 2 3 2 If 2 2 2 2 2 2 2 2 2 2 2 2 2 8 1 1 1 ____From part 1 8 3 24 3 i 2 i 2 i j 2 22 2 3 2 4 3 7 4 i 3 2 3 2 3 2 3 3 2 i 2 i 3 2 i 4 2 4 2 4 2 3 3 3 1 0 1 0 1 0 4 i 3 i 3 0 28 6 3 0 2 3 37 i 3 37 2 http www geocities com fatmuscle HSC 12
Maths Extension 2 - Polynomials  Example Roots are    ,    ,    for 2x3     4x2     3x     1 1.         1          1      ...
Maths Extension 2 Polynomials 5 i 4 P x is a cubic 2x3 4x2 3x 1 2x4 4x3 3x2 x x P x is now a quadratic Sub it in 2 4 2 4 2 4 4 2 i 2 i 4 2 i 3 4 i 4 37 2 4 2 i 4 3 4 3 4 3 4 74 97 2 3 2 3 2 3 2 2 3 i i 3 7 2 21 2 6 2 2 2 i j 2 2 2 2 2 2 2 i j 2 2 i j k 3 2 2 1 2 2 2 1 4 7 12 1 1 1 2 2 2 2 i j 2 i 2 1 4 1 4 j k 1 8 2 2 i j 2 2 2 2 2 2 3 i i j 3 i j k 2 3 3 1 2 2 9 2 http www geocities com fatmuscle HSC 13
Maths Extension 2 - Polynomials  5.      i  4  P x  is a cubic 2x3     4x2     3x     1 2x4     4x3     3x2     x x.P x  i...
Maths Extension 2 Polynomials Relationship Transformation methods Example If are the roots of 2x3 5x2 4x 6 0 form the equation whose roots are 2 2 2 Relationship Method Sum of the roots one at a time 2 2 2 Sum of the roots two at a time 2 2 2 2 2 2 Sum of the roots three at a time 2 2 2 2 2 5 2 5 4 4 4 2 8 8 8 3 24 The new equation is x3 5x2 8x 24 0 Transformation Method x _y 2x y y 2 2 2 _x 2 3 0 0 2 y y y 2 5 4 6 2 2 2 3 2 y y y 2 5 4 6 8 4 2 y3 5 y 2 2y 6 4 4 0 y 3 5 y 2 8 y 24 Change y for x 0 x 3 5 x 2 8 x 24 0 The Transformation Method is preferred http www geocities com fatmuscle HSC 14
Maths Extension 2 - Polynomials  Relationship   Transformation methods Example If    ,    ,    are the roots of 2x3     5x...
Maths Extension 2 Polynomials Example If are the roots of 2x3 3x2 x 5 0 form the equation whose roots are 1 1 C 2 2 2 B 2 2 2 A 1 D 2 2 2 E 2 2 2 F A 2x3 3x2 x 5 0 x 1 y 2 0 3 1 3 y 2 y3 1 2 y 3 y2 5 1 y 1 5 y 2 3 y y 2 5 y3 0 5 x3 x2 3x 2 B 2x3 3x2 x 5 0 y x 2 2 0 3 y 3 2 3 2y 8 3 y 2 2 3y 4 2 y 2 5 y 2 5 y 3 y 2 2 y 20 0 x 3 3 x 2 2 x 20 C 2x3 3x2 x 5 0 x y 2 2 y 2 3 3 y 2 2 y 2 5 2 y 2 y 2 y 2 3 y 2 y 2 y 2 5 0 2 y3 6y2 4y 8 3 y2 4y 4 y 2 5 2y3 9y2 13y 11 0 2 x 3 9 x 2 13 x 11 D 2x3 3x2 x 5 0 x y 2 2 y 2 3 3 y 2 2 y 2 5 2 y 2 y 2 y 2 3 y 2 y 2 y 2 5 0 2 y3 6y2 12y 8 3 y2 4y 4 y 2 5 2y3 15y2 37y 25 0 2 x 3 15 x 2 37 x 25 http www geocities com fatmuscle HSC 15
Maths Extension 2 - Polynomials  Example If    ,    ,    are the roots of 2x3   3x2   x     5   0, form the equation whose...
Maths Extension 2 Polynomials E 2x3 3x2 x 5 0 x y 3 2 2 y 2 3 y 2 y 2 5 0 3 1 2 y 2 3y y 2 5 1 1 1 3y 5 y 2 2 y 1 Square both sides 9y2 30y 25 y 4y2 4y 1 4y3 4y2 y 0 4y3 5y2 31y 25 0 4 x 3 5 x 2 31x 25 1 F 2x3 3x2 x 5 0 x y 0 2 y 3 3 y 2 y 5 2y3 3y2 y 5 2y3 3y2 y 5 0 2 x3 3x 2 x 5 http www geocities com fatmuscle HSC 16
Maths Extension 2 - Polynomials  E  2x3   3x2   x     5   0 x  y 3  2    2    y 2       3    y 2           y 2         5  ...
Maths Extension 2 Conics Conics General Equations of Conics Parabola Circle Ellipse Hyperbola Rectangular Hyperbola Equation Eccentricity Tangents Normals Foci Directrices Asymptotes Chord Chord of Contact General Cartesian x1 y1 Parametric a cos a sin Circle a cos b sin Ellipse a sec b tan Hyperbola http www geocities com fatmuscle HSC 1
Maths Extension 2     Conics  Conics   General Equations of Conics   Parabola, Circle, Ellipse, Hyperbola, Rectangular Hyp...
