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5761237
Calculus
Description
Mind Map on Calculus, created by Truong Anh Nguye on 17/06/2016.
Mind Map by
Truong Anh Nguye
, updated more than 1 year ago
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Created by
Truong Anh Nguye
about 8 years ago
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Resource summary
Calculus
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
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%
Integration
Improper integral
If the lim exists, then the improper integral converges
If the lim doesn't exists, then the improper integral diverges
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
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