762862
Description
Mind Map by abbeycropper, updated more than 1 year ago
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Created by abbeycropper
over 10 years ago
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Core 3
- Algebraic Fractions
- Long division
- The remainder theorem
- Long division
- Functions
- The domain is the set on which the rule acts
- The range is the set of results obtained by applying the rule
- A function cannot be one to many
- Composite functions
- g(f(x)) = f first, then g
- g(f(x)) = f first, then g
- Inverse Functions
- A function only as an inverse if it is a one to one
- The domain of the
original equation is the
range of the inverse and
vice versa
- From an equation:
1) swap x's and y's
2) make y the
subject
- From a graph: Reflect it in the line y=x
- A function only as an inverse if it is a one to one
- The domain is the set on which the rule acts
- Exponential and Log Functions
- lnx is the inverse of eˣ
- ln(eˣ) = x
- If y=eˣ then dy/dx = eˣ
- lnx is the inverse of eˣ
- Numerical Methods
- There is a change of sign in the
interval [a,b], therefore a root lies
between a and b
- There is a change of sign in the
interval [a,b], therefore a root lies
between a and b
- Transforming Graphs of Functions
- The Modulus Function |a|
- If y=|f(x)| draw the original graph and
anything below the x axis is reflected in
the x axis
- If y=f|(x)| draw the positive half
of the graph and reflect it in the
y axis
- If y=|f(x)| draw the original graph and
anything below the x axis is reflected in
the x axis
- f(x) +a - vertical up a units
- f(x+a) - horizontal left a units
- af(x) - stretch in y axis/multiply y co-ords by a
- f(ax) - stretch in x axis/multiply x co-ords by 1/a
- -f(x) - reflection in x axis
- f(-x) - reflection in y axis
- The Modulus Function |a|
- Differentiation
- The Product Rule
- If y=UV then dy/dx = Udv/dx + Vdu/dx
- If y=UV then dy/dx = Udv/dx + Vdu/dx
- The Quotient Rule
- If y=u/v then dy/dx = Vdu/dx - Udv/dx / v²
- If y=u/v then dy/dx = Vdu/dx - Udv/dx / v²
- If y=eˣ then dy/dx = eˣ
- If y=ef(x) then dy/dx = f'(x)ef(x)
- If y=lnx then dy/dx = 1/x
- Proof: If
y=lnx x=eʸ
dx/dy = eʸ
dy/dx = 1/eʸ
dy/dx = 1/x
- Proof: If
y=lnx x=eʸ
dx/dy = eʸ
dy/dx = 1/eʸ
dy/dx = 1/x
- If y=sinx then dy/dx = cosx
- If y=cosx then dy/dx = -sinx
- If y=tanx then dy/dx = sec²x
- If y=cosecx dy/dx = -cosecxcotx
- If y=secx dy/dx = tanxsecx
- If y=cotx then dy/dx = -cosec²x
- The Product Rule
- Trigonometry
- secθ = 1/cosθ
- cosecθ = 1/sinθ
- cotθ = 1/tanθ
- cosθ/sinθ = cotθ
- tan²θ + 1 = sec²θ
- sin²θ + cos²θ = 1
- 1 + cot²θ = cosec²θ
- Sin(A + B) = SinACosB + CosASinB
- Sin(A - B) = SinACosB - CosASinB
- Cos(A + B) = CosACosB - SinASinB
- Cos(A - B) = CosACosB + SinASinB
- Tan(A + B) = TanA + TanB/1 - TanATanB
- Tan (A - B) = TanA - TanB/1 + TanATanB
- Sin2A = 2SinACosA
- Cos2A = 1 - 2Sin²A or Cos2A = 2cos²A - 1
- Tan2A = 2TanA/1 - Tan²A
- secθ = 1/cosθ
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