Math 360 Notes

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(1)Additional Maths Notes (20 Oct 2014) Visit sleightofmath.com for latest notes, solutions and math tuition. Authors: Daniel and Samuel from Sleight of Math Ex 3.1. Table of Contents Ex 1.1. Simultaneous Equations ................................................. 5. Solve a Pair of Linear & Non-linear Eqns .............................. 5 Form Relation ................................................................................... 5 Ex 1.2. Sum and Product of Roots.............................................. 5. Sum & Product of Roots................................................................ 5 Form a Quadratic Equation from its Roots ........................... 5 Useful Formulae............................................................................... 5 Prove Identities involving Roots............................................... 5 Ex 1.3. Discriminant ........................................................................ 5. Complete the Square ...................................................................... 5 Sketch Quadratic graphs .............................................................. 6 Discriminant & Nature of Roots/Number of xintercepts/Number of Intersections ....................................... 6 Conditions for ax 2 + bx + c to be always positive or negative ............................................................................................... 6. Polynomials and Identities.............................................8. Definition of Polynomials ............................................................ 8 Multiply Polynomials .................................................................... 8 Find Unknown(s) in an Identity ............................................... 8 Ex 3.2. Division of Polynomials ...................................................8. Long Division .................................................................................... 8 Division Algorithm ......................................................................... 9 Ex 3.3. Remainder Theorem.........................................................9. Remainder Theorem ..................................................................... 9 Ex 3.4. Factor Theorem ..................................................................9. Factor Theorem ............................................................................... 9 Sum/Difference of Cubes ............................................................. 9 Ex 3.5. Cubic Polynomials and Equations ...............................9. Factorize Cubic Expressions ...................................................... 9 Form Cubic Polynomial ................................................................ 9 Ex 3.6. Partial Fractions .................................................................9. Quadratic Inequalities ..................................................... 6. Break into Partial Fractions ....................................................... 9. Solve Quadratic Inequality .......................................................... 6. Cover-up Rule ................................................................................. 10. Solve Simultaneous Inequalities ............................................... 6. Compare Coefficients .................................................................. 10. Form Quadratic Inequality from Solution............................. 6. Juggling ............................................................................................. 10. Ex 1.4. Ex 2.1. Surds ...................................................................................... 6. Surds Properties .............................................................................. 6. Proper & Improper fraction ..................................................... 10 Ex 4.1. Modulus Functions and their Graphs ...................... 10. Simplify Surds................................................................................... 7. Modulus Definition ...................................................................... 10. Rationalize Denominator ............................................................. 7. Modulus Properties ..................................................................... 10. Solve Surds Equation ..................................................................... 7. Solve Modulus Equations .......................................................... 10. Method of Difference ..................................................................... 7. Sketch y = f(|x|) ............................................................................ 10. Ex 2.2. Indices ................................................................................... 7. Law of Indices ................................................................................... 7 Ex 2.3. Index equations.................................................................. 7. Ex 4.2. Power Graphs ................................................................... 11. Sketch Power Graphs .................................................................. 11 Ex 5.1. Binomial Expansion of (𝟏 + 𝐛)𝐧 ............................... 11. Equality of Indices .......................................................................... 7. Factorial ............................................................................................ 11. Different Types of Manipulation ............................................... 8. Combination ................................................................................... 11. Exponential Functions..................................................... 8. Use Pascal’s Triangle ................................................................... 11. Sketch Exponential Functions.................................................... 8. Expand (1 + b)n ............................................................................ 11. Ex 2.4. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 1.

(2) Binomial Theorem Cross-applications ................................ 12 Ex 5.2. 𝐧. Binomial Expansion of (𝐚 + 𝐛) ................................12. Equate Coordinates ...................................................................... 17 Ex 8.2. Linear Law ......................................................................... 17. Expand (a + b)n ............................................................................ 12. Linearization................................................................................... 17. Use Tr+1 ............................................................................................ 12. Gradient & Y-intercept ............................................................... 17. Ex 6.1. Mid-point of a Line Segment .......................................12. Scale.................................................................................................... 17. Distance Formula ......................................................................... 12. Graphical Reading ........................................................................ 17. Gradient ........................................................................................... 12. Intersection ..................................................................................... 18. Find line ........................................................................................... 12 Use point on line/curve ............................................................. 13 Ratio of Diagonal Segments ..................................................... 13. Ex 9.1. Graphs of Parabolas of the Form 𝐲 𝟐 = 𝐤𝐱............. 18. Sketch y 2 = kx ............................................................................... 18 Ex 9.2. Coordinate Geometry of Circles ................................ 18. Use Vectors ..................................................................................... 13. Circle Equation .............................................................................. 18. Find Intersection .......................................................................... 14. Circle Equation Cross-applications ....................................... 19. Mid-point Formula ...................................................................... 14. Ex 10.1 Triangle Theorems ............................................................ 20. Ex 6.2. Parallel Lines.....................................................................14. Use Line Addition and Subtraction........................................ 20. Angle of Inclination ..................................................................... 14. Angle Properties of Line(s) ...................................................... 20. Parallel Lines.................................................................................. 14. Angle Properties of Triangles .................................................. 20. Collinearity ..................................................................................... 15. Congruency Tests ......................................................................... 20. Find Parallel Line ......................................................................... 15. Similarity Tests .............................................................................. 20. Ex 6.3. Perpendicular Lines .......................................................15. Perpendicular Lines .................................................................... 15. Mid-point Theorem ...................................................................... 21 Ex 10.2. Quadrilaterals Theorems ........................................ 21. Find Perpendicular Line ............................................................ 15. Definition & Properties of Quadrilaterals........................... 21. Find Perpendicular Bisector .................................................... 15. Prove Quadrilaterals ................................................................... 22. Ex 6.4. Areas of Triangles and Quadrilaterals.....................15. Shoelace Formula ......................................................................... 15 Ex 7.1. Ex 10.3. Circles Theorems........................................................ 22. Angle Properties of Circle.......................................................... 22. Introduction to Logarithms .........................................15. Chord Properties of Circle ......................................................... 22. Logarithm Definition .................................................................. 15. Tangent Properties of Circle .................................................... 22. Special Log Values........................................................................ 15 Convert between Log & Index Form .................................... 16 Ex 7.2. Ex 11.1. Trigo Ratios of Acute Angles .................................. 22. Special Angles................................................................................. 22. Laws of Logarithms ........................................................16. Convert between Degrees and Radians ............................... 23. Laws of Logarithm ....................................................................... 16. Complementary ∡s....................................................................... 23. Ex 7.3. Logarithmic Equations ..................................................16. Supplementary ∡s ........................................................................ 23. Equality of Logarithms............................................................... 16. Identify Quadrant ......................................................................... 23. Solve Log Equations .................................................................... 16. Find Basic Angle α ........................................................................ 23. Ex 7.4. 𝐱. Log and Eqns of the form 𝐚 = 𝐛 ..............................16. Find General Angle θ ................................................................... 23. x. Use ⊿ .................................................................................................. 23. Solve a = b .................................................................................... 16 Solve Index Equations ................................................................ 16 Ex 7.5. Ex 11.2. Trigo Ratios of any Angles ...................................... 23. Logarithmic Graphs ........................................................17. Trigo Function Definition .......................................................... 23. Draw Logarithmic Graphs ........................................................ 17. Use ⊿ in Quadrant(s) .................................................................. 23. Ex 8.1. Reducing Equations to Linear Form ........................17. Reciprocal Identities ................................................................... 24. Linearize .......................................................................................... 17. Negative Angles ............................................................................. 24. Form Non-linear Equation ....................................................... 17. ASTC Rule ......................................................................................... 24. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 2.

