[<< wikibooks] A-level Mathematics/OCR/C1/Appendix A: Formulae
By the end of this module you will be expected to have learned the following formulae:


== The Laws of Indices ==

  
    
      
        
          x
          
            a
          
        
        
          x
          
            b
          
        
        =
        
          x
          
            a
            +
            b
          
        
        
      
    
    {\displaystyle x^{a}x^{b}=x^{a+b}\,}
  

  
    
      
        
          
            
              x
              
                a
              
            
            
              x
              
                b
              
            
          
        
        =
        
          x
          
            a
            −
            b
          
        
      
    
    {\displaystyle {\frac {x^{a}}{x^{b}}}=x^{a-b}}
  

  
    
      
        
          x
          
            −
            n
          
        
        =
        
          
            1
            
              x
              
                n
              
            
          
        
      
    
    {\displaystyle x^{-n}={\frac {1}{x^{n}}}}
  

  
    
      
        
          
            (
            
              x
              
                a
              
            
            )
          
          
            b
          
        
        =
        
          x
          
            a
            b
          
        
      
    
    {\displaystyle \left(x^{a}\right)^{b}=x^{ab}}
  

  
    
      
        
          
            (
            
              x
              y
            
            )
          
          
            n
          
        
        =
        
          x
          
            n
          
        
        
          y
          
            n
          
        
      
    
    {\displaystyle \left(xy\right)^{n}=x^{n}y^{n}}
  

  
    
      
        
          
            (
            
              
                x
                y
              
            
            )
          
          
            n
          
        
        =
        
          
            
              x
              
                n
              
            
            
              y
              
                n
              
            
          
        
      
    
    {\displaystyle \left({\frac {x}{y}}\right)^{n}={\frac {x^{n}}{y^{n}}}}
  

  
    
      
        
          x
          
            
              a
              b
            
          
        
        =
        
          
            
              x
              
                a
              
            
            
              b
            
          
        
      
    
    {\displaystyle x^{\frac {a}{b}}={\sqrt[{b}]{x^{a}}}}
  

  
    
      
        
          x
          
            0
          
        
        =
        1
        
      
    
    {\displaystyle x^{0}=1\,}
  

  
    
      
        
          x
          
            1
          
        
        =
        x
        
      
    
    {\displaystyle x^{1}=x\,}
  


== The Laws of Surds ==

  
    
      
        
          
            x
            y
          
        
        =
        
          
            x
          
        
        ×
        
          
            y
          
        
      
    
    {\displaystyle {\sqrt {xy}}={\sqrt {x}}\times {\sqrt {y}}}
  

  
    
      
        
          
            
              x
              y
            
          
        
        =
        
          
            
              x
            
            
              y
            
          
        
      
    
    {\displaystyle {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}}
  

  
    
      
        
          
            a
            
              b
              +
              
                
                  c
                
              
            
          
        
        =
        
          
            a
            
              b
              +
              
                
                  c
                
              
            
          
        
        ×
        
          
            
              b
              −
              
                
                  c
                
              
            
            
              b
              −
              
                
                  c
                
              
            
          
        
        =
        
          
            
              a
              (
              b
              −
              
                
                  c
                
              
              )
            
            
              
                b
                
                  2
                
              
              −
              c
            
          
        
      
    
    {\displaystyle {\frac {a}{b+{\sqrt {c}}}}={\frac {a}{b+{\sqrt {c}}}}\times {\frac {b-{\sqrt {c}}}{b-{\sqrt {c}}}}={\frac {a(b-{\sqrt {c}})}{b^{2}-c}}}
  


== Polynomials ==


=== Parabolas ===
If f(x) is in the form 
  
    
      
        a
        (
        x
        +
        b
        
          )
          
            2
          
        
        +
        c
      
    
    {\displaystyle a(x+b)^{2}+c}
  

-b is the axis of symmetry
c is the maximum or minimum y valueAxis of Symmetry = 
  
    
      
        
          
            
              −
              b
            
            
              2
              a
            
          
        
      
    
    {\displaystyle {\frac {-b}{2a}}}
  


=== Completing the Square ===

  
    
      
        a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
        =
        0
        
      
    