Maths Extension 2 Conics Shapes From Phoenix Senior Maths Notepad Study Guides by George Fisher Eccentricity of Conics PS PM e Eccentricities Circle e 0 Ellipse e 0
Maths Extension 2     Conics  Shapes     From Phoenix Senior Maths Notepad Study Guides by George Fisher  Eccentricity of ...
Maths Extension 2 Conics General Equation of Conics Circle Basic x2 y 2 r 2 x h 2 y k 2 a 2 x 2 y 2 2 gx 2 fy c 0 Centre g f General Radius g2 f 2 c Basic _r _r _y h k _x Parametric General Form _x2 y2 4x 6y 10 0 P acos asin Complete the Square _a asin _acos x 2 2 y 3 2 10 0 Centre 2 3 Radius ______ Parabola Basic 4 9 10 3 x 2 4ay Vertex 0 0 Focus 0 a Directrix y a x h 2 4a y k Vertex h k SEE 3U Parabola NOTES http www geocities com fatmuscle HSC 3
Maths Extension 2     Conics  General Equation of Conics Circle Basic  x2   y 2   r 2   x     h  2     y     k  2   a 2 x ...
Maths Extension 2 Conics Ellipse P M Foci ae 0 Directrices x a e x2 y2 1 a2 b2 Major Axis A a 0 A a 0 Minor Axis B 0 b B 0 b S and S are the two foci D and D are the two Directrices P is a point on the ellipse M is the perpendicular distance from D to P _b is the minor axis _a is the major axis _b _a S S D D y a e B B 0 ae A A A ae 0 A ae 0 0 ae x a e AA BB _b2 a2 1 e2 B x a e B y a e AA BB _a2 b2 1 e2 http www geocities com fatmuscle HSC 4
Maths Extension 2     Conics  Ellipse      P  M  Foci    ae, 0  Directrices x      a e       x2 y2    1 a2 b2  Major Axis ...
Maths Extension 2 Conics Hyperbola D Foci ae 0 Directrices x a e x2 y2 1 a 2 b2 Asymptotes x2 y2 a2 D S D x D x D D S S D D S S ae 0 ae 0 xy c 2 x y a x y a S a e a e c c c c http www geocities com fatmuscle HSC 5
Maths Extension 2     Conics  Hyperbola      D     Foci    ae, 0  Directrices x      a e     x2 y2      1 a 2 b2  Asymptot...
Maths Extension 2 Conics Conics in Detail Circle Equation x2 y 2 r 2 x h 2 y k 2 a 2 Eccentricity Tangents P x1 y1 P a cos a sin Normals P x1 y1 P a cos a sin x 2 y 2 2 gx 2 fy c 0 0 xx1 yy1 a 2 x1 h x x1 y1 k y y1 a 2 x x1 x1 g y y1 y1 f 0 x cos y sin a x h cos y k sin a x g cos y f sin a xy1 yx1 0 y1 k x x1 x1 h y y1 0 x sin y cos 0 x h a cos sin y k a sin cos 0 http www geocities com fatmuscle HSC 6
Maths Extension 2     Conics  Conics in Detail Circle Equation  x2   y 2   r 2   x     h  2     y     k  2   a 2 Eccentric...
Maths Extension 2 Conics Circle Tangent x2 y2 a2 P x1 y1 OR P a cos a sin 2x 2 y dy dx dy dx 2y dy dx x 1 y1 xx1 x1 2 2 2 x1 y1 0 2x x y y y1 x x1 x h 2 y k 2 a 2 dy 2 x h 2 y k dx dy 2 y k dx 2 Equation of tangent P x1 y1 OR P h a cos k a sin 0 2 x h x h x h cos 1 y k y1 k sin cos y k a sin sin x h a cos 2 y k sin a sin x h cos a cos 2 y k sin x h cos a dy dx Gradient yy1 y1 xx1 yy1 x cos y sin a xx1 yy1 a 2 cos x1 sin y1 x h cos y k sin x 2 y 2 2 gx 2 fy c 0 x x1 x1 g y y1 y1 f 0 x g cos y f sin a a Gradient cos 2 sin 2 1 Equation of tangent P x1 y1 OR P g a cos f a sin Equation of tangent http www geocities com fatmuscle HSC 7
Maths Extension 2     Conics  Circle     Tangent x2   y2   a2  P x1 , y1   OR P a cos   , a sin       2x   2 y  dy dx dy d...
Maths Extension 2 Conics Circle Normal x2 y2 a2 dy dx y1 x1 y1 x x1 y1 xy1 xy1 yx1 0 y x y y1 x x1 x1 y x1 y1 yx1 P x1 y1 OR P a cos a sin Gradient y sin 1 x1 cos x sin y cos 0 P x1 y1 OR P h a cos k a sin x h 2 y k 2 a 2 y k x h y1 k y y1 x1 h x x1 y1 k x x1 x1 h y y1 0 dy dx Equation of normal y1 k x x1 x1 h y y1 0 x h a cos sin y k a sin cos 0 y1 k sin x1 h cos Equation of normal http www geocities com fatmuscle HSC 8
Maths Extension 2     Conics  Circle     Normal x2   y2   a2 dy dx y1 x1 y1 x     x1 y1 xy1 xy1     yx1   0  y x y     y1 ...