(3) Solve Trigo Eqn f(x) = k ........................................................... 24 Ex 11.3. Trigo Graphs .................................................................25. Solve Trigo Eqn f(x) = k by Graph........................................ 25. Ex 15.2. Increasing and Decreasing Functions ................ 31. Increasing/Decreasing function ............................................. 31 Ex 15.3. Rates of Change........................................................... 31. Range of Sine & Cosine............................................................... 25. Rate of Change ............................................................................... 31. Find Unknowns of Trigo Function af(bx) + c................... 25. Quantity & Constant Rate .......................................................... 31. Sketch Trigonometric Functions ........................................... 25 Use Symmetrical/Cyclical Nature of Trigo Graphs ........ 26 Inverse Trigo Function .............................................................. 27 Ex 12.1. Ex 15.4. Connected Rates of Change .................................... 32. Connected Rates of Change ...................................................... 32 Ex 16.1. Nature of Stationary Points .................................... 32. Simple Identities .........................................................27. Stationary Point/Value ............................................................... 32. Questions involving Identities ................................................ 27. 1st Derivative Test ....................................................................... 32. Ratio Identities .............................................................................. 27. 2nd Derivative Test...................................................................... 32. Pythagorean Identities............................................................... 28 Square Root of Trigo Function f(x) ....................................... 28 Ex 12.2. Further Trigo Eqns.....................................................28. Simplify to Tangent Eqn ............................................................ 28 Factorize Trigo Eqn ..................................................................... 28 Solve Trigo Eqn f(ax + b) = k................................................. 28 Ex 13.1. The Addition Formulae ............................................28. Addition Formulae ....................................................................... 28 Ex 13.2. The Double Angle Formulae ...................................29. Double ∡ Formulae...................................................................... 29 Half ∡ Formulae ............................................................................ 29 Ex 13.3. Ex 16.2. Maxima and Minima.................................................. 32. Maxima/Minima ............................................................................ 32 Ex 17.1. Derivatives of Trigo Functions.............................. 32. Derivatives of Trigonometric Functions ............................. 32 Ex 17.2. Derivatives of Exponential Functions ................ 32. Derivatives of Exponential Functions .................................. 32 Ex 17.3. Derivatives of Log Functions ................................. 33. Derivatives of Log functions..................................................... 33 Use Logarithmic Differentiation ............................................. 33 Ex 18.1. Indefinite Integrals .................................................... 34. Integral Rules ................................................................................. 34. The R-Formulae ..........................................................29. Find Integral from Derivative .................................................. 34. R-Formulae ..................................................................................... 29. Find Curve from Derivative ...................................................... 34. Ex 14.1. The Derivative and its Basic Rules .......................29. Derivative as Gradient ............................................................... 29. Integrals of Power Functions ................................................... 34 Ex 18.2. Definite Integrals........................................................ 35. Power Rule ...................................................................................... 30. Definite Integrals .......................................................................... 35. Constant Multiple Rule .............................................................. 30. Definite Integrals Rules .............................................................. 35. Sum/Difference Rule .................................................................. 30. Integrals of Modulus Functions .............................................. 35. Differentiation from First Principles.................................... 30 Ex 14.2. The Chain Rule.............................................................30. Chain Rule ....................................................................................... 30 Ex 14.3. Ex 18.3. Integrals of Trigo Functions................................... 35. Integrals of Trigonometric Functions .................................. 35 Ex 18.4. Integrals of Exponential Fns & 1/x ..................... 35. The Product Rule ........................................................30. Integrals of Exponential Functions ....................................... 35. Product Rule ................................................................................... 30. Integrals of 1/x & 1/(ax + b)................................................... 35. Ex 14.4. The Quotient Rule ......................................................31. Quotient Rule ................................................................................. 31 Ex 15.1. Tangents and Normals..............................................31. Find Tangent .................................................................................. 31 Find Normal.................................................................................... 31. Ex 19.1. Area by Integration ................................................... 36. Area by integration ...................................................................... 36 Ex 19.2. Area bounded by Curves ......................................... 36. Strategies to find area bounded by curves ......................... 36 Ex 20.1. Kinematics .................................................................... 36. Tangent Properties ...................................................................... 31. Kinematics Relation ..................................................................... 36. Normal Properties ....................................................................... 31. Implications of Kinematics Statements ............................... 37. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 3.

(4) Distance ............................................................................................ 37 Appendix 1. Geometric Formulae .............................................38. 2D Shapes ........................................................................................ 38 3D Shapes ........................................................................................ 38 Appendix 2. Trigonometric Identities .....................................39. Appendix 3. Calculus Formulae .................................................40. Differentiation ............................................................................... 40 Integration ...................................................................................... 40. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 4.

(5) Additional Math Notes (20 Oct 2014) Useful Formulae. Ex 1.1 Simultaneous Equations. α2 + β2 = (α + β)2 − 2αβ Solve a Pair of Linear & Non-linear Eqns. α−β. Step 1: Subject variable in linear eqn. = ±√(α − β)2. (α − β)2 = (α + β)2 − 4αβ. Step 2: Substitute it into non-linear eqn. α4 + β4 = (α2 + β2 )2 − 2(αβ)2 Form Relation Prove Identities involving Roots. Step 1: Assign variables. To form useful equations (to be substituted),. Step 2: Form relation between variables Ex 1.2 Sum and Product of Roots Sum & Product of Roots Step 1: Simplify to ax 2 + bx + c = 0. Step 3: Find SOR/POR. Product of roots = αβ =. b a. α. a. (iii). Apply power of n to (1). (ii) (1)2 :. α4 = (α − 3)2 α4 = α2 − 6α + 9 α4 = α2 − 6(α2 + 3) + 9 [use (1) to make α the subject] α4 = α2 − 6α2 − 18 + 9 α4 = −5α2 + 9 α4 + 5α − 9 = 0 (shown) ✓. α+β 2. Find roots and unknowns e.g. the equation x 2 − 4x + c = 0 has roots which differ by 2. Find the value of each root and c.. . To prove existence of positive & negative root, use αβ < 0 Convert to quadratic equation in y by substitution. −(1). α3 = α2 − 3α −(2) sub (1) into (2): α3 = (α − 3) − 3α α3 = −2α − 3 α3 + 2α + 3 = 0 (shown) ✓. β. . 2 3. Multiply αn to (1). (1) × α:. Given context e.g. the heights of two men satisfy 40x 2 − 138x + 119 = 0. Without solving the equation, find the average height of these two men. Average height =. . (ii). Solution (i) ∵ α is root, α2 = α − 3. c. Applications  Evaluate expressions involving its roots 2 2 e.g. find + . ∵ α is a root of ax 2 + bx + c = 0, aα2 + bα + c = 0 −(1). Question Given that α is a root of the equation x 2 = x − 3, show that (i) α3 + 2α + 3 = 0 (ii) α4 + 5α2 + 9 = 0. Step 2: State roots. Sum of roots = α + β = −. (i). Ex 1.3 Discriminant. 1 3. e.g. x − 2x + 3 = 0 has roots α & β. Complete the Square. 1. sub y = x 3 : 1. 1. y 2 − 2y + 3 = 0 has roots α3 & β3. k 2. k 2. 2. 2. x 2 + kx = (x + ) − ( ). Form a Quadratic Equation from its Roots Step 1: State roots Step 2: Find SOR/POR Step 3: Form equation x 2 − (SOR)x + (POR) = 0. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 5.

(6) Additional Math Notes (20 Oct 2014) Sketch Quadratic graphs Step 1: Step 2: Step 3: Step 4: Step 5:. Ex 1.4 Quadratic Inequalities. Express as y = a(x − h)2 + k Obtain turning point (h, k) Determine ∪ or ∩ −shape from a Sub x = 0 to find y-intercept Sub y = 0 to find x-intercept. Solve Quadratic Inequality Step 1: Simplify to ax 2 + bx + c vs 0, a > 0 Step 2: Factorize Step 3: Draw sign diagram (Arrange roots & alternate signs with + at left) + − + 𝑥1 𝑥2. Note: x = h is the line of symmetry e.g.. 𝑦 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Solve Simultaneous Inequalities. 𝑥. 𝑥1 𝑂2 (5, −3) To find x1 ,. Step 4: Find range of x. 2+x1 2. f(x) < g(x) < h(x) ⇒ f(x) < g(x) and = 5 ⇒ x1 = 8. Step 1: Split into 2 inequalities using ‘and’ Step 2: Solve each inequality. Discriminant & Nature of Roots/Number of xintercepts/Number of Intersections. Step 3: Take intersection of both solutions. Step 1: Simplify to ax 2 + bx + c = 0 (by substituting line into curve). Form Quadratic Inequality from Solution. Step 2: Use relation between b2 − 4ac & nature of roots/x-intercepts/intersections Discriminant 2. b − 4ac > 0 b2 − 4ac = 0 b2 − 4ac ≥ 0 b2 − 4ac < 0. Nature of roots 2 distinct ℝ 2 equal ℝ 2ℝ 0ℝ. g(x) < h(x). No. of x-intercepts/ intersections 2 1 (tangent) 1 or 2 (meet) 0. Conditions for ax 2 + bx + c to be always positive or negative ax 2 + bx + c > 0 for all x ↔ a > 0,. b2 − 4ac < 0. ax 2 + bx + c < 0 for all x ↔ a < 0,. b2 − 4ac < 0. Step 1: Simplify inequality to ax 2 + bx + c vs 0 Step 2: Use 2 conditions (i) a > 0 or a < 0 (ii) b2 − 4ac < 0. ⇒ k(x − x1 )(x − x2 ) < 0. x1 < x < x2. x < x1 or x > x2 ⇒ k(x − x1 )(x − x2 ) > 0 Ex 2.1 Surds Surds Properties For a > 0 and b > 0, . √a × √b = √ab. . √a √b. . √a × √a = a. =√. a. b. Notation n For √x, n ≡ index x ≡ radicand √ n. √x. ≡ radical sign or radix or root symbol n ≡ surd (if √x is irrational) n. Note: For √x and x < 0, Even n results in non-real number Odd n results in real number 3 e.g. √−4 does not exist but √−8 exists. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 6.

(7) Additional Math Notes (20 Oct 2014) Simplify Surds. Ex 2.2 Indices. Factor out largest square number Law of Indices. e.g. √45 = √9 × 5 = 3√5 Prime factorize (for more challenging numbers) e.g. √540 = √22 × 33 × 5 = 2 × 31.5 × √5 = 2 × 3√3 × √5 = 6√15 Rationalize Denominator 1. . a0 = 1. . a−n =. . a n = ( √a) = √am. . (am )n = amn. . am × an = am+n. m. √a √a = a √a √a 1 1 1 × = 2 a√h + b√k a√h − b√k a h − b 2 k. n. = am−n. an. . m. n. am. . ×. 1 an. Same Base. an × bn = (ab)n a n. an. . =( ). bn. Same Power. b. When you multiply/divide terms, identify common base/power. Solve Surds Equation. 1. 1. Square both sides √a = b ⇒ a = b2. e.g.. Equate rational & irrational terms a + b√k = c + d√k ⇒ a = c, b = d. e.g. ( √√a3 + b 2 + b) ( √√a3 + b 2 − b). 33 ×30 ×93 2. (common base is 3). 273 3. 3. 1. Note: Check the answer mentally by substituting it into the original equation. e.g. √6 − 5x = −x 6 − 5x = x2 2 x + 5x − 6 = 0 (x + 6)(x − 1) = 0 x = −6 or x = 1 (rej) When x = 1, LHS = √6 − 5 = 1 RHS = −1 LHS ≠ RHS. (common power is ) 3. When you add/subtract terms, identify highest common factor e.g. 8x+2 −34(23x ) 3 x+2 = (2 ) −2 × 17(23x ) = 23x+6 −17(23x+1 ) (HCF is 23x+1 ) = 23x+1 (25 − 17) = 23x+1 (15) Note: Equations involving even power functions may have multiple solutions e.g. x 4 = 16 ⇒ x = 2 or x = −2. If you cannot simplify to √a = b or a + b√k = c + d√k, consider solving surds equation by substitution e.g. 2x + 3√x + 1 = 0 sub u = √x: 2u2 + 3u + 1 = 0. Ex 2.3 Index equations Equality of Indices ax = an , for a > 0, a ≠ 1 ⇒x =n. Method of Difference Step 1: Break each term into partial sums Step 2: Arrange partial sums vertically Step 3: Cancel diagonally. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 7.

