FORMULA SHEET

FORMULA SHEET

  1. The Rules for Indices
  2. and
  3. =  or

 

  1. The Log Rules

 
 
 

  1. Rules for finding Derivatives

 

  • Power rule:

If then               =
 

  • Exponential rule

If            then                =
 

  • Log rule

If       then                =
 

  • Product or Multiplication rule

If  (or )   then        =
 
 

  • Division or Quotient Rule

if   (or )                    then
 

  • Chain Rule or Function of Function Rule

         If                                     then          =  ×

  1. Rules for integration

(a) Power Rule
 
 
(b) Constant rule:
,
where  is a constant
 
(c) Log rule
 
 

  1. d) Exponential Rule

 
 
 
Evaluating definite integral:
 
Consumer Surplus (CS) =
Producer Surplus (PS) =
 
 
 
 

  1. General form of linear and non-linear (quadratic) equations

 
Linear/straight line:
: slope : intercept
Quadratic:
 

  1. Quadratic formula

To find the roots of any quadratic equation, ,
 

  1. Arithmetic series/progression

The nth term of the arithmetic sequence is:
: the first term of the sequence
: common difference between numbers
 
The sum of the first n terms,, of an arithmetic series is given by the formula:
=
 
 
 

  1. Geometric series/progression

The nth term of a geometric series is
 
: the first term of the sequence
: common multiple/ratio between numbers
 
The sum of the first  terms of a geometric series is given by
=
 

  1. Annuity
  2. The future value of an ordinary annuity:

=
      A0:  payment; :   interest rate;  :   time period
 

  1. The present value of an annuity:

=
 

  1. The value of a sinking fund, payments A0, made at the start of each year:

=
 
 

  1. Expanding Squares

1)
2)
 

  1. Difference of two squares

 

  1. Dividing two fractions

  =  ×
 

  1. Important economic definitions
  • Total Revenue (TR) = P×Q
  • Total Cost (TC) = Fixed cost (FC) + Variable cost (VC)
  • Average Cost (AC) =
  • Average Revenue (AR) =
  • Marginal Cost (MC) =
  • Marginal Revenue (MR) =
  • Profit = TR – TC
  • At the break-even, TR = TC or Profit = 0
  • At the equilibrium, Pd = Ps = P and Qd = Qs = Q
  1. Important economic definitions (cont.)

If price discrimination is not permitted, then P1 = P2 = P. The overall demand is the sum of the two separate demands: Q = Q1 + Q2
 

  1. The method for finding optimum points of a function,
  2. Solve the equation to find the turning point(s),
  3. If then the function has a minimumat

If then the function has a maximumat
 

  1. The method for finding optimum points of a function
  2. Solve the simultaneous equations

 
and
to find the turning points,

  1. Let Δ =
  • if > 0 and > 0 and Δ > 0 at , then the function has a minimum at
  • if < 0 and < 0 and Δ > 0 at , then the function has a maximum at
  • The point is a point of inflection if both second derivatives have the same sign but Δ < 0
  • The point is a saddle point if the second derivatives have different signs and Δ < 0
  • If Δ = 0 then there is no conclusion
  1. Profit maximization via MR and MC
    • Profit is maximized when MR = MC and
    • Profit is minimized when MR = MC and

 

  1. Constrained Optimization and Lagrange Multipliers

To find the optimum values of a function, , subject to a constraint,  , define the Lagrangian function, , where
where λ is called a Lagrange multiplier.
 

  1. Solution of differential equations of the form
  2. integrate both sides of the differential equation with respect to This gives the general solution.
  3. If conditions are given for and , substitute these values into the general solution and solve for the arbitrary constant, .
  4. Substitute this value of into the general solution to find the particular solution.

 

  1. Matrices
  • Evaluating a 2×2 determinant:

=
 

  • Evaluating a 3×3 determinant:

=(×   – × + ×
 
 

  • To write a system of equations in matrix form:

 
 
 
× =
 
This is known as  format,
 
where  = ,  =  and  =
 

  • Inverse matrix method involves solving using =

 

  • Gauss – Jordan Elimination

Write down the augmented matrix as
 
Transform the above augmented matrix as
(This is known as row echelon form)
 
In this form the solutions can be read off immediately.
 

  • There are three elementary row operations used to achieve the row echelon form:
  1. Swap the positions of two rows
  2. Multiply (or divide) each element of a row by a nonzero constant
  3. Replace a row by the sum of itself and a constant multiple of another row of the matrix.

 
 

  • To find the inverse of a matrix: Using Gauss-Jordan method

 
Write down the augmented matrix as
 
Transform the above augmented matrix as
 
 
The original matrix , is now reduced to the identity/unit matrix.
The inverse of  is given by the transformed unit matrix.  That is,
=
 
 
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