Maths Extension 2 Conics Ellipse Equation x2 y2 1 a2 b2 b 2 a 2 1 e x ae 0 a x e xx1 yy1 2 1 a2 b Eccentricity Foci Directrices Tangents P x1 y1 P a cos b sin x cos y sin 1 a b y mx c c2 a2m2 b2 a2 x b2 y a2 b2 x1 y1 Normals P x1 y1 P a cos b sin Chord P a cos b sin Q a cos b sin Chord of Contact T x0 y0 ax by a2 b2 cos sin y x sin cos cos b 2 a 2 2 xx0 yy0 2 1 a2 b Other Stuff Angles in an ellipse The tangent at a point P on the ellipse is equally inclined to the focal chords through P The segment of the tangent to an ellipse between the point of contact and the directrix subtends at right angles at the corresponding focus P S Area PS PS 2a S ab http www geocities com fatmuscle HSC 9
Maths Extension 2     Conics  Ellipse Equation  x2 y2    1 a2 b2 b 2   a 2  1     e   x       ae,0   a x    e xx1 yy1   2 ...
Maths Extension 2 Conics Area of the Ellipse by integration y2 b2 a A y dx a a b2 x 2 b 2 a 2 2 a a 2b 1 a a 2b a x2 a2 y 2 y 1 x2 a2 b2 x2 b 2 a 2 b2 b2 x2 a2 a2 x2 a2 2b a 2 2 a a x a 2b 1 2 a a 2 a2 x2 Is a Semicircle Area of Ellipse ab Gradient of an Ellipse x2 y2 a 2 b2 dy 2 x 2 y dx 2 a2 b dy dx 1 0 2 x b2 a2 2 y Implicit Differentiation b2 x a2 y http www geocities com fatmuscle HSC 10
Maths Extension 2     Conics  Area of the Ellipse by integration y2 b2  a  A       y dx    a  a  b2 x 2 b     2 a    2    ...
Maths Extension 2 Conics Chords in an Ellipse A chord through the centre is a diameter Eg PR C P A chord through a focus is a focal chord Eg CD S S Q A focal chord that is perpendicular to the major axis is the Latus Rectum Eg PQ R D Chord P a cos b sin Q a cos b sin b sin b sin m a cos a cos 2a sin b cos a sin 2 2 2 2 2b cos sin sin 2 2 OR b cot Gradient of Chord a 2 b cos y b sin x a cos a sin xb cos ab cos cos ay sin ab sin sin 2 2 2 2 y x sin cos b 2 a 2 2 2 cos cos sin sin 2 2 cos 2 y x sin cos cos b 2 a 2 2 Equation of chord http www geocities com fatmuscle HSC 11
Maths Extension 2     Conics  Chords in an Ellipse A chord through the centre is a diameter Eg. PR  C P  A chord through a...
Maths Extension 2 Conics Ellipse Tangent dy dx Gradient of Tangent xb 2 ya 2 At x1 y1 x1b 2 y1a 2 b cos a sin 2 y y1 b x 2 1 x x1 a y1 b 2 xx1 b 2 x1 2 2 b 2 x1 a 2 y1 2 At a cos b sin a 2 yy1 a 2 y1 b 2 xx1 a 2 yy1 2 x1 y 12 2 a b xx1 yy1 2 a2 b xx1 yy1 1 2 2 a b x cos y sin 1 a b xx1 yy1 2 1 a2 b Ellipse Normal dy ya 2 dx xb 2 y a2 1 2 x1b Gradient of Normal At x1 y1 At a cos b sin a 2b sin b 2 a cos y y1 y1a 2 x x1 x1b 2 a 2 y1 x a 2 y1 x1 a 2 y1 x b 2 x1 y1 y1 a 2 x b 2 x1 b 2 x1 y b 2 x1 y1 b 2 x1 y a 2 x1 y1 x1 b 2 y a 2 y1 2 a x b2 x1 a 2 x b2 y x1 y1 a2 x b2 y a2 b2 x1 y1 Equation of tangent 2 b y a2 y1 a 2 b2 ax by a2 b2 cos sin Equation of normal http www geocities com fatmuscle HSC 12
Maths Extension 2     Conics  Ellipse     Tangent dy dx        Gradient of Tangent  xb 2 ya 2  At  x1, y1   x1b 2 y1a 2 b ...
Maths Extension 2 Conics Hyperbola Equation x2 y2 1 a 2 b2 b2 a 2 e2 1 x ae 0 a x e xx1 yy1 1 a 2 b2 Eccentricity Foci Directrices Tangents P x1 y1 P a sec b tan x sec y tan 1 a b Normals P x1 y1 a 2 x b2 y a2 b2 x1 y1 P a sec b tan Chord P a sec b tan Q a sec b tan Chord of Contact T x0 y0 Gradient of a Hyperbola x2 y2 1 a 2 b2 dy 2 x 2 y dx 0 2 a2 b dy 2 x b2 2 dx a 2y ax by a 2 b2 sec tan x y cos cos a 2 2 b 2 xx0 yy0 2 1 a2 b Implicit Differentiation b2 x a2 y http www geocities com fatmuscle HSC 13
Maths Extension 2     Conics  Hyperbola Equation  x2 y2      1 a 2 b2 b2   a 2 e2     1 x       ae,0   a x    e xx1 yy1   ...