(8) Additional Math Notes (20 Oct 2014) Different Types of Manipulation Manipulate Simplify Express Evaluate Show Solve Given. Find Unknown(s) in an Identity. Key Words Complex to simple In terms of … Find numerical value Work towards distinct characteristic Equation Consider rearranging given equation.. Ex 2.4 Exponential Functions Sketch Exponential Functions y = ax , a > 1 (slopes up) x. y=a , 0<a<1 (slopes down). 𝑦 1 𝑂 𝑦 1 𝑂. 𝑥. 𝑥. Note: y = ax → 0 for 0 < a < 1 Ex 3.1 Polynomials and Identities Definition of Polynomials.  . Substitute Compare coefficients. Tip: Sub values of x that makes a factor zero e.g. a(x − 2) + b = 5 − 3x sub x = 2: a(0) + b = 5 − 3(2) b = −1 ✓ compare x:. a = −3 ✓. Ex 3.2 Division of Polynomials Long Division Step 1: Surface out hidden terms and express polynomial in powers of decreasing integers Repeat step 2-5 until Deg(R) < Deg(divisor) Step 2: Divide Step 3: Multiply Step 4: Subtract Step 5: Bring down e.g. (3x 2 − 2x + 5) ÷ (x + 2). Polynomials must have  non-negative power  integer power Multiply Polynomials Expand using strategic alignment e.g. (x + 1)(x 2 + x + 1) = x3. +x 2 +x 2. +x +x. +1. Find coefficient using selective multiplication e.g. Find coefficient of x 2 in (x 2 + x + 1)(x 2 + 2x + 3). © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 8.

(9) Additional Math Notes (20 Oct 2014) Division Algorithm. Ex 3.5 Cubic Polynomials and Equations. Quotient Q(x) Divisor g(x) Dividend f(x) ⋮ Remainder R(x). Factorize Cubic Expressions Step 1: Guess factor (factor thm) (x − α). . Dividend = Divisor × Quotient + Remainder. . Dividend. . Deg(Dividend). . Deg(Remainder) < Deg(Divisor). Divisor. = Quotient +. Step 2: Compare x 3. Remainder. (x − α)(px 2 + ⬚ + ⬚). Divisor. = Deg(Divisor) + Deg(Quotient). Step 3: Compare x 0 (x − α)(px 2 + ⬚ + r). For quadratic divisor, f(x) ÷ (px 2 + qx + r) ⇒ R(x)= ax + b ⇒ f(x) = (px 2 + qx + r)Q(x) + ax + b. Step 4: Compare x 2 (x − α)(px 2 + qx + r) Step 5: Compare x (optional) (x − α)(px 2 + qx + r). Ex 3.3 Remainder Theorem Remainder Theorem. Form Cubic Polynomial. b. Polynomial f(x) ÷ (ax + b) ⇒ R = f (− ) a. Tip:. Insert value of x that makes the divisor zero. Note: If polynomial is not given, use division algorithm Ex 3.4 Factor Theorem Factor Theorem b. Polynomial f(x) has factor (ax + b) ⇔ f (− ) = 0 a. Tip:. Insert value of x that makes the factor zero. Sum/Difference of Cubes a3 ± b3 = (a ± b)(a2 ∓ ab + b2 ). . Given 1 distinct root x1 , f(x) = k(x − x1 )3. . Given 2 distinct roots x1 and x2 , f(x) = k(x − x1 )2 (x − x2 ) or k(x − x1 )(x − x2 )2. . Given 3 distinct roots x1 , x2 and x3 , f(x) = k(x − x1 )(x − x2 )(x − x3 ). Ex 3.6 Partial Fractions Break into Partial Fractions Step 1: Convert to proper fraction Step 2: Factorize denominator Step 3: Break into partial fraction forms Denominator Form ax + b (ax + b)2 x2 + c2. A ax+b A ax+b Ax+B. B. + (ax+b)2. x2 +c2. Step 4: Solve for unknowns  Cover-up rule  Substitution  Compare coefficients. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 9.

(10) Additional Math Notes (20 Oct 2014) Proper & Improper fraction. Cover-up Rule. Proper Fraction: Deg(Numerator) < Deg(Denominator). To solve for unknown with linear factors, Step 1: Insert root Step 2: Cover up linear factor (that becomes zero) Step 3: Equate unknown (highest power) e.g.. f(x) (x−x1 )(x−x2 )2. A. B. C. = (x−x ) + (x−x ) + (x−x 1. Improper Fraction: Deg(Numerator) ≥ Deg(Denominator). 2. 2). 2. Ex 4.1 Modulus Functions and their Graphs. ‘insert x1 , cover up (x − x1 ) and equate A’ x = x1 :. f(x) | ( )(x−x2 )2 x=x. Modulus Definition. =A 1. |x| = {. ‘insert x2 , cover up (x − x2 ) and equate C’ x = x2 :. f(x) | (x−x1 )( )2 x=x 2. e.g. Given a > 2, simplify |3 − 2a| a >2 −2a < −4 3 − 2a < −1 <0 |3 ⇒ − 2a| = −(3 − 2a) = 2a − 3. Step 1: Clear fractions by multiplying denominator Step 2: Simplify to polynomial of descending power Step 3: Compare coefficients x2 +2x+15. 5. Bx+C. x. x2 +3 2. = +. x(x2 +3) 2. x≥0 x<0. Tip: Use given condition to determine if |f(x)| = f(x) or − f(x). =C. Compare Coefficients. e.g.. x −x. x + 2x + 15 = 5(x + 3) +(Bx + C)x 2 = 5x + 15 +Bx 2 + Cx = (5 + B)x 2 + Cx + 15. . |−a| = |a|. Compare coefficients: x 2 : 1 = 5 + B ⇒ B = −4 x: C=2. . |ab| = |a||b|. . | |. . |an | = |a|n. . |a|2 = |a2 | = a2. ∴. x2 +2x+15 x(x2 +3). 5. −4x+2. x. x2 +3. = +. Modulus Properties. = |b|. Tip: ‘differ’ is the trigger word to use modulus. Juggling Step 1: Copy denominator to numerator Step 2: Multiply to match term with highest power Step 3: Add to balance e.g. 4x 2 + 3 x2 − 2 (x 2 − 2) … 2 x −2 4(x 2 − 2) … 2 x −2 4(x 2 − 2) + 11 = x2 − 2 11 =4+ 2 x −2. |a|. a. b. e.g. A differs from B by 10 ⇒ |A − B| = 10 Solve Modulus Equations |f(x)| = g(x) ⇒ f(x) = g(x) or f(x) = −g(x). ‘copy (𝑥 2 − 2) to the to numerator’. Note: Check the answer by mentally substituting it into the original equation.. ‘Multiply 4 to match term with highest power’. Tip:. ‘add 11 to numerator to balance’ ‘Divide each term in numerator by denominator’. Consider squaring both sides to remove mod |f(x)| = k [f(x)]2 = k 2. Sketch y = |f(x)| Step 1: Sketch y = f(x) Step 2: Reflect negative part of f(x) in x-axis. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 10.

(11) Additional Math Notes (20 Oct 2014) Use Pascal’s Triangle. Ex 4.2 Power Graphs. To create table, n Binomial coefficients 0 1 1 1 1 2 1 2 1 Step 1: Insert 1 at the sides. Sketch Power Graphs For y = ax n , a > 0 Integer Integer Negative Positive Even Odd Even Odd (≠ 1) 1 1 y= 2 y= y = x2 y = x3 x x. Step 2: Form numbers inside triangle by adding the 2 numbers above it Step 3: Use formula (1 + b)n = (1st coeff)b0 +(2nd coeff)b1 + ⋯ + (last coeff)bn. Rational Rational 0<n<1. n<0. Expand (1 + b)n n n n n (1 + b)n = ( ) b0 + ( ) b1 + ⋯ + ( ) br + ⋯ + ( ) bn 0 1 r n n r =1 +nb + ⋯ + ( ) b + ⋯ + bn r. n>1. (↓ at ↓ rate) (↑ at ↓ rate) (↑ at ↑ rate) Ex 5.1 Binomial Expansion of (1 + b)n Factorial . n! = n × (n − 1) × … × 2 × 1 = n × (n − 1)!. . 0! = 1. To divide between factorials, Step 1: Expand bigger factorial till smaller factorial Step 2: Strike out smaller factorials e.g.. 7! 5!. =. 7×6×5!. (n+1)! (n−2)!. 5!. =. = 7 × 6 = 42. (n+1)n(n−1)(n−2)! (n−2)!. = (n + 1)n(n − 1). Combination n n!  ( ) = (n−r)!r! r n  ( )=1 0 n  ( )=n 1 n n(n−1)  ( )= 2 2 n n(n−1)(n−2)  ( )= 3! 3 n n(n−1)(n−2)(n−3)  ( )= 4! 4. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 11.

