Core 3

Core 3

1. Algebraic Fractions

   • Long division
   • The remainder theorem

2. 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
   • 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

3. Exponential and Log Functions

   • lnx is the inverse of eˣ
   • ln(eˣ) = x
   • If y=eˣ then dy/dx = eˣ

4. Numerical Methods

   • There is a change of sign in the interval [a,b], therefore a root lies between a and b

5. 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
   • 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

6. Differentiation

   • The Product Rule
     • 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=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
   • 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

7. 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

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