Maths Extension 2 Conics Hyperbola Tangent dy b2 x 2 dx a y Gradient of Hyperbola At x1 y1 b 2 x1 a 2 y1 b sec a tan y y1 x x1 b 2 x1 a 2 y1 2 b 2 xx1 b 2 x1 b 2 xx1 a 2 yy1 xx1 yy1 a 2 b2 xx1 yy1 a 2 b2 At a sec b tan 2 a 2 yy1 a 2 y1 2 2 b 2 x1 a 2 y1 2 2 x y 12 12 a b 1 x sec y tan 1 a b xx1 yy1 1 a 2 b2 Hyperbola Normal dy a2 y 2 dx b x a2 y 2 1 b x1 a tan b sec 2 y y1 a y 2 1 x x1 b x1 a 2 y1 x a 2 y1 x1 a 2 x1 y1 b 2 x1 y1 x1 y1 a 2 b 2 a b 2 a 2 b2 2 a2 x b2 y x1 y1 Equation of tangent Gradient of Normal At x1 y1 At a sec b tan b 2 x1 y b 2 x1 y1 a 2 xy1 b 2 x1 y a 2 xy1 b 2 x1 y a2 x b2 y x1 y1 a 2 b2 ax by sec tan Equation of normal http www geocities com fatmuscle HSC 14
Maths Extension 2     Conics  Hyperbola     Tangent dy b2 x   2 dx a y  Gradient of Hyperbola At  x1, y1   b 2 x1 a 2 y1 b...
Maths Extension 2 Conics Rectangular Hyperbola Equation Eccentricity Foci e 2 a 2 a a a x 2 x y a y x x y axis Directrices Asymptotes Tangents P x1 y1 P cp c p x2 y2 a2 xy c 2 xy1 yx1 2c 2 x p 2 y 2cp Normals P x1 y1 xx1 yy1 x1 y1 P a cos b sin p 3 x py c p 4 1 Chord P x1 y1 Q x2 y2 c 2 x x1 x2 y c 2 x1 x2 2 2 c P cp c Q cq q p x pqy c p q Chord of Contact T x0 y0 xy0 yx0 2c 2 Other Stuff The area of the triangle bounded by a tangent 2c2 and the asymptotes is a constant Intersection of 2 tangents P and Q 2 xpq 2c p q p q http www geocities com fatmuscle HSC 15
Maths Extension 2     Conics  Rectangular Hyperbola Equation Eccentricity Foci  e  2 a 2       a,   a   a x    2   x   y  ...
Maths Extension 2 Conics Gradient of a Rectangular Hyperbola xy c 2 y c2 x dy c2 2 dx x c Chord P cp c Q cq q p y c p x cp c x cp q cp xc xc c 2 p c2 q p q c q c p cq cp y c cq cp p c2q cqy cpy c2 p 0 pq 2 y p 2 qy cq 2 cp 2 qx px pqy q p c q 2 p 2 x q p pqy c p q x x pqy c p q Equation of Chord http www geocities com fatmuscle HSC 16
Maths Extension 2     Conics  Gradient of a Rectangular Hyperbola xy   c 2 y c2   x dy c2      2 dx x             c Chord ...
Maths Extension 2 Conics Rectangular Hyperbola Tangent dy c2 2 dx x 2 y1 2 x1 1 2 p 1 p2 y x p2 y xy1 yx1 2c 2 At cp c p x cp c p p 2 y pc 2cp Equation of tangent x p 2 y 2cp Rectangular Hyperbola Normal dy x2 2 dx c 2 x 12 y1 Gradient of Normal At x1 y1 At cp c p p2 p2 p x cp 2 c p 2 x cp p y x cp At x1 y1 y c p x cp y c p p 2 x cp 3 y p 3 x cp 4 py c 2 xx1 yy1 x1 y1 2 c p p 3 x py c p 4 1 Equation of Normal http www geocities com fatmuscle HSC 17
Maths Extension 2     Conics  Rectangular Hyperbola     Tangent dy c2      2 dx x 2 y1      2 x1 1      2 p      1 p2     ...
Maths Extension 2 Integration Standard Integrals Function Derivative Product Integrate by Parts Partial Fractions A B A Bx C Completing the Square Substitution Normal Trigonometry T Formula Reduction Formula http www geocities com fatmuscle HSC 1
Maths Extension 2 - Integration           Standard Integrals Function   Derivative Product     Integrate by Parts Partial ...
Maths Extension 2 Integration INTEGRALS c is a constant x n 1 c _________ n 1 n 1 x n dx 1 dx x ln x c ax b n dx f x dx f x 1 Standard Integral ax b n 1 c a n 1 ln f x c 1 ln ax b c a e ax dx 1 ax e c a x a dx ax c ln a ax b dx 1 x a2 x a 2 1 2 x 2 a 2 2 dx ln x x 2 a 2 c dx ln x x 2 a 2 c 1 dx a2 1 x a ln c 2a x a 1 dx x2 1 a x ln c 2a a x http www geocities com fatmuscle HSC 2
Maths Extension 2 - Integration  INTEGRALS  c is a constant  x n  1   c _________ n        1   n  1         x n dx        ...
Maths Extension 2 Integration 1 sin ax c a cos ax dx sin ax dx 1 cos ax c a sec 2 ax dx 1 a x 2 2 1 1 tan ax c a dx x sin 1 c a dx x x cos 1 c __OR__ sin 1 c a a a 2 x2 1 dx 2 a x2 1 cot ax c a cos ec ax dx 2 sec ax tan ax 1 x tan 1 c a a dx cos ecax cot ax dx 1 sec ax c a 1 cos ecax c a Integration by special properties Theorems notes all these are with respect to dx a 0 a a a a a f a x dx f x dx f x dx 2 f x dx ___if f x is an EVEN function f x dx 0 _________ __if f x is an ODD function 0 a 0 http www geocities com fatmuscle HSC 3
Maths Extension 2 - Integration  1 sin ax   c a         cos ax dx            sin ax dx  1       cos ax   c a         sec 2...
