Calculus

Calculus

Linear algebra
1. Matrices
  1. Special matrices
    1. Square matrices
    2. Triangular matrices
    3. Diagonal matrices
    4. Identity matrices
    5. Null matrices
  2. Transposed matrices
  3. 3 elementary row operations
    1. Interchanging 2 rows
    2. Adding a multiple of 1 row to another
    3. Multiplying a row by a nonzero scalar
  4. Basic operations
    1. Addition, Subtraction
    2. Scalar multiplying
    3. Matrix multiplying
    4. 5 properties
      1. AB = BA
      2. A(B+C) = AB + AC
      3. A(BC) = (AB)C
      4. A + 0 = A
      5. A.I = A
2. Dterminants and the inverse matrix
  1. 7 Properties
    1. |A| = |A |
    2. Interchanging 2 rows or 2 columns => |B| = -|A|
    3. 2 identical rows or columns => D = 0
    4. 1 row/column is a multiple of the other row/column => |A| = 0
    5. x any row/column by a => x D by a
    6. x every elements by a => x D by a times n
    7. |AB| = |A||B|
  2. Inverse matrix
    1. A is an invertible matrix <=> |A| = 0
    2. A = adj(A)/|A|
3. Systems of linear equations
  1. Cramer system
  2. Using inverse matrix
  3. Applications of the system linear equations
    1. Equilibrium in goods market
    2. IS - LM model
    3. Input - output model
Function of 1 variable
1. A function is increasing if : V x1, x2 C D : x1<x2 => f(x1)>f(x2)
2. A function is decreasing if : V x1, x2 C D : x1<x2 => f(x1)<f(x2)
3. Some functions in economic
  1. TC = a + bQ
  2. Q = a + bP
  3. Q = c + dP
  4. y = p(x)/q(x)
  5. TR = pQ
4. Limits and continuity
  1. 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
    1. Lim f(x) = Lim f(x) = L => Lim f(x) = L
  2. y is continuous at x=a if
    1. f(a) exists
    2. Lim f(x) exists
    3. Lim f(x) = f(a)
5. Derivatives
  1. Find min, max values
    1. x is a critical value of f(x)
      1. x0 C D and f'(x0) = 0
      2. x0 C D and f'(x0)
    2. f(c) is a relative min value
      1. f'(c) = 0 and f''(c) > 0
    3. f(c) is a relative max value
      1. f'(c) = 0 and f''(c) <0
  2. Application
    1. Average function
      1. Ay = F(x)/x
    2. Marginal function
      1. My = f'(x)
      2. My(x0) = f'(x0) ; at x = x0 when increases 1 unit then y increases f'(x0) unit
    3. Elasticity
      1. E = D'(p) P/D(p)
      2. At P = a when price increases 1% then demanded quantity increases E%
Integration
1. Improper integral
  1. If the lim exists, then the improper integral converges
  2. If the lim doesn't exists, then the improper integral diverges
Functions of several variables
1. Finding partial derivatives
  1. 1) Consider the function x.y
  2. 2) Suppose that y is fixed
  3. 3) Take the first derivative with respect to x
2. Second-order partial derivatives
  1. Hessian matrix
    1. 1. Negative definite (concave down)
        D1 = a11 > 0 and D2 = |A| < 0
      2. Positive definite (concave up)
        D1 = a11 > 0 and D2 = |A| > 0
3. Derivatives of implicit function
  1. y'(x) = - F'(x)/F'(y)
4. Maximum - minimum problems z = f(x,y)
  1. 1) Find f'x, f'y, f''xx, f''xy, f''yy, f''yx
  2. 2) Solve f'x=0 ; f'y=0. (a,b) represent the solution
  3. 3) D= f''xx.f''yy-f''xy.f''yx
  4. 4) Then
    1. a/ has a max at (a,b) if D>0 ; f''xx<0
    2. b/ has a min at (a,b) if D>0 ; f''xx>0
    3. c/ has neither at (a,b) if D<0
    4. d/ isn't aplication if D=0

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