(12) Additional Math Notes (20 Oct 2014) Binomial Theorem Cross-applications App 1: App 2: App 3:. Ex 5.2 Binomial Expansion of (a + b)n. Multiply selectively Substitute value/terms Compare coefficients. Expand (a + b)n n n n (a + b)n = ( ) an−0 b0 + ( ) an−1 b1 + ⋯ + ( ) an−n bn 0 1 n. Question. =1. 8. 1. +nan−1 b. + ⋯ + bn. (i) Expand (2 − x) (1 + x) in ascending powers of x, as 2. Use Tr+1 n Tr+1 = ( ) an−r br r To find particular term, Step 1: Simplify to (a + b)n. far as the term in x 3 . (ii) Hence estimate the value of 1.9 × (1.05)8 Solution Multiply selectively (i). 1. (2 − x) (1 + x). Step 2: Use Tr+1  Pull out x. 8. 2. = (2 − x)(1 +4x + 7x 2 + 7x 3 + ⋯ ) [by Binomial Thm] = 2 +8x +14x 2 +14x 3 −x −4x 2 −7x 3 + ⋯ = 2 +7x +10x 2 +7x 3 + ⋯ ✓.  constant. 1. 2. ⇒ power = 0. Step 4: Insert r into Tr+1. Substitute values (ii) 1.9 × (1.05)8 = (2 − 0.1) (1 + 0.5)8 = [2 − (0.1)] [1 + (0.1)]. Step 3: Find r  Equate power n  middle term ⇒ r =. Ex 6.1 Mid-point of a Line Segment 8. Distance Formula. 2. = 2 +7(0.1) +10(0.1)2 +7(0.1)3 + ⋯ [sub x = 0.1 into (i)]. √(x1 − x2 )2 + (y1 − y2 )2. ≈ 2.807 ✓. Gradient y1 − y2 m= x1 − x2. Question The first three terms in the expansion, in ascending powers of x of (1 + 2x)n are 1 + 16x + ax 2 . Find n and a. Solution Compare coefficients (1 + 2x)n = 1 + 2nx + 2n(n − 1)x 2 + ⋯ ≈ 1 + 16x + ax 2 Compare x: Compare x 2. (by Binomial Thm) (given). 2n = 16 ⇒ n = 8 ✓ 2n(n − 1) = a Sub n = 8: 2(8)(8 − 1) = a ⇒ a = 112 ✓. Find line If y-intercept is not given, Step 1: Find point Step 2: Find gradient Step 3: Find line y − y1 = m(x − x1 ) If y-intercept is given, Step 1: State y-intercept Step 2: Find gradient Step 3: Find line y = mx + c Note: For m = 0, horizontal line y = c For m → ∞, vertical line x=a. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 12.

(13) Additional Math Notes (20 Oct 2014) Use point on line/curve. Use Vectors. (x1 , y1 ) lies on y = f(x) ⇒ y1 = f(x1 ) (form eqn) ⇒ (x1 , f(x1 )) (express coordinates in only 1 variable) e.g. (2,1) lies on y = kx − 2 ⇒ 1 = k(2) − 2 −2k = −3 3 k = 2. e.g. A(a1 , a2 ) lies on y = 2x ⇒ A(a1 , 2a1 ) Ratio of Diagonal Segments By similar triangles, B F A: B (diagonal) A E = C: D (horizontal) C D = E: F (vertical). ⃗⃗⃗⃗⃗ AB. ⃗⃗⃗⃗⃗ + OB ⃗⃗⃗⃗⃗ = AO ⃗⃗⃗⃗⃗ − OA ⃗⃗⃗⃗⃗ = OB. Question. 𝑦 𝐴(0,9). Given A is (0,9) & C is (6,3) and AB: BC = 2: 1, find the coordinates of B.. 2. 𝐵 𝐶(6,3) 𝑂 1 𝑥. Solution ⃗⃗⃗⃗⃗ = OA ⃗⃗⃗⃗⃗ OB ⃗⃗⃗⃗⃗ = OA. ⃗⃗⃗⃗⃗ +AB +. 2 2+1 2. ⃗⃗⃗⃗⃗ AC. ⃗⃗⃗⃗⃗ = OA. ⃗⃗⃗⃗⃗ − OA ⃗⃗⃗⃗⃗ ) + (OC. ⃗⃗⃗⃗⃗ = OA. 2 ⃗⃗⃗⃗⃗ − 2 OA ⃗⃗⃗⃗⃗ + OC. 3 3. 3. 1 ⃗⃗⃗⃗⃗ + 2 OC ⃗⃗⃗⃗⃗ = OA 3 1. 3. 2 6 0 = ( ) + ( ) 3 9 3 3 4 =( ) 5 ∴ B(4,5) ✓. 𝑦. Question. 𝐵(8,8) A is (0,6), B is (2, −2) and D is 𝐴(0,6) (2, −2). AB is parallel to DC and 𝑂 AB: DC = 1: 2. 𝐷(2, −2) Find the coordinates of C. 𝐶 𝑥. Solution ⃗⃗⃗⃗⃗ OC = ⃗⃗⃗⃗⃗ OD = ⃗⃗⃗⃗⃗ OD. ⃗⃗⃗⃗⃗ +DC ⃗⃗⃗⃗⃗ +2AB. = ⃗⃗⃗⃗⃗ OD. ⃗⃗⃗⃗⃗ − OA ⃗⃗⃗⃗⃗ ) +2(OB 8 0 ⃗⃗⃗⃗⃗ = OD +2 [( ) − ( )] 8 6 8 ⃗⃗⃗⃗⃗ = OD +2 ( ) 2 2 16 = ( ) +( ) −2 4 18 =( ) 2 ∴ C is (18,2) ✓. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 13.

(14) Additional Math Notes (20 Oct 2014) Question A is (2,4) and B is (6,10). ∡ACB and ∡MDB are 90°. AC: MD = 3: 1. Find the coordinates of M. 𝑦. 𝐵(6,9). 𝐶 3. 𝐷 1. M=( 𝑀. 𝑂 𝐴(3,3) Solution ⊿ ACB ~ ⊿ MDB. AB MB. = =. AC MD 3. (corr. sides or ~ △ s). 1. =3 AB = 3MB ⇒ AM: MB = 2: 1 ⃗⃗⃗⃗⃗⃗ = OA ⃗⃗⃗⃗⃗ OM ⃗⃗⃗⃗⃗ = OA ⃗⃗⃗⃗⃗ = OA. ⃗⃗⃗⃗⃗⃗ +AM 2 + ⃗⃗⃗⃗⃗ AB 3 2. ⃗⃗⃗⃗⃗ ) + (OB − OA 3. 1. ⃗⃗⃗⃗⃗ + 2 OB ⃗⃗⃗⃗⃗ = OA 3 1. 3. 2 6 3 = ( ) + ( ) 3 3 3 9 5 =( ) 7 (5,7) ∴ M is ✓. Find Intersection Solve a pair of equations. Mid-point Formula. 𝑥. x1 + x2 y1 + y2 ) , 2 2. To find endpoint, Step 1: Denote endpoint Step 2: M = (. x1 +x2 y1 +y2 2. ,. 2. ). Step 3: Equate coordinates To find curve traced by mid-point, Step 1: Find midpoint M Step 2: Let M = (x, y) Step 3: Equate coordinates Step 4: Connect x & y Shapes Parallelogram ABCD A B M D C iso.△ ABC with AB = AC A B C Circle with diameter AB A B. Implications MAC = MBD. MBC = Foot of ⊥ from A to BC MAB = Centre. Question A is (2,4) and B is (6,10). AC: MD = 2: 1. Given the diagram below, find the coordinates of M. 𝑦 𝐵(6,10) 𝐷 𝐶 1 𝑀 2 𝑂 𝐴(2,4) 𝑥 Solution M = MAB = (. 2+6 4+10 2. ,. 2. ) = (4,7) ✓. Ex 6.2 Parallel Lines Angle of Inclination m = tan θ (wrt positive x − axis) Parallel Lines l1 ∥ l2 ⇔ m1 = m2. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 14.