Maths Extension 2 Integration Trigonometric Identities sin2 cos2 1 sin2 1 cos2 cos2 1 sin2 tan2 sec2 1 sec2 tan2 1 sec2 tan2 1 1 cot2 cosec2 1 cosec2 cot2 cot2 cosec2 1 sin tan cos cos cot sin cos2 1 1 cos 2 2 sin2 1 1 cos 2 2 sin 2t 1 t2 cos 1 t 1 t2 tan 1 1 t2 T Formula Substitution 2 t tan 2 a 2 x 2 Let x a sin x 2 a 2 Let x a sec Trigonometric Substitution a 2 x 2 Let x a tan http www geocities com fatmuscle HSC 4
Maths Extension 2 - Integration  Trigonometric Identities sin2     cos2     1    sin2     1     cos2     cos2     1     si...
Maths Extension 2 Integration STANDARD INTEGRALS Basic Form n 1 x 5 dx 1 x dx or x 1 dx ax b dx 2x 3 dx n 6 f x dx f x 2x x 2 1 dx 1 x dx or x 1 dx 1 ax b dx 3x 2 dx 1 e ax dx e 3x dx a 3 x dx x dx x c _________ n 1 n 1 General Integral x6 c 6 ln x c 1st special function x dx n ax b n 1 c a n 1 Linear function 2 x 3 7 c 14 ln f x c Function and Derivative ln x 2 1 c Logarithm and Exponential ln x c 1st special function 1 ln ax b c a 1 ln 3 x 2 c 3 1 e ax c a 1 3x e c 3 ax c ln a 3x c ln 3 http www geocities com fatmuscle HSC 5
Maths Extension 2 - Integration  STANDARD INTEGRALS Basic Form n  1     x  5  dx  1     x  dx or     x    1 dx       ax   ...
Maths Extension 2 Integration Trigonometric Functions sin ax dx sin 3x dx sin 3 x dx 1 cos ax c a cos 3 x c 3 sin 2 x sin x dx 1 cos x sin x dx sin x cos x sin x dx 2 2 cos 2 x sin x dx Let u cos x du sin x dx du dx sin x sin x dx cos x cos x cos ax dx cos 3x dx cos x dx 3 u 2 sin x du sin x u3 3 cos 3 x c 3 1 sin ax c a sin 3 x c 3 cos 2 x cos x dx 1 sin x cos x dx cos x sin x cos x dx 2 2 sin 2 x cos x cos x dx sin x Let u sin x du cos x dx du dx cos x u 2 cos x du cos x u3 3 sin 3 x sin x c 3 http www geocities com fatmuscle HSC 6
Maths Extension 2 - Integration  Trigonometric Functions      sin ax  dx      sin 3x  dx      sin  3  x dx  1       cos ax...
Maths Extension 2 Integration sec 2 ax dx sec 2 3x dx tan x dx tan 2 tan 3 1 tan ax c a 1 tan 3 x c 3 sin x dx _________Function and Derivative cos x ln cos x c sec 2 x 1 dx x dx x dx tan x x c tan 2 x tan x dx sec x 1 tan x dx sec x tan x tan x dx_____Let tan x u 2 2 tan 2 x ln cos x c 2 http www geocities com fatmuscle HSC 7
Maths Extension 2 - Integration      sec  2  ax dx      sec  2  3x dx      tan x dx     tan  2      tan  3  1 tan ax   c a...
Maths Extension 2 Integration u f x du f x dx Function and Derivative f x f x n dx f x dx f x f x n sin x cos x dx f x 1 f x 2 dx 1 5 n 1 e f x c sin f x c sin n 1 x c n 1 tan 1 f x c ln ln x c 4 x dx du sin x dx du dx sin x sin 4 sin x cos 4 x dx 1 cos x sin x cos x dx 2 cos x cos x sin x cos x dx 1 cos x sin x 2 cos x sin x cos x sin x dx x 3 x 2 x 4 dx 2 split the integral take out constants complete the square standard integrals 2 2 4 2 4 4 f x n 1 c ln f x c f x e dx f x cos f x dx x ln x dx sin x cos 4 6 8 cos5 x 2 cos7 x cos9 x c 5 7 9 1 2 x 2 4 2x 2 2 x 4 dx 1 2x 2 1 x 2 2 x 4 dx 4 x 2 2 x 4 dx 2 1 2x 2 1 x 2 2 x 4 dx 4 x 1 2 5 dx 2 1 2 x 1 5 c ln x 2 2 x 4 ln 2 5 x 1 5 http www geocities com fatmuscle HSC 8
Maths Extension 2 - Integration  u   f  x  du   f     x dx  Function and Derivative        f   x    . f     x   n       dx...
Maths Extension 2 Integration uv dx uv vu dx Integrating by parts x cos x dx Let u x du 1 dx ln x dx dv cos x dx v sin x Let u ln x du 1 dx x dv 1 dx v x Let u du dx x sin x sin x dx Let v dv dx x sin s cos x c ln x 1 dx x ln x 1 dx x ln x x c These integrals have only one step other integrals have multi step e x Let du dx Let du dx cos x dx I u ex ex u ex ex dv cos x dx v sin x dv sin x dx v cos x e x sin x e x sin x dx e x sin x e x cos x e x cos xdx I e x sin x e x cos x I 2I e x sin x e x cos x 1 I e x sin x e x cos x c 2 e x Let du dx Let du dx sin x dx I u ex ex u ex e x dv sin x dx v cos x dv cos x dx v sin x e x cos x e x cos x dx e x cos x e x sin x e x sin xdx I e x cos x e x sin x I 2I e x sin x e x cos x 1 I e x sin x e x cos x c 2 http www geocities com fatmuscle HSC 9
Maths Extension 2 - Integration      uv  dx   uv         vu   dx  Integrating by parts      x cos x dx  Let u   x du  1 dx...
