Calculus

Calculus

1. Linear algebra

   • Matrices
     • Special matrices
       • Square matrices
       • Triangular matrices
       • Diagonal matrices
       • Identity matrices
       • Null matrices
     • Transposed matrices
     • 3 elementary row operations
       • Interchanging 2 rows
       • Adding a multiple of 1 row to another
       • Multiplying a row by a nonzero scalar
     • Basic operations
       • Addition, Subtraction
       • Scalar multiplying
       • Matrix multiplying
       • 5 properties
         • AB = BA
         • A(B+C) = AB + AC
         • A(BC) = (AB)C
         • A + 0 = A
         • A.I = A
   • Dterminants and the inverse matrix
     • 7 Properties
       • |A| = |A |
       • Interchanging 2 rows or 2 columns => |B| = -|A|
       • 2 identical rows or columns => D = 0
       • 1 row/column is a multiple of the other row/column => |A| = 0
       • x any row/column by a => x D by a
       • x every elements by a => x D by a times n
       • |AB| = |A||B|
     • Inverse matrix
       • A is an invertible matrix <=> |A| = 0
       • A = adj(A)/|A|
   • Systems of linear equations
     • Cramer system
     • Using inverse matrix
     • Applications of the system linear equations
       • Equilibrium in goods market
       • IS - LM model
       • Input - output model

2. Function of 1 variable

   • A function is increasing if : V x1, x2 C D : x1<x2 => f(x1)>f(x2)
   • A function is decreasing if : V x1, x2 C D : x1<x2 => f(x1)<f(x2)
   • Some functions in economic
     • TC = a + bQ
     • Q = a + bP
     • Q = c + dP
     • y = p(x)/q(x)
     • TR = pQ
   • Limits and continuity
     • As x approaches a, the limit of f(x) is the L if the limit from the left and right exist and both lim are L
       • Lim f(x) = Lim f(x) = L => Lim f(x) = L
     • y is continuous at x=a if
       • f(a) exists
       • Lim f(x) exists
       • Lim f(x) = f(a)
   • Derivatives
     • Find min, max values
       • x is a critical value of f(x)
         • x0 C D and f'(x0) = 0
         • x0 C D and f'(x0)
       • f(c) is a relative min value
         • f'(c) = 0 and f''(c) > 0
       • f(c) is a relative max value
         • f'(c) = 0 and f''(c) <0
     • Application
       • Average function
         • Ay = F(x)/x
       • Marginal function
         • My = f'(x)
         • My(x0) = f'(x0) ; at x = x0 when increases 1 unit then y increases f'(x0) unit
       • Elasticity
         • E = D'(p) P/D(p)
         • At P = a when price increases 1% then demanded quantity increases E%

3. Integration

   • Improper integral
     • If the lim exists, then the improper integral converges
     • If the lim doesn't exists, then the improper integral diverges

4. Functions of several variables

   • Finding partial derivatives
     • 1) Consider the function x.y
     • 2) Suppose that y is fixed
     • 3) Take the first derivative with respect to x
   • Second-order partial derivatives
     • Hessian matrix
       • • Negative definite (concave down)
           • D1 = a11 > 0 and D2 = |A| < 0
         • Positive definite (concave up)
           • D1 = a11 > 0 and D2 = |A| > 0
   • Derivatives of implicit function
     • y'(x) = - F'(x)/F'(y)
   • Maximum - minimum problems z = f(x,y)
     • 1) Find f'x, f'y, f''xx, f''xy, f''yy, f''yx
     • 2) Solve f'x=0 ; f'y=0. (a,b) represent the solution
     • 3) D= f''xx.f''yy-f''xy.f''yx
     • 4) Then
       • a/ has a max at (a,b) if D>0 ; f''xx<0
       • b/ has a min at (a,b) if D>0 ; f''xx>0
       • c/ has neither at (a,b) if D<0
       • d/ isn't aplication if D=0