Core 3

abbeycropper
Mind Map by abbeycropper, updated more than 1 year ago
abbeycropper
Created by abbeycropper almost 6 years ago
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Mind Map on Core 3, created by abbeycropper on 04/16/2014.

Resource summary

Core 3
1 Algebraic Fractions
1.1 Long division
1.2 The remainder theorem
2 Functions
2.1 The domain is the set on which the rule acts
2.2 The range is the set of results obtained by applying the rule
2.3 A function cannot be one to many
2.4 Composite functions
2.4.1 g(f(x)) = f first, then g
2.5 Inverse Functions
2.5.1 A function only as an inverse if it is a one to one
2.5.2 The domain of the original equation is the range of the inverse and vice versa
2.5.3 From an equation: 1) swap x's and y's 2) make y the subject
2.5.4 From a graph: Reflect it in the line y=x
3 Exponential and Log Functions
3.1 lnx is the inverse of eˣ
3.2 ln(eˣ) = x
3.3 If y=eˣ then dy/dx = eˣ
4 Numerical Methods
4.1 There is a change of sign in the interval [a,b], therefore a root lies between a and b
5 Transforming Graphs of Functions
5.1 The Modulus Function |a|
5.1.1 If y=|f(x)| draw the original graph and anything below the x axis is reflected in the x axis
5.1.2 If y=f|(x)| draw the positive half of the graph and reflect it in the y axis
5.2 f(x) +a - vertical up a units
5.3 f(x+a) - horizontal left a units
5.4 af(x) - stretch in y axis/multiply y co-ords by a
5.5 f(ax) - stretch in x axis/multiply x co-ords by 1/a
5.6 -f(x) - reflection in x axis
5.7 f(-x) - reflection in y axis
6 Differentiation
6.1 The Product Rule
6.1.1 If y=UV then dy/dx = Udv/dx + Vdu/dx
6.2 The Quotient Rule
6.2.1 If y=u/v then dy/dx = Vdu/dx - Udv/dx / v²
6.3 If y=eˣ then dy/dx = eˣ
6.4 If y=ef(x) then dy/dx = f'(x)ef(x)
6.5 If y=lnx then dy/dx = 1/x
6.5.1 Proof: If y=lnx x=eʸ dx/dy = eʸ dy/dx = 1/eʸ dy/dx = 1/x
6.6 If y=sinx then dy/dx = cosx
6.7 If y=cosx then dy/dx = -sinx
6.8 If y=tanx then dy/dx = sec²x
6.9 If y=cosecx dy/dx = -cosecxcotx
6.10 If y=secx dy/dx = tanxsecx
6.11 If y=cotx then dy/dx = -cosec²x
7 Trigonometry
7.1 secθ = 1/cosθ
7.2 cosecθ = 1/sinθ
7.3 cotθ = 1/tanθ
7.4 cosθ/sinθ = cotθ
7.5 tan²θ + 1 = sec²θ
7.6 sin²θ + cos²θ = 1
7.7 1 + cot²θ = cosec²θ
7.8 Sin(A + B) = SinACosB + CosASinB
7.9 Sin(A - B) = SinACosB - CosASinB
7.10 Cos(A + B) = CosACosB - SinASinB
7.11 Cos(A - B) = CosACosB + SinASinB
7.12 Tan(A + B) = TanA + TanB/1 - TanATanB
7.13 Tan (A - B) = TanA - TanB/1 + TanATanB
7.14 Sin2A = 2SinACosA
7.15 Cos2A = 1 - 2Sin²A or Cos2A = 2cos²A - 1
7.16 Tan2A = 2TanA/1 - Tan²A
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