(15) Additional Math Notes (20 Oct 2014) Collinearity. Ex 6.4 Areas of Triangles and Quadrilaterals. If A, B & C are collinear, mAB = mAC (any two line segments). Shoelace Formula Area of triangle 1 x1 x2 x3 x1 = |y y y y | 2 1 2 3 1 1 = (x1 y2 + x2 y3 + x3 y1 2 −x2 y1 − x3 y2 − x1 y3 ). Find Parallel Line Step 1: Find point/y-intercept Step 2: Find gradient (m1 = m2 ) Step 3: Find ∥ line y − y1 = m(x − x1 ) y = mx + c. Area of quadrilateral 1 x1 x2 x3 x4 x1 | | 2 y1 y2 y3 y4 y1 1 = (x1 y2 + x2 y3 + x3 y4 + x4 y1 2 −x2 y1 − x3 y2 − x4 y3 − x1 y4 ). Ex 6.3 Perpendicular Lines Perpendicular Lines l1 ⊥ l2 ⇔ m1 . m2 = −1 −1 m1 =. Area of polygon 1 x1 x2 … xn x1 | = |y y … y 2 1 2 n y1 1 = [(sum of products ↘) 2 −(sum of product ↙)]. m2. Shapes Rhombus ABCD A B. Implications → mAC ⊥ mBD. . D C P is equidistant from A and B A P B. → AB⊥ intersects P  . Find Perpendicular Line Step 1: Find point/y-intercept Step 2: Find gradient (m1 =. −1 m2. Coordinates should be in anti-clockwise order to have positive output. On the contrary, if coordinates are in clockwise order, the output is negative. Use modulus if unsure anti-clockwise or clockwise Zero area implies points are collinear 1. App 1:. To find △ angle, use △ area = ab sin C. App 2:. To find △ height (⊥ distance from point to line),. 2. 1. use △ Area = (base)(height). ). 2. Step 3: Find ⊥ line y − y1 = m(x − x1 ) y = mx + c. Ex 7.1 Introduction to Logarithms Logarithm Definition. Find Perpendicular Bisector. A logarithm must have (i) base > 0 (ii) base ≠ 1 (iii) arg > 0. Step 1: Find mid-point (MAB ) Step 2: Find gradient. (. −1 mAB. ). Step 3: Find ⊥ bisector (AB⊥ ) −1 (x − x1 ) y − y1 =. Special Log Values. mAB. Tip: To find the line equidistant to points A & B, find the perpendicular bisector of AB. log a a = 1 ‘Same base & argument results in output of 1’ log a 1 = 0 ‘argument of 1 results in output of 0’. If the 2 points have the same x or y-coordinate, ⊥ bisector = average of the other coordinates. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 15.

(16) Additional Math Notes (20 Oct 2014) Solve Log Equations. Convert between Log & Index Form x = log a y ⇔ y = ax. Step 1: Use laws of log. Step 1: Identify base. Laws of log. Step 2: Connect base to opp. side of eqn Step 3: Switch form keeping base e.g. Log → Index Base is a. log a y = x Connect base a to x y = ax Switch from log to index form. Keep the base a, therefore power is x. Index → Log Base is a. Connect base a to y = log a y Switch from index to log form. Keep the base a, therefore argument is x.. ax = y x. Action. Change-of-base law convert to common base Power law. move coefficient to power. Product law/ Quotient law. combine to single log. Step 2: Remove log  Equality of log  Change to index form Step 3: Check log conditions Note: Use substitution u = log a x if you cannot simplify to log 𝑎 𝑥 = 𝑏 Ex 7.4 Log and Eqns of the form ax = b. Ex 7.2 Laws of Logarithms. Solve ax = b To solve ax = b, log both sides. Laws of Logarithm Product Law. log a xy = log a x + log a y x. Quotient Law. log a. Power Law. log a x r = r log a x. Change-of-Base Law. log a b =. y. = log a x − log a y. logc b logc a. Ex 7.3 Logarithmic Equations Equality of Logarithms log a x = log a n. ⇔x=n. =. 1 logb a. Solve Index Equations Method 1: Method 2: Method 3: Method 4:. ax = an ⇒ x = n ax = b (log both sides) Convert to log form Substitution. Step 1: Use laws of indices to simplify to ax = an or b  When multiply/divide terms, identify common base/power  When add/subtract terms/identify highest common factor Step 2: Remove base  Equality of indices (ax = an ⇒ x = n) (ax = b)  Log both sides  Convert to log form If you cannot simplify to ax = an or b, use substitution u = ax e.g.. 9(3x )2 + 1 = 10(3x ). e.g.. x 2 − 8x −2 = 7. 3. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 3. 16.

(17) Additional Math Notes (20 Oct 2014) Equate Coordinates. Ex 7.5 Logarithmic Graphs. The first and second coordinates are not necessarily x and y respectively!. Draw Logarithmic Graphs 𝑦. y = log a x, a>1 (slopes up). Question. 𝑂 1. 𝑦 y = log a x, 0<a<1 (slopes down) 𝑂 1. 𝑥. If C(9, −8) lies on the graph of y√x against x, find the value of y corresponding to c.. 𝑥. Solution Equate 1st coordinate: Equate 2nd coordinate:. Note: For base > 1, there is an inverse relation between base & rate of increase. x =9✓ y√x = −8 y√9 = −8 8 y =− ✓ 3. For a > b > c > 1, 𝑦 𝑦 = log 𝑎 𝑥 𝑦 = log 𝑏 𝑥 𝑂 𝑥 𝑦 = log 𝑐 𝑥 1. Ex 8.2 Linear Law Linearization Step 1: Simplify to Y = mX + c Step 2: Complete table Gradient & Y-intercept. Ex 8.1 Reducing Equations to Linear Form. Step 1: State 2 points: (i) On y-axis (ii) Halfway-down. Linearize Contains x&y X&Y ✓ m&c ✘. Contains constants ✘ ✓. Step 2: Equate gradient & Y-intercept Scale a. 1. b. b. e.g. if ax 2 + by 3 = 1, then y 3 = − x 2 + a. 1. b. b. i.e. Y = y 3 , X = x 2 , m = − , c = b+x a. e.g. if y = e. 1. b. a. a. , then ln y = x + 1. b. a. a. i.e. Y = ln y , X = x, m = , c =. Step 1: Estimate Y-intercept Y1 = mX1 + c c = Y1 − mX1 Step 2: State domain & range Step 3: Find X & Y interval X −X X-Interval = last 1st 10 Ylast −Y1st. Y-Interval = 12 (Round down to 1, 2, 25 or 5). To find unknowns, Step 1: Linearize to axes variables Step 2: Equate gradient & Y-intercept or use points on line (whichever is given). Step 4: State X & Y scale Graphical Reading. Step 1: Find point/gradient/ Y-intercept. Step 1: Simplify to (X or Y) Step 2: Identify point Step 3: Equate (Y or X) & solve for desired variable. Step 2: Form linear equation Y = mX + c (If Y-intercept is given) Y − Y1 = m(X − X1 ) (If Y-intercept is not given). Note: Graphical reading is reliable only within the data range (interpolation) & not reliable outside the data range (extrapolation). Form Non-linear Equation. Step 3: Form non-linear eqn by replacing X & Y with axes variables. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 17.

(18) Additional Math Notes (20 Oct 2014) Intersection. Ex 9.2 Coordinate Geometry of Circles. Step 1: Work towards 2 curves on each side Step 2: Plot 2nd curve & use intersection. Circle Equation . Standard form (x − a)2 + (y − b)2 = r 2. Ex 9.1 Graphs of Parabolas of the Form y = kx. . General form. x 2 + y 2 + 2gx + 2fy + c = 0. Sketch y 2 = kx. . Centre. (a, b) = (−g, −f). . Radius. r = √g 2 + f 2 − c. 2. 2. 𝑦. y = kx, k>0 𝑂. Note: It appears to be a counter-intuitive convention that g comes before f in the formula. 𝑥. Note: y = 0 is the line of symmetry. Trigger/Setup. Action. Question Given the graph y 2 = 2x, draw a suitable line to solve x 2 − 8x + 9 = 0.. 2 points. Find ⊥ bisector of chord where centre lies on. Centre & point. Use distance formula to find radius.. Solution x 2 − 8x + 9 x 2 − 6x + 9 (x − 3)2 ⇒y=x−3. Diameter. Use midpoint formula. =0 = 2x = 2x or y = −(x − 3) ✓. Touches Sketch graph. Deduce horizontal/vertical coordinates, centre, radius or line point on circle. (see example) Right angle triangle drawn. Use Pythagoras’ Theorem. 0,1 or 2 intersections. Use discriminant.. Touches another circle. Connect centres with a line.. Line is tangent to circle. Identify right angle (tan ⊥ rad) Find normal.. If the centre cannot be found from the approaches above (or only 1 coordinate can be deduced),  use given information about centre (if any) e.g. centre C(h, k) lies on line y = f(x) ⇒ C is (h, f(h)) e.g. centre C(h, k) is 6 units away from point A(1,2) ⇒ √(h − 1)2 + (k − 2)2 = 6 . © Daniel & Samuel Math Tuition 📞9133 9982. insert parameters into (x − h)2 + (y − k)2 = r 2 and solve for unknowns by elimination.. sleightofmath.com. 18.

(19) Additional Math Notes (20 Oct 2014) Examples of sketching graph to deduce information Touch axis Given: centre (3, −2), touches x-axis Deduce: Cut axis Given: Deduce:. 𝑂. radius = 2. 𝑥 2 (3, −2) 𝑦. Cuts y-axis at −2 and −5 y − coordinate of centre −1+(−5) = = −3 2. Touch line(s) Given: Touches x = 2 & x = 8 Deduce:. 𝑦. radius =. 8−2 2. 𝑂 𝑥 −1 𝐶(𝑐1 , −3) −5. 2. 3 𝐶(5, 𝑐2 ) 𝑂𝑥=2 𝑥=8𝑥. Pythagoras Theorem  Find length of PT, given radius is √13. C(2, −1). Find ⊥ line  Find tangent/normal at point of contact e.g. Find AB. T. P(3, −10) 2√10. Idea: Find PC by distance formula PT = √PC 2 − CT 2 (Pythagoras’ thm) . Use Distance Formula  Find radius  To check if point A lies within circle, compare distance between A and centre with the radius Use Midpoint Formula  Given that A(2,3) and B(4,5) are points on the circle, the find the centre.  Given A is (2,3), the centre C is (4,5) and AB is the diameter of the circle, find the point B. Circle Equation Cross-applications. P(9,2). Find Intersection Point  Find point on circle  Find point of contact between tangent & normal  Find centre where line through centre meets perpendicular bisector of chord. 𝑦. =3. x − coordinate of centre 2+8 = =5. Use Discriminant  Find number of intersections between line & circle (you can also compare the perpendicular distance with the radius to determine the number of points of intersection)  Find unknown c in line eqn given line is tangent to circle. AB. Find AC 𝑦. C(1, −4). Find ⊥ bisector  Whenever two points on circle are given, consider finding the perpendicular bisector. The perpendicular bisector of the chord passes through the centre of the circle. 6B r = 5 A C 2 𝑥 𝑂 Idea: AC = √r 2 − AB 2 (Pythagoras’ thm). Use Properties of Circle (refer to Ex 10.3) Solve System of Equations  To find circle equation given 3 points on the circle, insert the points into general form of circle. 𝑂 A B ⊥ bisector of chord. tan ⊥ rad. Complete the Square  Convert general form to standard form. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 19.