    {\displaystyle ax^{2}+bx+c=0\,}
   becomes 
  
    
      
        a
        
          
            (
            
              x
              +
              
                
                  b
                  
                    2
                    a
                  
                
              
            
            )
          
          
            2
          
        
        −
        
          
            
              b
              
                2
              
            
            
              4
              a
            
          
        
        +
        c
      
    
    {\displaystyle a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c}
  


=== The Quadratic Formula ===
The solutions of the quadratic 
  
    
      
        a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
        =
        0
      
    
    {\displaystyle ax^{2}+bx+c=0}
   are: 
  
    
      
        x
        =
        
          
            
              −
              b
              ±
              
                
                  
                    b
                    
                      2
                    
                  
                  −
                  4
                  a
                  c
                
              
            
            
              2
              a
            
          
        
      
    
    {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
  
The discriminant of the quadratic 
  
    
      
        a
        
          x
          
            2
          
        
        +
        b
        x
        +
        c
        =
        0
      
    
    {\displaystyle ax^{2}+bx+c=0}
   is 
  
    
      
        
          b
          
            2
          
        
        −
        4
        a
        c
      
    
    {\displaystyle b^{2}-4ac}
  


== Errors ==

  
    
      
        A
        b
        s
        o
        l
        u
        t
        e
         
        e
        r
        r
        o
        r
        =
        v
        a
        l
        u
        e
         
        o
        b
        t
        a
        i
        n
        e
        d
        −
        t
        r
        u
        e
         
        v
        a
        l
        u
        e
      
    
    {\displaystyle Absolute\ error=value\ obtained-true\ value}
  

  
    
      
        R
        e
        l
        a
        t
        i
        v
        e
         
        e
        r
        r
        o
        r
        =
        
          
            
              a
              b
              s
              o
              l
              u
              t
              e
               
              e
              r
              r
              o
              r
            
            
              t
              r
              u
              e
               
              v
              a
              l
              u
              e
            
          
        
      
    
    {\displaystyle Relative\ error={\frac {absolute\ error}{true\ value}}}
  

  
    
      
        P
        e
        r
        c
        e
        n
        t
        a
        g
        e
         
        e
        r
        r
        o
        r
        =
        r
        e
        l
        a
        t
        i
        v
        e
         
        e
        r
        r
        o
        r
        ×
        100
      
    
    {\displaystyle Percentage\ error=relative\ error\times 100}
  


== Coordinate Geometry ==


=== Gradient of a line ===

  
    
      
        m
        =
        
          
            
              
                y
                
                  2
                
              
              −
              
                y
                
                  1
                
              
            
            
              
                x
                
                  2
                
              
              −
              
                x
                
                  1
                
              
            
          
        
      
    
    {\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}
  


=== Point-Gradient Form ===
The equation of a line passing through the point 
  
    
      
        
          (
          
            
              x
              
                1
              
            
            ,
            
              y
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \left(x_{1},y_{1}\right)}
   and having a slope m is 
  
    
      
        y
        −
        
          y
          
            1
          
        
        =
        m
        
          (
          
            x
            −
            
              x
              
                1
              
            
          
          )
        
      
    
    {\displaystyle y-y_{1}=m\left(x-x_{1}\right)}
  .


=== Perpendicular lines ===
Lines are perpendicular if 
  
    
      
        
          m
          
            1
          
        
        ×
        
          m
          
            2
          
        
        =
        −
        1
      
    
    {\displaystyle m_{1}\times m_{2}=-1}
  


=== Distance between two points ===

  
    
      
        d
        =
        
          
            (
            
              x
              
                2
              
            
            −
            
              x
              
                1
              
            
            
              )
              
                2
              
            
            +
            (
            
              y
              
                2
              
            
            −
            
              y
              
                1
              
            
            
              )
              
                2
              
            
          
        
      
    
    {\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}
  


=== Mid-point of a line ===

  
    
      
        
          (
          
            
              
                
                  
                    
                      x
                      
                        1
                      
                    
                  
                  +
                  
                    
                      x
                      
                        2
                      
                    
                  
                
                2
              
            
            ;
            
              
                
                  
                    
                      y
                      
                        1
                      
                    
                  
                  +
                  
                    
                      y
                      
                        2
                      
                    
                  
                
                2
              
            
          