Maths Extension 2 Integration 1 2 3 4 Split to Standard Integrals A B A Bx C Completing the Square x 1 1 dx x 1 x 1 1 1 dx x 1 x ln x 1 c Partial Fractions x 2 x 1 dx 1 x 2 x 1 dx 1 x 2 x 1 When 1 1 x 1 1 x 2 A B x 2 x 1 A x 1 B x 2 ___0___ __ 3B __ 3A A B x 2 x 1 dx 1 1 1 1 x 2 3 x 1 dx 3 1 ln x 2 ln x 1 c 3 1 x 2 ln c 3 x 1 P x Q x ______ P x Q x 3x 2 2 x 1 dx 2x2 x 1 x 2 dx LONG DIVISION 2 x 3 r7 2 x 2 2x x 1 x 4 x 0 x 1 A 1 __17 __1 __5 2 x 1 3 1 2 dx 2x 1 31 3 1 2 dx 2 2x 1 3 7 x ln 2 x 1 c 2 2 7 2x 3 dx x 2 dx x 2 3x 7 ln x 2 c ____ 2 x 2 4 x ___ _________ 3 x 1 _________ 3x 6 _____________7 1 4x 4 x x 2 1 dx 1 4x 3 2 Bx C dx x2 1 1 x 2 dx 4 x x 1 ln 4 x 1 ln x 2 1 c 2 A x 2 1 17A A 2A B 1 Bx c 4 x 0 C4 0 Bx 3 C 0 A 4 x http www geocities com fatmuscle HSC 10
Maths Extension 2 - Integration  1. 2. 3. 4.  Split to Standard Integrals A, B A, Bx   C Completing the Square x  1 1     ...
Maths Extension 2 Integration Finding x in terms of u Let u du dx dx Substitution 1 12 4 x x 4 dx Let u x du 1 dx 2 x dx 2 x du x u2 dx Let u x 1 du 2u 2x dx udu dx x 2 2 x du Changing the Limits Let u 2 x2 u 2 1 1 x tan 2 4 2 3 1 2 tan 1 2 tan 2 3 4 12 x 3 udu u2 x IMPLICIT DIFFERENTIATION x x 12 x 4 12 x2 1 x 1 du 4 u2 1 u 2 tan 1 2 2 u tan 1 2 2 2 x3 1 4 u u 2 3 u 2 2 3 u tan 1 2 2 3 4 12 x du u 1 du 2 2 u3 u c 3 1 x2 1 x2 1 x2 1 c 3 http www geocities com fatmuscle HSC 11
Maths Extension 2 - Integration  Finding x in terms of u Let u   du       dx     dx    Substitution  1  12       4   x    ...
Maths Extension 2 Integration Trigonometry Substitution a2 x2 x a 2 Let x a sin Let x a sec Let x a tan 2 a2 x2 x 4 x2 dx Let x 2 sin dx 2 cos d dx 2 cos d Trig Identity 4 sin 2 2c cos d 4 4 sin 2 sin 2 cos 8 d 4 1 sin 2 4 sin cos x 2 sin x sin 2 2 d cos 2 4 1 1 cos 2 d 2 x sin 1 2 sin 2 c 2 2 x 2 sin 1 2 sin cos c 2 2 x 4 x 1 x c 2 sin 2 2 2 2 x 2 4 x2 2 x x 4 x 2 sin 1 c 2 2 http www geocities com fatmuscle HSC 12
Maths Extension 2 - Integration  Trigonometry Substitution a2     x2 x    a 2  Let x   a sin    Let x   a sec   Let x   a ...
Maths Extension 2 Integration t tan ________ 2 T Formula sin 2t 1 t2 1 t2 cos 1 t2 tan 2 2t 1 t2 4 d 3 5 cos dt 1 2 sec d 2 2 2dt d sec 2 2 2dt d 1 tan 2 2 4 2dt 3 5 1 t 3 1 t 8 2t 2dt d 1 t2 0 2dt 1 t2 d 4 t 1 t 2 1 t 2 2 8 dt 2 5 1 t 1 t 2 1 t2 2 8 4 2 2 dt dt 4 1 dt 4 t2 1 2 t 4 4 ln 2 2 2 t 0 2 t 4 ln 2 t 0 ln 3 By Partial Fractions 4 A 2 t 4 0 B 1 t 2 4 4 t 1 2 B 2 t 4B 4 4A A 1 t 2 0 dt 1 2 t 2 t ln 2 t ln 2 t 4 0 2 t 4 ln 2 t 0 http www geocities com fatmuscle HSC 13
Maths Extension 2 - Integration  t   tan       ________                  2  T-Formula sin       2t 1  t2  1    t2 cos     ...
