Mathematical Models Flashcards
Axioms
Mathematical statements that serve as starting points for other statements and are logically derived.
Tou can’t prove something using nothing
Axioms are unproved truths 1+1=2
Continuous Quantitative Variables
infinite number of values between 2 values (i.e. decimal points)
Discrete Quantitative Variables
a finite number of values between 2 values (i.e. no decimal points)
Functions
You put X into a given function and get Y out of it.
A weekly salary (X) is a function of the Hourly Pay and number of Hours Worked.
HP x HW = X(the function)
X and Y are the axis of th graph and the function is the plotted point
ex.
f(x) = a². also means y = a²
Theorem
Statements logically derived from or proven by one or more axioms or previous statements.
Linear Equations
f(x) = ax + b
Quadratic Equation
f(x) = ax² + bx + c
Exponential Equation
f(x) = a^{x}
Logistic Equation
f(x) = a ÷ (c + b^{x})
Logarithmic Equation
f(x) = log,{a}(x)
Trigonomic Equation
f(x) = cos(x) f(x) = sin(x)
Polynomial Equation
ax + bx² + cx³
Vertical Shift
f(x)±b
If b is positive+ it moves up
If b is negarive- it moves down
Horizontal Shift
f(x±c)
If C is positive+ it moves left
If C is negative- it moves right
Vertical Stretch/Squeeze
d*f(x)
If d is more than 1 or less than -1 it STRETCHES
If d is a decimal it SQUEEZES
Horizontal Stretch/Squeeze
f(e*x)
If e is a decimal it STRETCHES
If e is more than 1 or less then -1 it SQUEEZES
Linear Model
Population increases by a fixed amount each generation.
Most commonly used to describe population of inanimate objects like garbage.
Common Difference
In each transition period the population grows by a fixed amount (d)
Linear Models
Common Ratio
Fixed factor (r) the population is multiplied by
In each generation the population is equal to population in previous generation, multiplied by common Ratio
Exponential Model
Exponential Model
Constant rate of growth.
Logistic Model
Growth linked to amount of space available.
Once hitting the carrying capacity, there can be no more growth possible to be sustained.
Elbow Room
The difference between carrying capacity and the current population size.
Rate of growth is directly linked to the amount of elbow room available to be inhabited
