Mathematical Models Flashcards
What is a mathematical model?
A mathematical description - often by means of a function of an equation - of a real world phenomenon.
What is the purpose of a mathematical model?
To understand the phenomenon and, perhaps, to make predictions about future behaviour.
Explain the process of mathematical modelling.
Given a real-world problem, the first stage is to formulate a mathematical model.
The second stage is to apply the mathematics that we know in order to derive mathematical conclusions.
The third stage is to take those conclusions and interpret them as information about the original real-world phenomenon - by way of offering explanations or making predictions.
The fourth and final step is to test our predictions by checking against raw real data.
What is the slope-intercept form of an equation?
y OR f(x) = mx + c where m is the slope of the line and c is the y-intercept.
What is a power function?
A function of the form f(x) = x^a, where a is a constant.
What is the shape of the graph f(x) = x^n where n is a positive integer?
If n is even, then f(x) = x^n is an even function and its graph is similar to the parabola y = x^2 (as n increases, the graph gets flatter and steeper).
If n is odd, then f(x) = x^n is an odd function and its graph is similar to y = x^3 (as n increases, the graph gets flatter at 0 and steeper).
What is the shape of the graph x^(1/n)?
If n is even, it is similar to the upper half of the parabola x = y^2 (passes through the point (1, 1)).
If n is odd, it is similar to the graph of y = cube root(x) (passes through (1, 1) and (-1,-1)).
What is an algebraic functions?
A function f is called an algebraic function if it can be constructed using algebraic operations - addition, subtraction, multiplication, division, taking roots - starting with polynomials.
What type of function is automatically an algebraic function?
A rational function.
What is an important property of the sine and cosine functions? (period)
They are periodic functions and have a period of 2pi.
Therefore, sin(x + 2pi) = sinx and cos(x + 2pi) = cosx
When is the tangent function undefined?
When cosx = 0, because tanx = sinx/cosx.
Therefore, when x = +-pi/2, +-3pi/2, …
What is an important property of the tangent function? (period)
It has a period of pi.
Therefore, tan(x + pi) = tanx
What are the reciprocals of the sine, cosine and tangent functions, respectfully?
Cosecant, secant and cotangent.
What is an exponential function?
The functions of the form f(x) = a^x where the base a is a positive constant.
