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Vector, Ordinary Differential Equation, Matrix, Hypothesis Testing,…
Vector
Properties of Vector
If two vector
v <v1,v2,v3>
and
w <w1,w2,w3>
are
equivalent
, then
v1=w1 , v2=w2 , v3=w3
kv = <kv1 , kv2 ,kv3> #k is any scalar
P1P2→ , P1 = initial point (x1,y1,z1), P2= terminal point (x2,y2,z2)
<v1,v2,v3> = < x1 - x2, y1- y2 , z1 - z2 >
v+w = <v1+w1 , v2+w2 , v3+w3>
DOT Product
If u= <u1, u2, u3>, v= <v1, v2, v3>
u.v = u1v1 + u2v2 + u3v3
u.v =||u|| ||v|| cos θ if u ≠ 0 and v ≠ 0 or = 0 if u = 0 or v = 0
If u.v =0, vector u & v are perpendicular.
u= ( v / ||v|| ) (unit vector)
Equation
Plane Equation
a(X-X₀) + b (Y - Y₀) + c (Z - Z₀) = 0
Parametric Equation
X= (X₀ + at) ; Y= (Y₀+ bt) ; Z = (Z₀+ ct)
Symmetric Equation
t = [ (X-X₀) / a ] = [ (Y- Y₀) / b ] = [(Z- Z₀) / c]
General Solution
Equation need 1 point and 1 vector
If 3 points given
Form 2 vector with the 3 points given
Generate 1 vector with cross product
Use vector and common point to substitute into formula
Cross Product
||u x v || = ||u|| ||v|| sin θ
u x v is perpendicular to both u and v
Norm (length) of a Vector
Formula
||v|| = √ (v1² + v2²) or ||v|| = √ (v1² + v2² + v3²)
Norm =1 (unit vector)
Unit Vector
u = v / ||v||
Standard Unit Vector
i = <1,0,0> , j = <0,1,0> k = <0,0,1>
v = <v1, v2, v3> = v1i +v2j +v3k
Ordinary Differential Equation
Classification
Order
The order of the highest-order derivative in a differential equation is called the order of the equation
forth-order partial differential equation.
second-order ordinary differential equation
Type
Partial Differentiation Equation
Ordinary Differential Equation
Linear or Nonlinear
Characterized by two properties
The dependent variable y and all its derivatives are of the first degree; that is, the power of each term involving y is 1.
Each coefficient depends on only the independent variable x.
Second Order Differential
Non homogenous (r(x) not equal to 0)
Basic rule,
Sum rule
Modification rule
If a term in your choice for yp contains terms that duplicate terms in yh , then that yp must be multiplied by x^n, where n is the smallest positive integer that eliminates that duplication
HOMOGENEOUS EQUATIONS
For second-order homogeneous linear equations, a general solution will be of the form
Homogenous when (r(x) = 0), where c1 not equal to 0, then
Distinct real roots ,
Repeated roots,
Complex Conjugate,
Exact Differential Equations
differential expression:
M (x, y)dx + N(x, y)dy is an exact differential in a region R of the xy-plane if there is a function F(x, y)
Linear Differential Equations
First-Order Differential Equations
Method of solution : Separable Equation
Integrate on both sides with respect to x
Matrix
System of Linear Equation
General Concept
If Equation is
parallel
to each other (no solution)
If Equation is
perpendicular
to each other (1 solution)
If Equation is
in one line
(infinitely many solutions)
Matrix Equation
AX=B
Steps to solve
Step 1: Change equation to matrix equation
Step 2: Find cofactor of A
Step 3: Find det(A)
Find inverse of A
Solve using formula
X= (inverse of A) (B)
Inverse Matrix
2x2 Inverse Matrix
n x n Inverse Matrix
adj is the transpose of cofactor:
Determinant [ det(A) / |A| ]
Determinant of 2x2
Determinant of n x n
Step1 : choose a row
Step 2: Get cofactor of that row
Example of Cofactor:
Step 4: Sum up all the Cofactor
Step 3: det(A) = Cofactor x |A|
Size of matrix [m x n] m=row; n = column
Hypothesis Testing
Concept
A statistical hypothesis is a conjecture or claim concerning one ore more populations
Null hypothesis denoted by Ho is a claim concerning population parameter that is initially said as true
Alternative hypothesis, denoted by H1 is assertion that is contrary to Ho
5 Steps in hypothesis testing
State research hypothesis as null or alternate hypothesis
Collect data to test hypothesis
Execute statistical test
Reject or accept null hypothesis
Present findings
Hypothesis Testing about the Variance
Hypothesis Testing about the Proportion
Hypothesis Testing about the Mean
Example:
Regression
Coefficient of Determination
The sample coefficient of determination, r2, represents the proportion of the total variation of the variable Y that can be explained by a linear relationship with the values of X.
Higher the value of r2, the more successful is the simple linear regression model in explaining y variation.
Coefficient of Correlation
Correlation is used to measure the strength of linear relation between X and Y by means of a single number called a correlation coefficient.
2 Possible conclusions from hypothesis testing
reject Ho or
fail to reject Ho
