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Straight Line (2) Composite Functions (1) Higher Past Papers by Topic 2015 Onwards Higher Past Papers by Topic 2015 Onwards www.mathsrevision.com Differentiation (2) Recurrence Relations (1) Polynomials (3) Integration (1) Trigonometry (1) The Circle (1) Vectors (2) Logs & Exponential (1) Wave Function (1) Summary of Higher Thursday, 14 May 2015Thursday, 14 May 2015Thursday, 14 May 2015Thursday, 14 May 2015 Created by Mr. Lafferty Maths Dept.

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Straight Line Composite Functions Higher Mindmaps by Topic Higher Mindmaps by Topic www.mathsrevision.com Differentiation Recurrence Relations Polynomials Integration Trigonometry The Circle Vectors Logs & Exponential Wave Function Thursday, 14 May 2015Thursday, 14 May 2015Thursday, 14 May 2015Thursday, 14 May 2015Created by Mr. Lafferty Maths Dept. Main Menu

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Straight Line y = mx + c m = gradient c = y intercept (0,c) m = tan θ θ Possible values for gradient m > 0 m < 0 m = 0 m = undefined Distance between 2 points For Perpendicular lines the following is true. m 1.m 2 = -1 Parallel lines have same gradient Form for finding line equation y – b = m(x - a) (a,b) = point on line Terminology Median – midpoint Bisector – midpoint Perpendicular – Right Angled Altitude – right angled m 1.m 2 = -1 Mindmaps

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flip in x-axis flip in y-axis f(x) - + + - Graphs & Functions y = -f(x) y = f(-x) y = f(x) ± k y = f(kx) Move vertically up or downs depending on k Stretch or compress vertically depending on k y = kf(x) Stretch or compress horizontally depending on k f(x) y = f(x ± k) Move horizontally left or right depending on k Remember we can combine these together !! Mindmaps

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Composite Functions Similar to composite Area A complex function made up of 2 or more simpler functions =+ f(x) = x 2 - 4 g(x) = 1 x x DomainRange y = f(x) 1 y 1 x 2 - 4 Restriction x 2 - 4 ≠ 0 (x – 2)(x + 2) ≠ 0 x ≠ 2x ≠ -2 But y = f(x) is x 2 - 4 g(f(x)) g(f(x)) = f(x) = x 2 - 4 g(x) = 1 x x DomainRange y = g(x) f(g(x)) y 2 - 4 Restriction x 2 ≠ 0 But y = g(x) is f(g(x)) = 1 x 1 x 2 - 4 Rearranging 1 x2x2 Mindmaps

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Adding Basics before Differentiation / Integration Working with fractions Indices Surds Subtracting Multiplication Division Format for Differentiation / Integration Mindmaps

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Differentiation of Polynomials f(x) = ax n then f’x) = anx n-1 Derivative = gradient = rate of change Graphs f’(x)=0 f’(x)=0 Stationary Pts Max. / Mini Pts Inflection Pt Nature Table 25 + 0 - x f’(x) Max Gradient at a point Equation of tangent line Straight Line Theory Leibniz Notation Mindmaps

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Differentiations Polynomials Stationary Pts Mini / Max Pts Inflection Pts Rate of change of a function. Harder functions Use Chain Rule Meaning Graphs Rules of Indices Gradient at a point. Tangent equation Straight line Theory Factorisation Real life Trig Mindmaps

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Polynomials Functions of the type f(x) = 3x 4 + 2x 3 + 2x +x + 5 b 2 -4ac > 0 Real and distinct roots Degree of a polynomial = highest power Discriminant of a quadratic is b 2 -4ac Completing the square f(x) = a(x + b) 2 + c If finding coefficients Sim. Equations b 2 -4ac = 0 Equal roots b 2 -4ac < 0 No real roots Tangency 1 4 5 2-2 -2-4-2 2101 (x+2) is a factor since no remainder Easy to graph functions & graphs f(x) =2x 2 + 4x + 3 f(x) =2(x + 1) 2 - 2 + 3 f(x) =2(x + 1) 2 + 1 Factor Theorem x = a is a factor of f(x) if f(a) = 0 Mindmaps

