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From Wikipedia

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:

  • a first-degree polynomial function of one variable;
  • a map between two vector spaces that preserves vector addition and scalar multiplication.

Analytic geometry

In analytic geometry, the term linear function is sometimes used to mean a first-degree polynomialfunction of one variable. These functions are known as "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.

Such a function can be written as

f(x) = mx + b
(y-y1) = m(x-x1)
0= Ax + By + C

(called slope-intercept form), where m and b are realconstants and x is a real variable. The constant m is often called the slope or gradient, while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.

Examples of functions whose graph is a line include the following:

  • f_{1}(x) = 2x+1
  • f_{2}(x) = x/2+1
  • f_{3}(x) = x/2-1.

The graphs of these are shown in the image at right.

Vector spaces

In advanced mathematics, a linear function means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

For example, if x and f(x) are represented as coordinate vectors, then the linear functions are those functions f that can be expressed as

f(x) = \mathrm{M}x,

where M is a matrix. A function

f(x) = mx + b

is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.


Factorial

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5 ! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \ 0! is a special case that is explicitly defined to be 1. The factorial operation

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.

Example

In numerical analysis, different decompositions are used to implement efficient matrix algorithms.

For instance, when solving a system of linear equations Ax=b, the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrixL and an upper triangular matrixU. The systems L(Ux)=b and Ux=L^{-1}b require fewer additions and multiplications to solve, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly the QR decomposition expresses A as QR with Q a unitary matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = c, and the system Rx = c is solved by "back substitution". The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.

Decompositions related to solving systems of linear equations

LU Reduction

Block LU decomposition

Rank factorization

  • Applicable to: square, symmetric, positive definite matrix A
  • Decomposition: A=U^TU, where U is upper triangular with positive diagonal entries
  • Comment: the Cholesky decomposition is a special case of the symmetric LU decomposition, with L=U^T.
  • Comment: the Cholesky decomposition is unique
  • Comment: the Cholesky decomposition is also applicable for complex hermitian positive definite matrices
  • Comment: An alternative is the LDL decomposition which can avoid extracting square roots.
  • Applicable to: m-by-n matrix A
  • Decomposition: A=QR where Q is an orthogonal matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n
  • Comment: The QR decomposition provides an alternative way of solving the system of equations Ax=b without inverting the matrix A. The fact that Q is orthogonal means that Q^TQ=I, so that Ax=b is equivalent to Rx=Q^Tb, which is easier to solve since R is triangular.

Singular value decomposition


From Yahoo Answers

Question:I have to do a math reflection due tomorrow and I dont know what to write. It has to be 500 words or I can do a poster. Please help me, its due tomorrow.

Answers:http://en.wikipedia.org/wiki/Exponential_growth#Examples you can start with that , and you need to re-express them in your own words .

Question:1) Suppose that f is an exponential function with f (5) = 4 and f (8) = 15. What is the growth factor for f? 2) A certain quantity has a yearly growth factor of 1.12. What is its monthly growth factor? (Round your answer to two decimal places.) So yeah wtf how do you do this?!?!

Answers:In both cases, these use the basic exponential growth formula (like calculating compounding interest in a bank account): y=K((1+g)^x). Where y is the final amount after growth, K is the initial value, g is the growth rate per period, and x is the number of growth periods. For example, if you deposit $100 in a savings account for 3 years and the interest rate (growth rate) is 1% per year, after three years, you would have: y=100((1+.01)^3) or $103.03 in your account. For #1, you have to do a little algebraic manipulation but it is solvable. 1st case: y=4=K(1+g)^5 -- you don't know the initial value or growth rate 2nd case: y=15=K(1+g)^8 -- again, you don't know values for K or g at this point, you have two equations and two unknowns that can be solved by elimination or substitution. For this work, let's solve both equations for K and eliminate K since we are looking for the value of g. 1st case: K=4/(1+g)^5, and 2nd case: K=15/(1+g)^8; since K=K, you can eliminate K by writing these equations equal to each other: 4/(1+g)^5=15/(1+g)^8, now simplify and solve for g. (1+g)^8/(1+g)^5=15/4 >>> (reduce the left side since these are the same group raised to different exponents) >>> (1+g)3=15/4 now the tricky part; you have to take the cubed root of both side to be able to isolate g >>> 1+g = (15/4)^(1/3) and solve for g >>> g=(15/4)^(1/3) - 1 do the math and you find that g = .5536 and if you want to check your work, solve for K and try recalculating the original answers (btw K=.4419) The second problem is easier. again, it follows y=K(1.12)^x for annual, x=1 for a monthly growth, you will have y=K(g)^12 (12 months in a year) set these equal and solve for g: (1.12)^1=(g)^12 , take the twelfth root of both sides (I hope you have a scientific calculator) >>> g = 1.12^(1/12) or g = 1.0095 or 1.01 I hope this helps. Memorize the generic formula on top and you should always be able to do these types of problems.

Question:its currently 7:15 and I have tons of homework. I just thought Id quickly ask you guys for these definitons because I cant spend to long searching online because I have to go to bed early cuz I wake up at 3:00 am ( cuz I dont have car to get to school,I bike 8 miles every day to school) Im really sick of getting 4 hours of sleep, can sumone just help out a little? I need these definitons (dont have to answer them all, I dont expect you to waste your time on me) a) Hypothesis - b) Dependent variable - c) Independent variable - d) Control - e) Mass - f) Volume - g) Density - h) Miscible - i) Immiscible - j) Density Column - 2) Measure mass and volume of liquids and solids using the correct units. (Write down how you measure the mass and volume.) 3) Calculate density using the correct units. (Write down the equation for density.) 4) Name at least 10 properties of matter. Im trying to get this done (and study it), plus a resume, 3 math pages, + 2 chaps. of a book

Answers:hypothesis-a possible explaniation or sollution to the problem dependent variable-the variable that is measured in an experiment independent variable-the factor that is changed in an experiement

Question:1- Power of a Power Property 2-Power of a Product Property 3-A Nonzero number to the zero power is 1. Give an example:____ 4-a^-n is the reciprocal of a^n : a^-n = 1/a^n "a" cannot = zero 5-Quotient of Powers Property 6-Power of a Quotient Property 7- Scientific Notation 8-Exponential Growth 9-Exponential Growth Thanks! =D

Answers:Dear,I searched the whole net..but couldn't find any as they are not proper mathematical terms. But I know what a Scientific Notation is. Scientific notation is a mathematical format used to write very large and very small numbers; this system avoids using a lot of zeros by using powers (exponents). Hope this helps!!

From Youtube

Math Definitions : What Is a Factor in Math? :In math, all numbers can be decomposed into factors and the product of the factors equals the value of the number. Discover the definition of factors inmath with tips from a mathematics tutor in this free video on math lessons. Expert: Ken Au Bio: Ken Au is a math teacher and tutor for middle school through college levels. Au holds several international patents and has published numerous technical papers. Filmmaker: Mark Bullard

Math Definitions : What Is a Proportion? :In math, one quantity is proportional to another quantity by a scale factor, which is a constant. Discover the definition of a proportion inmath with tips from a mathematics tutor in this free video on math lessons. Expert: Ken Au Bio: Ken Au is a math teacher and tutor for middle school through college levels. Au holds several international patents and has published numerous technical papers. Filmmaker: Mark Bullard