          )
        
      
    
    {\displaystyle \left({\frac {{x_{1}}+{x_{2}}}{2}};{\frac {{y_{1}}+{y_{2}}}{2}}\right)}
  


=== General Circle Formulae ===

  
    
      
        A
        r
        e
        a
        =
        π
        
          r
          
            2
          
        
        
      
    
    {\displaystyle Area=\pi r^{2}\,}
  

  
    
      
        C
        i
        r
        c
        u
        m
        f
        e
        r
        e
        n
        c
        e
        =
        2
        π
        r
        
      
    
    {\displaystyle Circumference=2\pi r\,}
  


=== Equation of a Circle ===

  
    
      
        
          
            (
            
              x
              −
              h
            
            )
          
          
            2
          
        
        +
        
          
            (
            
              y
              −
              k
            
            )
          
          
            2
          
        
        =
        
          r
          
            2
          
        
      
    
    {\displaystyle \left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}}
  , where (h,k) is the center and r is the radius.


== Differentiation ==


=== Differentiation Rules ===
Derivative of a constant function:
  
    
      
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        
          (
          c
          )
        
        =
        0
      
    
    {\displaystyle {\frac {dy}{dx}}\left(c\right)=0}
  

The Power Rule:
  
    
      
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        
          (
          
            x
            
              n
            
          
          )
        
        =
        n
        
          x
          
            n
            −
            1
          
        
      
    
    {\displaystyle {\frac {dy}{dx}}\left(x^{n}\right)=nx^{n-1}}
  

The Constant Multiple Rule:
  
    
      
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        c
        f
        
          (
          x
          )
        
        =
        c
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        f
        
          (
          x
          )
        
      
    
    {\displaystyle {\frac {dy}{dx}}cf\left(x\right)=c{\frac {dy}{dx}}f\left(x\right)}
  

The Sum Rule:
  
    
      
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        
          
            [
            
              
                
                  f
                  
                    (
                    x
                    )
                  
                  +
                  g
                  
                    (
                    x
                    )
                  
                
              
            
            ]
          
        
        =
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        f
        
          (
          x
          )
        
        +
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        g
        
          (
          x
          )
        
      
    
    {\displaystyle {\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)+g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)+{\frac {dy}{dx}}g\left(x\right)}
  

The Difference Rule:
  
    
      
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        
          
            [
            
              
                
                  f
                  
                    (
                    x
                    )
                  
                  −
                  g
                  
                    (
                    x
                    )
                  
                
              
            
            ]
          
        
        =
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        f
        
          (
          x
          )
        
        −
        
          
            
              d
              y
            
            
              d
              x
            
          
        
        g
        
          (
          x
          )
        
      
    
    {\displaystyle {\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)-g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)-{\frac {dy}{dx}}g\left(x\right)}
  


=== Rules of Stationary Points ===
If 
  
    
      
        
          f
          ′
        
        
          (
          c
          )
        
        =
        0
      
    
    {\displaystyle f'\left(c\right)=0}
   and 
  
    
      
        
          f
          ″
        
        
          (
          c
          )
        
        <
        0
      
    
    {\displaystyle f''\left(c\right)<0}
  , then c is a local maximum point of f(x). The graph of f(x) will be concave down on the interval.
If 
  
    
      
        
          f
          ′
        
        
          (
          c
          )
        
        =
        0
      
    
    {\displaystyle f'\left(c\right)=0}
   and 
  
    
      
        
          f
          ″
        
        
          (
          c
          )
        
        >
        0
      
    
    {\displaystyle f''\left(c\right)>0}
  , then c is a local minimum point of f(x). The graph of f(x)  will be  concave up on the interval.
If 
  
    
      
        
          f
          ′
        
        
          (
          c
          )
        
        =
        0
      
    
    {\displaystyle f'\left(c\right)=0}
   and 
  
    
      
        
          f
          ″
        
        
          (
          c
          )
        
        =
        0
      
    
    {\displaystyle f''\left(c\right)=0}
   and 
  
    
      
        
          f
          ‴
        
        
          (
          c
          )
        
        ≠
        0
      
    
    {\displaystyle f'''\left(c\right)\neq 0}
  , then c is a local inflection point of f(x).