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Recurrence Relations next number depends on the previous number a > 1 then growth a < 1 then decay Limit exists when | a | < 1 + b = increase - b = decrease Given three value in a sequence e.g. U 10, U 11, U 12 we can work out recurrence relation U n+1 = a U n + b | a | < 1 | a | > 1 a = sets limit b = moves limit U n = no effect on limit Limit L is equal to (1 - a) L = b U 11 = a U 10 + b U 12 = a U 11 + b Use Sim. Equations Mindmaps

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g(x) Integration of Polynomials IF f’(x) = ax n Then I = f(x) = Integration is the process of finding the AREA under a curve and the x-axis Area between 2 curves Finding where curve and line intersect f(x)=g(x) gives the limits a and b Remember to work out separately the area above and below the x-axis. Remember to change sign to + if area is below axis. f(x) A= ∫ f(x) - g(x) dx b a

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Special case Mindmaps

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Trig Formulae and Trig equations Addition Formulae sin(A ± B) = sinAcosB cosAsinB cos(A ± B) = cosAcosB sinAsinB Double Angle Formulae sin2A = 2sinAcosA cos2A = 2cos 2 A - 1 = 1 - 2sin 2 A = cos 2 A – sin 2 A 3cos 2 x – 5cosx – 2 = 0 Let p = cosx 3p 2 – 5p - 2 = 0 (3p + 1)(p -2) = 0 p = cosx = 1/3 cosx = 2 x = no sol n x = cos -1 ( 1/3) x = 109.5 o and 250.5 o C A S T 0o0o 180 o 270 o 90 o xoxo 44 2 The exact value of sinx sinx = 2sin(x/2)cos(x/2) sinx = 2 (¼ + √(4 2 - 1 2 ) ) sinx = ½ + 2√15) Mindmaps

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Trigonometry sin, cos, tan Basic Strategy for Solving Trig Equations Basic Graphs 360 o 1 0 1 0 360 o 1 0 180 o 90 o sin x cos x Exact Value Table 1.Rearrange into sin = 2.Find solution in Basic Quads 3.Remember Multiple solutions Amplitude Period Amplitude Period Complex Graph 90 o 2 1 0 180 o 270 o 360 o 3 y = 2sin(4x + 45 o ) + 1 Max. Value =2+1= 3 Mini. Value = -2+1 -1 Period = 360 ÷4 = 90 o Amplitude = 2 degreesradians ÷180 then X π ÷ π÷ π then x 180 C A S T 0o0o 180 o 270 o 90 o Period tan x Amplitude Period Mindmaps

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P Q B A Vector Theory Magnitude & Direction Vectors are equal if they have the same magnitude & direction a Notation Component form Unit vector form Basic properties same for subtraction Scalar product Magnitude scalar product 2 vectors perpendicular if Component form Addition

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B Vector Theory Magnitude & Direction Points A, B and C are said to be Collinear if B is a point in common. C A Section formula properties n m a c b O A B C Angle between two vectors θ a b Tail to tail Mindmaps

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y x y x (1,0) (a,1) (0,1) Logs & Exponentials Basic log rules y = ab x Can be transformed into a graph of the form y = log a x log A + log B = log AB log A - log B = log B A log a 1 = 0 log a a = 1 log (A) n = n log A Basic log graph Basic exponential graph (1,a) y = a x a 0 = 1 a 1 = a log y = x log b + log a C = log am = log b (0,C) log y x y = ax b Can be transformed into a graph of the form log y = b log x + log a C = log am = b (0,C) log y log x To undo log take exponential To undo exponential take log Y = bX + CY = (log b) X + C Y = mX + CY = mX + C Mindmaps

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Wave Function transforms f(x)= a sinx + b cosx into the form f(x) = a sinx + b cosx compare to required trigonometric identity Process example a and b values decide which quadrant OR f(x) = k sin(x + β) = k sinx cos β + k cosx sin β Compare coefficients a = k cos β b = k sin β Square and add then square root gives Divide and inverse tan gives Related topic Solving trig equations Write out required form Mindmaps

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