(20) Additional Math Notes (20 Oct 2014) Similarity Tests. Ex 10.1 Triangle Theorems Use Line Addition and Subtraction A. B. C. SSS 3 ∝ sides. AB + BC = AC AC − AB = BC. Question (prove product of sides) Given △ ABC ~ △ DEF, prove that AB × DF = AC × DE. Question Given AB = CD, prove AC = BD A B. Solution Whenever you encounter product of multiple line Given △ ABC ~ △ DEF, segments, consider using the prove that property of similar triangles: AB × DF = AC × DE ratios of corresponding sides are equal.. C D. Solution AB = CD AB + (BC) = CD + (BC) AC = BD ✓ Angle Properties of Line(s) a b. SAS AA 2 ∝ sides, 2 eq. ∡ 1 included ∡. b. a. a. b a. b. AB AC. …. Identify which line segments in the above product correspond to the triangle ABC. AB and AC. AB Take ratio at the left. Note the AC sequence. AB is 12 and AC is 13. 12 over 13.. =. Use same sequence on the other triangle DEF at the right. 12 is DE DE and 13 is DF. Take ratio at DF the right.. ab. ∡s in line opp. ∡ int. ∡ corr. ∡ alt. ∡  . Prove straight lines by ∡s in line = 180° Prove parallel lines by int.∡, corr. ∡ & alt. ∡. AB AC. DE DF. Angle Properties of Triangles b a c ∡s in △ = 180° . a. bc. ext. ∡ = sum of int. opp. ∡s. ab ⊿. ab. c ab. AB × DF = AC × DE [proven] ✓. iso.△ eq.△. Prove equal sides/angles using iso.△ & eq.△ Congruency Tests. Question (Prove relation/ratio of line segments) Given △ ABC ~ △ DEF & DE: EF = 1: 2, prove that 1 AB = BC (or AB: BC = 1: 2) 2. Solution AB BC. = =. SSS 3 eq. sides. SAS AAS RHS 2 eq. sides, 2 eq. ∡s, 1 rt ∡, 1 included ∡ 1 corr. 1 eq. hyp, sides 1 eq. side Note: Order of Points matter e.g. △ ABC ≅ △ XYZ is not the same as △ ACB ≅ △ XYZ . Cross multiply.. DE EF 1 2 1. ⇒ AB = BC ✓ 2. ⇒ AB: BC = 1: 2 ✓. Prove equal sides/angles using congruent △s. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 20.

(21) Additional Math Notes (20 Oct 2014) Question (Use ratio of area of similar triangles) Given △ ABC ~ △ EDC, B 1 E BC: CD = 1: 2 & C area of △ ABC = x, A 2 D find the area of △ DEC Solution 2 2. Area of △ DEC = ( ) x = 4x ✓ 1. [use. Mid-point Theorem D = MAB , E = MAC 1 ⇒ DE ∥ BC, DE = BC 2. A2. l. 2. = ( 1) ] l2. Definition & Properties of Quadrilaterals Kite Quad. with two pairs of equal adjacent sides  ∡s between unequal sides are equal (angle)  One diagonal bisects the other (diagonal)  Longer diagonal bisects ∡s (diagonal)  Diagonals are ⊥ (diagonal) Note: Concave kite have interior ∡s > 180°. A D B. A1. Ex 10.2 Quadrilaterals Theorems. E C. Trapezium Quad. with exactly one pair of parallel sides  supplementary interior ∡s Parallelogram Quad. with two pairs of parallel sides  Opp. sides are equal  Opp. ∡s are equal  interior ∡s are supplementary  Diagonals bisect each other. (side) (angle) (angle) (diagonal). Rectangle Quad. with four right angles  Opp. sides are parallel  Opp. sides are equal  Diagonals bisect each other  Diagonals are equal. (side) (side) (diagonal) (diagonal). Rhombus Quad. with four equal sides  Opp. sides are parallel  Supplementary interior ∡s  Diagonals bisect ∡s Diagonals are ⊥ bisector of each other. (side) (angle) (diagonal) (diagonal). Square Quad. with four equal sides & four right angles  Diagonals bisect angles (diagonal)  Diagonals are equal (diagonal)  Diagonals are ⊥ bisector of each other (diagonal). © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 21.

(22) Additional Math Notes (20 Oct 2014) Tangent Properties of Circle. Prove Quadrilaterals Parallelogram  2 pairs of ∥ sides  2 pairs of equal & opp. sides  1 pair of equal & ∥ sides  2 pairs of equal opp. ∡s  Diagonals bisect each other. (definition) (side) (side) (angle) (diagonal). Rectangle  4 right ∡s  Parallelogram + 1 right ∡. (definition) (angle). Q b. P. a. R tangents from ext. point. alt. segment tan ⊥ rad thm. Ex 11.1 Trigo Ratios of Acute Angles Special Angles. Rhombus  4 equal sides (definition)  Parallelogram + eq. adj. sides (side)  Parallelogram + bisecting diagonals (diagonal)  Parallelogram + ⊥ diagonals (diagonal). Table. Square  4 equal sides & 4 right ∡s  Rectangle + eq. adj sides  Rhombus + 1 right ∡. tan θ 0. 0° 30° 45° 60° 90° 0 sin θ 0 cos θ 1. (definition) (side) (angle). π. π. π. π. 6 1. 4. 3. 2. √2 2. √3 2 1. 1. 2 √3 2 1 √3. √2 2. 2. 0. 1 √3 ∞. Triangle. √2. Trapezium  Parallel opposite sides. O. 1. (definition). Kite  2 pairs of equal adjacent sides (definition). 45° 1 Unit circle. 30°. 2. √3. 60°. 60°. 1. Ex 10.3 Circles Theorems Angle Properties of Circle a. b. a b. a ∡ in semicircle. b a. ∡ at centre ∡s in same ∡s in opp. = 2 ∡ at segment segment circumference. Chord Properties of Circle. A. O B. C. ⊥ bisector of chord passes through centre . A C. X. B. O Y D. Equal chords are equidistant from centre. Equal arcs results in equal chords. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 22.

(23) Additional Math Notes (20 Oct 2014) Find Basic Angle α. Convert between Degrees and Radians π rad = 180° . To convert from degrees to radians, multiply. . To convert from radians to degrees, multiply. π 180° 180°. Tip: Track the unit conversion to avoid the mistake of multiplying the wrong fraction e.g. 60° = 60°. ×. π. 1. 180°. = π 3. [rad]. [deg] = [deg] × [deg] = [rad]. 60° = 60°. ×. [deg] = [deg] ×. 180° π [deg] [rad]. = =. 10800 π [deg]2 [rad]. π. Step 1: Add or subtract 360° until 0° ≤ θ ≤ 360° Step 2: Use table Quadrant α 1 θ 2 180° − θ 3 θ − 180° 4 360° − θ Find General Angle θ Quadrant 1 2 3 4. ✓ ✓. ✘ ✘. θ α 180° − α 180° + α 360° − α. Use ⊿ Step 1: Draw ⊿ Step 2: Find all 3 sides (by Pythagoras’ Thm). Complementary ∡s. Ex 11.2 Trigo Ratios of any Angles. . sin(90° − θ) = cos θ. . cos(90° − θ) = sin θ. . tan(90° − θ) =. 1 tan θ. Supplementary ∡s. Trigo Function Definition y. . sin θ =. . cos θ =. . tan θ =. r = √x 2 + y 2. r x r y. r θ. y x. . sin(180° − θ) = sin θ. . cos(180° − θ) = − cos θ. . . tan(180° − θ) = − tan θ. Angles measured anti-clockwise from the positive x-axis are positive. On the contrary, angles measured clockwise from the positive x-axis are negative.. Identify Quadrant Step 1: Add or subtract 360° until 0 ≤ θ ≤ 360° Step 2: Use table Angle 0° < θ < 90° 90° < θ < 180° 180° < θ < 270° 270° < θ < 360°. Quadrant 1 2 3 4. © Daniel & Samuel Math Tuition 📞9133 9982. x. Use ⊿ in Quadrant(s) Step 1: Identify quadrant Step 2: Draw ⊿ in quadrant Step 3: Find coordinates. sleightofmath.com. 23.