Maths Extension 2 Integration Reduction Formula Show 2 In 0 cos n x dx_______n 2 In n 1 In 2 n 1 In In cosn 1 x cos1 x dx Let n 1 u cos x du n 1 cos n 2 sin x dx dv cos x dx v sin x Take out constant and expand sin x cos n 1 2 x n 1 cos n 2 x sin 2 x dx 0 0 n 1 cosn 2 x 1 cos2 x dx n 1 cosn 2 x dx n 1 cosn x dx n 1 In 2 n 1 In Find I6 2 I6 0 cos6 x dx I6 6 1 I6 2 6 1 I6 I6 5I4 5I6 6I6 5I4 6I6 5 3I2 3I4 6I6 15I2 15I4 6I6 15I2 15 3 I2 4 15 6I6 4 I2 I6 5 I2 8 I4 3I2 3I4 4I4 3I2 I4 3 I2 4 2 I2 0 cos 2 x dx I2 I2 I6 sin 2 x 2 sin x 2 0 1 2 5 16 http www geocities com fatmuscle HSC 14
Maths Extension 2 - Integration  Reduction Formula Show      2  In      0 cos n x dx_______n     2  In    n     1 In-2    ...
Maths Extension 2 Volumes Volumes Slicing Volume of Pyramid Relationships Ratios Slicing Revolution Cross Sectional Area Shells Washer Method Torus Quadratic Equation Basic Volumes Formulae Basic b V y 2 dx a b V x 2 dy a General b V A z dz a http www fatmuscle cjb net 1
Maths Extension 2  Volumes  Volumes    Slicing    Volume of Pyramid    Relationships   Ratios    Slicing    Revolution    ...
Maths Extension 2 Volumes Slicing Volume of a Pyramid h h h k k k b a b a k Find the volume of one slice k k a h b h ak h Volume of one slice k V k bk h Volume of entire pyramid h V dk 0 h ak bk dk h h 0 ab h 2 k dk h2 0 h ab k 3 2 h 3 0 abh 3 3h 2 1 abh 3 http www fatmuscle cjb net 2
Maths Extension 2  Volumes  Slicing Volume of a Pyramid  h h  h  k  k  k                 b a  b  a     k      Find the vol...
Maths Extension 2 Volumes Relationship 4 h 5 l 3 4 h 5 12 w l 3 h 10 5 w 12 10 Volume of One Slice y 4 h 0 y 12 h 5 y mx b l y 3 y 10 m rise run h 0 h 5 b int ercept x h 8h 4 5 w 7h 3 5 V w l h Volume of entire object 5 V 8h 7 h 3 dh 4 5 0 5 5 4 2 14h 65h 75 dh 25 0 5 4 14h3 65h 2 75h 25 3 2 0 283 http www fatmuscle cjb net 3
Maths Extension 2  Volumes  Relationship 4 h 5 l  3  4 h 5  12  w l  3 h 10 5  w  12  10  Volume of One Slice y 4 h 0 y   ...
Maths Extension 2 Slicing Shape The base of a certain solid is x 2 y 2 9 Plane sections are perpendicular to the y are equilateral triangles on side being the solid base Find the volume Volume of 1 slice 2y Volumes axis and 2y 2y Volume of one slice V 1 2 y 2 y sin 60 x 2 3 y 2 x Total Volume x2 y 2 3 V 3 y 2 dx y 3 2 9 9 x2 3 9 x dx 3 2 3 3 2 3 9 x 2 dx 3 3 x3 2 3 9 x 3 3 36 3 http www fatmuscle cjb net 4
Maths Extension 2  Slicing Shape The base of a certain solid is x 2   y 2   9 . Plane sections are perpendicular to the y ...
Maths Extension 2 Shells y f x Volumes R r Area of one strip A f x dx x f x Volume of One Shell V R 2 r 2 f x x x x f x x 2 x x x x f x 2 2 2 2 2 2 x x f x 2 x x f x Volume of Total b V 2 x f x dx a b 2 xy dx a http www fatmuscle cjb net 5
Maths Extension 2  Shells y   f  x    Volumes  R r  Area of one strip A   f   x  dx     x  f  x    Volume of One Shell V  ...
Maths Extension 2 Volumes Find the volume of the solid obtained by revolving the region x y 2 x 0 y 0 about the y axis x y y b V 2 xy dx 2 2 x a 2 2 x 2 x dx 0 2 2 2 x x 2 dx 2 0 2 2 x 2 x3 2 3 0 2 8 2 4 3 8 cubic units 3 2 2 Find the volume bounded by the graphs y x 3 y 0 within the range of 1 x 3 rotated about the y axis b V 2 xy dx a 3 2 xx3 dx 1 3 2 x 4 dx 1 3 x5 2 5 1 243 1 2 5 5 484 cubic units 5 http www fatmuscle cjb net 6
Maths Extension 2  Volumes  Find the volume of the solid obtained by revolving the region x   y   2 , x   0 , y   0 about ...
Maths Extension 2 Find the volume of the solid obtained by revolving the ellipse V 2 xy dx a y2 b2 a 2 x 2 b a 2 x 2 dx a 0 V x2 y2 1 about the y axis a2 b2 x2 y2 a2 b2 b 4 b a 2 2 x a x dx a 0 u a2 x2 du 2x dx dx du 2x 4 b 0 du x u 2x a a Volumes y 2 x a u 0 x 0 u a2 1 1 x2 a2 b2 b2 x2 a2 b2a 2 b2 x 2 a2 b2 2 a 2 x 2 a b 2 a x2 a 2 b 0 u du a a 2 y 0 2 b 2u 2 a 3 2 a 3 4 b 2 2 a 3a 4 ba 2 cubic units 3 3 http www fatmuscle cjb net 7
Maths Extension 2  Find the volume of the solid obtained by revolving the ellipse  V    2       xy.dx a  y2 b2  a    2    ...