(24) Additional Math Notes (20 Oct 2014) Question. Reciprocal Identities. Given that tan A = −. 5 12. and that tan A and cos A have. opposite signs, find the value of each of the following. (i) sin(−A) (ii) cos(−A) π. (iii) tan ( − A). . sec θ =. . csc θ =. . cot θ =. 1 cos θ 1 sin θ 1 tan θ. 2. Negative Angles. Solution Thought Process Step 1: Identify quadrants Observe that ratio for tan is negative. 5 tan A = − < 0 Tan is only positive in 1st or 3rd quad. 12 Therefore, it is in 2nd or 4th quad. ⇒ 2nd or 4th quad. tan A & cos A have In 3rd quad., only tan is positive opp. signs In 4th quad., only cos is positive ⇒ 3rd or 4th quad. Therefore, it is in 3rd or 4th quad. Take overlap of above deductions. Therefore it is in 4th quadrant.. ∴ 4th quadrant Step 2: Draw ⊿ in quadrant. cos(−θ) = cos(θ). . sin(−θ) = − sin(θ). . tan(−θ) = − tan(θ) ASTC Rule. sin is + S A all are + tan is +T C cos is + All trigo functions can be converted to trigo function of basic angle with positive or negative sign depending on ASTC rule. e.g. sin(210°) = − sin(30°) Solve Trigo Eqn f(x) = k Quadrants method Step 1: Find α = f −1 (|k|) & identify quadrants Step 2: State interval Step 3: Find x using quadrants. Draw ⊿ in 4th quadrant.. 12 r. . −5. Step 3: Find coordinates 5 y tan A = − = 12. 180° − α 2 1 α 180° + α 3 4 360° − α. y. tan A = by definition. x. π−α2 1α π + α 3 4 2π − α. x. y = −5,. Equate numerator, 𝑦 = −5. y-coordinate is negative in 4th quad. Equate denominator, 𝑥 = 12. x-coordinate is positive in 4th quad.. x = 12,. r = √122 + (−5)2 Find hypotenuse r by Pythagoras’ Theorem = 13 y. 5. r x. 13 12. r. 13. sin A = = − cos A = =. ,. Find other trigo ratios to serve as useful inputs. The rest of the question makes use of the 3 basic trigo ratios: sin A , cos A & tan A.. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 24.

(25) Additional Math Notes (20 Oct 2014) Sketch Trigonometric Functions. Ex 11.3 Trigo Graphs. Step 1: Simplify to y = af(bx) + c Solve Trigo Eqn f(x) = k by Graph. Step 2: Find amplitude & period Sin/Cos Tan |a| Amplitude Nil. Graphical method When α = 0° or 90°, i.e.. Period. sin f(x) = 0, ±1. tan f(x) = 0 Step 1: State interval Step 2: Find x using graph y = cos x y = tan x. 𝑦. 𝑦. 𝑦. 1. 1. 1. 180° 90°. 360°. 180°. 𝑥. 270°. −1. 90°. 270°. 180°. 𝑥. b. b. y = sin x. 360°. −1. π. Step 3: Complete table and sketch graph Domain x1 ≤ x ≤ x2 Axis with y = c ± |a| Amplitude Shape ±sin/cos/tan x2 −x1 Cycle T. cos f(x) = 0, ±1. y = sin x. 2π. 270°. 90°. 𝑥 1. −1. −1. Range of Sine & Cosine. 1. 𝑦. 1. 1. 𝑥. −1. 90° 180° 270° 360°. 𝑥. 90° 180° 270° 360°. 𝑥. −1. 90° 180° 270° 360°. 𝑥. −1. sin x = −1 at x = 270° sin x = 1 at x = 90°. Max. 𝑥. sin x = 0 at x = cos x = 0 at x = tan x = 0 at x = 0°, 180°, 360° 90°, 270° 0°, 180°, 360° sin x = −1 cos x = −1 Min Nil at x = 270° at x = 180° sin x = 1 cos x = 1 Max Nil at x = 90° at x = 0°, 360°. y = cos x. 𝑦. Min. 𝑦. 0. −1 ≤ sin x ≤ 1 −1 ≤ cos x ≤ 1 y = sin x. 90° 180° 270° 360°. y = tan x. 𝑦. 90° 180° 270° 360°.  . y = cos x. 𝑦. 360°. cos x = −1 at x = 180° cos x = 1 at x = 0°, 360°. Find Unknowns of Trigo Function af(bx) + c Sine/Cosine. Tangent. max A = |a|. c. c. A = |a| min T=. 360°. T=. b. Amplitude A = |a| =. max−min. 360°. Period. T=. Axis. c= 2 = min + A = max − A. 2. 180° b. Period T =. 180° b. b max+min. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 25.

(26) Additional Math Notes (20 Oct 2014) Question. Use Symmetrical/Cyclical Nature of Trigo Graphs. Sketch y = 3(1 − 2 cos 4x) for 0° ≤ x ≤ 270°. Symmetrical Question Given α & β are roots of 3 cos x + 2 = 2 where 3 < k < 4. Find β in terms of α, given that α < β. Solution Step 1: Simplify to 𝐲 = 𝐚𝐟(𝐛𝐱) + 𝐜 y = 3(1 − 2 cos 4x) = 3 − 6 cos 4x = −6 cos 4x + 3. Solution y. Step 2: Find amplitude & period A = |−6| = 6 T=. 360° 4. 𝑦 = 3 cos 𝑥 + 2. 5. = 90°. 2. Step 3: Complete table and sketch graph Domain 0° ≤ x ≤ 270° Follow the Axis with y=3±6 sequence from Amplitude top down to Shape −cos sketch the graph. 270−0 =3 Cycle 90. 𝑦. 𝑂. 𝜋. 2𝜋. x. -1 ∵ x = π is line of symmetry, α+β =π 2. β = 2π − α ✓ Cyclical. 𝑂. 270°. Mark the endpoint of domain, 270°. 𝑥. 𝑦 9 3 𝑂. 270°. −3. 𝑥. Mark the axis 3. Add and subtract 6 to get max 9 and min −3.. Question Given that α is the smallest positive root of the equation √2 cos 4x = −3.1 tan 2x, where 0° ≤ x ≤ 360°, state the other roots in terms of α. Solution 𝑦 𝑦1 = −3.1 tan 2𝑥 √2. 𝑦 9. Draw 1 cycle of negative cosine.. 3 𝑂. 270°. −3. 𝑂 −√2. 𝑥. 90°. 𝑥 = 45°. 𝑦2 = √2 cos 4𝑥 𝑥 180°. 𝑥 = 135°. ∵ Period = 90°, ⇒ x = α, α + 90°, α + 180°, α + 270° ✓. 𝑦 9 3 𝑂. −3. © Daniel & Samuel Math Tuition 📞9133 9982. 270°. There are 3 cycles in total. Draw 2 more. 𝑥. sleightofmath.com. 26.

(27) Additional Math Notes (20 Oct 2014) Note: If basic angles cannot be found, use ⊿ in quadrant. Inverse Trigo Function. Question (Evaluate compound inverse trigo functions). Principal values π. π. 2. 2. . − ≤ sin−1 x ≤. . 0 ≤ cos −1 x ≤ π π. . − < tan. −1. 2. x<. 1. Find the exact value of sin [cos −1 (− )] 5. Solution π. Step 1: Identify quadrant. 2. 1. Let A = cos −1 (− ) 5 ⇒ 2nd quadrant. Step 1: Identify quadrant Step 2: Find basic angle Step 3: Find general angle. Since 0 ≤ cos −1 x ≤ π, it is in the 1st or 2nd quad. Because of the negative sign 1 of − , it is in 2nd quad. 5. Step 2: Draw ⊿ in quad.. Question 1. Evaluate cos −1 (− ) without using the calculator. 2. 𝑦 5 −1. Solution. ✘ ✘✘. 1. cos −1 (− ) 2. π. = (π − ) 3. =. 2π 3. ✓. 𝜋. Step 3: Find coordinates. 3. cos A = − =. Thought process Step 1: Identify quadrant 0 ≤ cos −1 x ≤ π Strike out 3rd and 4th quadrants. ✘✘ 1. Input − is negative. 2 1st quadrant always corresponds to positive ratios. Strike out 1st quadrant. Step 2: Find basic angle Mentally use table of special angles and ignore the negative 1. sign of − , 0 cos θ 1. 2 π. π. π. π. 6. 4. 2. √3 2. √2 2. 3 1. ⇒ basic angle =. ✘ ✘✘. x. 5. r. x. cos A = by trigo definition r. x = −1,. Equate numerator: x = −1. x-coordinate is negative in 2nd quad.. r=5. Equate denominator: r = 5. r is always positive.. y = √52 − (−1)2 = √24 = √4 × 6 = 2√6. Find y by Pythagoras’ Thm. 1. sin [cos −1 (− )] y. 5 2√6. r. 5. = sin A = =. y. sin A = by trigo definition ✓. r. Ex 12.1 Simple Identities. ✘ ✘✘ 𝜋 3. 0. 2. 1. Questions involving Identities . Simplify using identities. . Evaluate using identities. . Prove identities. π. Ratio Identities. 3. Step 3: Find general angle General angle is the angle wrt the positive x-axis. ACW is positive. CW is negative. π General angle = π − 3. © Daniel & Samuel Math Tuition 📞9133 9982. ✘ ✘✘ 𝜋. . tan θ =. . cot θ =. 3. sleightofmath.com. sin θ cos θ cos θ sin θ. 27.