Maths Extension 2 Washer Method Area of one slice A z R 2 r 2 R2 r 2 R r R r x2 x1 x2 x1 Volumes R r Solve the quadratic formula to find x2 and x1 Where x2 x1 b b 2 4ac 2a Washer Taurus Find the volume of the region bounded by the parabola y x 2 and the lines y 0 x 2 Volume of One Slice V 4 x 2 y x 2 is the larger radius 22 4 Total Volume 4 2 V 4 x dy 0 4 4 y dy 0 4 y2 4 y 2 0 8 http www fatmuscle cjb net 8
Maths Extension 2  Washer Method Area of one slice A z       R 2       r 2      R2     r 2       R   r   R     r         x...
4 Unit Maths Mechanics Newton s Second Law of Motion Fnet is defined as the sum of forces The direction of forces need to be taken into account Equations of motion Horizontal resisted motion Vertical resisted motion Sometimes It is necessary to describe two motions Terminal velocity Terminal velocity is a constant velocity reached by a projectile moving vertically downwards Circular motion Angular Tangential Thompson Ly 1
4 Unit Maths     Mechanics Newton   s Second Law of Motion.                                      Fnet is defined as the su...
4 Unit Maths Mechanics Polar Through calculations it can be shown that If there is more than one particle in the system isolation is required Pendulum motion If the particle is touching any object such as a sphere or a cone the particle will be subjected to a normal force Banked motion Going quick When an object moves quickly it moves up the bank Thompson Ly 2
4 Unit Maths     Mechanics  Polar                                                                                         ...
4 Unit Maths Mechanics Going slow When an object moves slowly it moves down the bank Ideal velocity Ideal velocity occurs when the object is subjected to no friction Slipping Where is the coefficient of friction Thompson Ly 3
4 Unit Maths     Mechanics Going slow     When an object moves slowly, it moves down the bank.                            ...
Maths Extension 2 Harder Maths Extension 1 3U Harder Maths Extension 1 Harder 3U Inequalities Induction http fatmuscle cjb net 1
Maths Extension 2     Harder Maths Extension 1  3U   Harder Maths Extension 1   Harder 3U   Inequalities   Induction  http...
Maths Extension 2 Harder Maths Extension 1 3U Inequalities Proof a 2ab b 2 0 0 a2 b2 2 a 2 b 2 2ab True if True if Example 1 PROVE a b 2 a b 2 a b a b 4ab a 2 2ab b 2 a b 4ab 2 a b 2 4ab Example 2 If a b c d are 0 then a 4 b 4 c 4 d 4 4abcd PROVE a 2 b 2 c 2 ab bc ca a 2 b2 b2 c 2 c2 a 2 2 a 2 b2 c 2 a2 b2 c2 2ab 2bc 2ca ab bc ca ab bc ca Using a 2 b 2 2ab Example 3 The arithmetic mean The geometric mean PROVE a b 2 a b 2 4 a b 2 a 2 2ab b 2 a 2 2ab b 2 a b 2 True if True if a b 2 ab ab ab 4ab 4ab 0 0 a b a b http fatmuscle cjb net 2
Maths Extension 2     Harder Maths Extension 1  3U   Inequalities Proof a     2ab   b 2     0    0 a2   b2 2     a 2   b 2...
Maths Extension 2 Harder Maths Extension 1 3U Example 4 1 a b 1 a b Are they equal ab a 2ab b ab a 2 ab b 2 2 if 2 1 1 a b 1 1 a b b a ab 2 a b 0 0 Prove by contradiction 1 1 1 a b a b b a 1 ab a b 2 a b ab Since a b 2 a b 2 2ab ab http fatmuscle cjb net 3
Maths Extension 2     Harder Maths Extension 1  3U   Example 4 1 a b 1 a b  Are they equal   ab  a   2ab   b     ab a 2   ...
Maths Extension 2 Harder Maths Extension 1 3U Induction Example 1 f x xe x f x xe x e x f x xe x e x e x f x f n x Prove for n 0 LHS f 0 x xe x Assume n k f k x e x x k xe x ke x f k 1 x xe x e x ke x e x x 0 e x x 1 e x x 2 e x x 3 e x x n RHS e x x 0 xe x Assume n k 1 f k 1 x e x x k 1 xe x e x ke x http fatmuscle cjb net 4
Maths Extension 2     Harder Maths Extension 1  3U   Induction Example 1 f  x     xe x f    x     xe x   e x f      x     ...
Maths Extension 2 Harder Maths Extension 1 3U Example 2 Recurrence Induction A sequence of terms U n n 1 2 3 Tn U n U n 4U n 1 5U n 2 2U n 3 Initial conditions U1 3 U 2 1 U 3 0 Show that by induction U n 2n 1 3n 5 n 4 5 n 1 2 3 Consider S n 2n 1 3n 5 S 1 21 1 3 1 5 1 3 5 3 S 2 22 1 3 2 5 2 6 5 1 S 3 4 9 5 0 Assume S k 2k 1 3k 5 Prove S k 1 2k 1 1 3 k 1 5 2k 3k 3 5 2k 3k 2 LHS 4U k 1 5U k 2 2U k 3 4 2k 1 3k 5 5 2k 2 3 k 1 5 2 2 k 3 3 k 2 5 2k 1 12k 20 5 2k 2 15k 40 2k 2 6 22 2 k 2 5 2 2 2 2 3k 2 RHS 2k 3k 2 2 3k 2 k Hence if n k True for n k 1 True for n 1 2 3 must be true for all positive integer n http fatmuscle cjb net 5
Maths Extension 2     Harder Maths Extension 1  3U   Example 2 Recurrence Induction A sequence of terms U n , n   1, 2, 3,...