(28) Additional Math Notes (20 Oct 2014) Pythagorean Identities. Factorize Trigo Eqn. . sin2 θ + cos 2 θ = 1. . tan2 θ + 1. = sec 2 θ. . cot 2 θ + 1. = csc 2 θ. Question Given that sin x + sin y = a and cos x + cos y = a, where a ≠ 0, express sin x + cos x in terms of a.. . Take out common factor 2 sin x cos x − sin x = 0 sin x (2 cos x − sin x) = 0. . Express in factor form cos 2 x − cos x − 2 =0 (cos x − 2)(cos x + 1)= 0. . Factorize by grouping 3 sin x tan x − 12 sin x −2 tan x + 8 3 sin x (tan x − 4) −2(tan x − 4) (tan x − 4)(3 sin x − 2). Solution sin x + sin y = a ⇒ sin y = a − sin x cos x + cos y = a ⇒ cos y = a − cos x. −(1) −(2) . (1)2 + (2)2 : sin2 y + cos 2 y= (a − sin x)2 + (a − cos x)2 2. = (a − sin x) + (a − cos x). 1. tan x =. sin x + cos x. −(−2)±√(−2)2 −4(1)(−2). = 1 ± √3. 2(1). Solve Trigo Eqn f(ax + b) = k. = 2a2 − 2a(sin x + cos x) + sin2 x + cos 2 x. 0. If unable to factorize, use quadratic formula tan2 x − 2 tan x − 2 = 0. 2. = (a2 − 2a sin x + sin2 x) +(a2 − 2a cos x + cos 2 x). Quadrants Method Step 1: Find α & identify quadrants. = 2a2 − 2a(sin x + cos x) + 1. Step 2: Adjust interval. = 2a2 − 2a(sin x + cos x). Step 3: Find ax + b & x using quadrants. = a − (sin x + cos x). 180° − α 2 1 α 180° + α 3 4 360° − α. =a. Square Root of Trigo Function f(x) . f(x) ≥ 0: √[f(x)]2 = f(x). . f(x) < 0: √[f(x)]2 = −f(x). The output of square root is positive by definition e.g. √sin2 x = sin x. for 0 < x < 90°. √sin2 x = − sin x for 180 < x < 270°. Graphical Method When α = 0° or 90° i.e. sin f(x) = 0, ±1 cos f(x) = 0, ±1 tan f(x) = 0 Step 1: Adjust interval Step 2: Find ax + b & x using graphs y = sin x. Ex 12.2 Further Trigo Eqns. y = cos x y = tan x. 𝑦. 𝑦. 𝑦. 1. 1. 1. 180° 90°. Simplify to Tangent Eqn. =0 =0 =0. 360°. 270°. −1. 180°. 𝑥. 90°. 360°. 270°. −1. 180°. 𝑥. 90°. 360°. 270°. 𝑥. −1. Step 1: Separate sin & cos to opp. sides of eqn Step 2: Divide by cos x. Ex 13.1 The Addition Formulae. e.g. a sin θ + b cos θ = 0 a sin θ = −b cos θ. Addition Formulae. tan θ. =−. © Daniel & Samuel Math Tuition 📞9133 9982. b a. . sin(A ± B) = sin A cos B ± cos A sin B. . cos(A ± B) = cos A cos B ∓ sin A sin B. . tan(A ± B) =. sleightofmath.com. tan A±tan B 1∓tan A tan B. 28.

(29) Additional Math Notes (20 Oct 2014) Ex 13.2 The Double Angle Formulae. Ex 13.3 The R-Formulae. Double ∡ Formulae. R-Formulae. . a sin θ ± b cos θ = R sin(θ ± α). sin 2A = 2 sin A cos A 1 ⇒ sin A cos A = sin 2A. a cos θ ± b sin θ = R cos(θ ∓ α). 2. . . cos 2A = cos 2 A − sin2 A = 2 cos 2 A − 1 = 1 − 2 sin2 A ⇒ cos 2 A =. 1+cos 2A. ⇒ sin2 A =. 1−cos 2A. tan 2A =. Tip:. . R = √a2 + b 2. . α = tan−1 ( ). . min = −R, max = R. b a. 2. Ex 14.1 The Derivative and its Basic Rules. 2. 2 tan A. Derivative as Gradient. 1−tan2 A. The gradient of the curve y = f(x) at (x1 , y1 ) is. As cos 2A has 3 possible outputs, the output that eliminates 1 is often chosen. Note: Most of the time,. Question. dy dx. dy. |. dx x=x1. is a function of x.. To find the gradient we need the x-input.. Show 1 + cos 2A = 2 cos 2 A. Question Calculate the gradient(s) of the curve at the point(s) where y is given. y = 2x 2 + 3x, y = 2.. Solution LHS = 1 + cos 2A Do not use: 2 2 [(i) cos 2A = cos A − sin A] 2 (ii) cos 2A = 1 − 2 sin A. = 1 + (2 cos 2 A − 1). Solution y = 2x 2 + 3x dy dx. = 2 cos 2 A = RHS. = 4x + 3. At y = 2, 2x 2 + 3x =2 2 2x + 3x − 2 = 0 (2x − 1)(x + 2) = 0. Half ∡ Formulae 1. 1. 2. 2. To find sin A, use cos A = 1 − 2 sin2 ( A) 1. 1−cos A. 2. 2. ⇒ sin A = ±√ 1. To find cos A, use cos A = 2. 1 2 cos 2 ( A) 2. 1. 1+cos A. 2. 2. ⇒ cos A = ±√. 1. 1 2 1 1−tan2 ( A) 2 −b±√b2 −4ac. 2. 2a. 1. To find tan A, use tan A = 2. ⇒ tan A =. © Daniel & Samuel Math Tuition 📞9133 9982. x= dy. 1 2. or x = −2. |. dx x=1. =5. 2. dy. −1. |. dx x=−2. = −5. 2 tan( A). sleightofmath.com. 29.

(30) Additional Math Notes (20 Oct 2014) Power Rule. Differentiation from First Principles. d n (x ) = nx n−1 dx. f ′ (x) = lim. Useful shortcuts. Question. d. . 1. (√x) = dx 2√x d. . 1. 1. dx x. x2. ( )=−. d dx d. Solution. dx d dx. . d. [x(x + 1)] = ( (. f(x) = √x f(x + δx) = √x + δx. 2x2 +4x x x2 +2x x−1. d. )=. )=. dx. dx d. dx. (x 2 + x) = 2x + 1. f ′ (x). (2x + 4) = 2. (x + 3 +. 3. = lim. e.g.. (√x) =. dx d 1. ( ). =. dx x. . d dx d dx. 1 2. ) (long division). x−1. =. (x −1 ). =−. d. [. 2x+1. ]=. dx x(x+1). d. 1. ( +. dx x. 1. Constant Multiple Rule d dx. [kf(x)] = k. d dx. [f(x)]. Sum/Difference Rule d dx. [f(x) ± g(x)] = f ′ (x) ± g′(x). √x+δx−√x δx. ⋅. √x+δx+√x √x+δx+√x. x+δx−x. δx→0 δx(√x+δx+√x). 2√x 1. δx. = lim. δx→0 δx(√x+δx+√x). x2. )=−. x+1. δx. = lim. = lim. Breaking into partial fractions e.g.. = lim. δx→0. 1. (x ). f(x+δx)−f(x). δx→0. Use law of indices d. δx. Find the derivative of f(x) = √x from first principles. Consider simplifying before differentiating  Multiply or divide e.g.. f(x+δx)−f(x). δx→0. 1. δx→0 (√x+δx+√x). 1 x2. 1. − (x+1)2. = =. 1. ∵(. √x+√x 1 2√x. as δx → 0, ) √x + δx → √x. ✓. Ex 14.2 The Chain Rule Chain Rule d dx. [fg(x)] = f′g(x). × g′(x). = Diff Outer × Diff Inner Ex 14.3 The Product Rule Product Rule d dx. [f(x)g(x)] = f(x) g ′ (x) +f ′ (x) g(x) Keep Diff. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. Diff Keep. 30.

(31) Additional Math Notes (20 Oct 2014) Tangent Properties. Ex 14.4 The Quotient Rule  . Quotient Rule Diff Bottom Top d. [. f(x). dx g(x). ]=. g(x). Diff Top Bottom. f′ (x) −f(x) [g(x)]2. Normal Properties. g′ (x). Square Bottom. For the case of. k f(x). d dx. 1. [(2x−3)2 ] =. d dx. [(2x − 3)−2 ]. 2. = − (2x−3)3 =. d dx. (2x − 3). ×2. 4 − (2x−3)3. Consider converting to proper fraction first if it is an improper fraction d dx. (. Normal intersects curve. . mnorm =. , it is preferable to use chain rule instead. = −2(2x − 3)−3 ×. e.g.. . 3x2 +x+3. d. x2 +1. dx. )= = = =. −1 f′ (x1 ). Ex 15.2 Increasing and Decreasing Functions. of quotient rule. e.g.. Tangent intersects curve mtan = f ′ (x1 ). (3 +. (x2 +1)⋅. x. For increasing function,. . For decreasing function,. dy dx dy dx. >0 <0. Applications  Determine whether a function is increasing or decreasing  Find the range of values of x for which a function is increasing or decreasing Ex 15.3 Rates of Change. d d (x) −x⋅ (x2 +1) dx dx (x2 +1)2. (x2 +1)⋅1. Rate of Change. −x⋅2x. dy. (x2 +1)2. dt. ≡ rate of change of y wrt t. −2x2. Consider adding/subtracting between related rates. (x2 +1)2 1−x2. = (x2. . ). x2 +1. (x2 +1). Increasing/Decreasing function. Question. +1)2. Water is entering a container at a constant rate of 5 cm3 /s Water is leaking from the container at a constant rate of 1 cm3 /s. Find the net rate of water flow into the container.. Ex 15.1 Tangents and Normals Find Tangent. Solution Net rate = 5 − 1 = 4cm3 /s. Step 1: Find point Step 2: Find gradient Step 3: Find tangent y − y1 = f ′ (x1 )(x − x1 ). Quantity & Constant Rate Quantity = (constant rate) × time. Find Normal Step 1: Find point Step 2: Find gradient Step 3: Find normal −1 (x − x1 ) y − y1 = ′ f (x1 ). Note: The tangent and normal are perpendicular to each other at the point of contact. © Daniel & Samuel Math Tuition 📞9133 9982. sleightofmath.com. 31.

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