Functions

• Function


  A machine or operator that takes an input (denoted x), does something to that input, and produces an output (denoted f(x))
• Variable x


  The most commonly used input into a function
• Notation f


  The function's name
• Functions are quite useful in describing a quantity in terms of other quantities
• The distance covered by a moving car depends on the speed of the car and the time it has traveled
• Relation


  A non-empty subset of the Cartesian product X x Y
• Function f from X to Y

  A relation from X to Y such that every x in X is associated with exactly one element y in Y
• The domain of f is X and the co-domain is Y
• Linear functions


  Have the form y = ax + b
• Exponential functions


  Have the form y = b^x, where b > 0 and b ≠ 1
• Exponential functions grow much faster than linear functions as x becomes large
• Many quantities grow exponentially, such as populations
• Deterministic model


  Behavior is an inevitable consequence of antecedent sufficient causes, so the output is fully determined by the parameter values and initial conditions
• Probabilistic/Stochastic model


  Behavior has different chances of occurrence, possessing some inherent randomness, so the same parameter values and initial conditions will lead to an ensemble of different outputs
• Deterministic or Stochastic?


  • Stock market
  • Patient's EKG, EEG, blood pressure or temperature
  • Rolling a pair of dice
  • Calculating your savings account balance
  • Typhoons
• Calculus is the branch of mathematics that studies continuous change, which can be observed in the behaviour of functions
• Deterministic model


  Behavior is an inevitable consequence of antecedent sufficient causes, output is fully determined by the parameter values and the initial conditions
• Probabilistic or stochastic model

  Behavior has the property of different chances of occurrence, possesses some inherent randomness, the same set of parameter values and initial conditions will lead to an ensemble of different outputs
• Physical phenomena can often be modelled by functions or a set of function that describe various states of the function at a given time
• The main ideas of the calculus took a long time to be developed, the ancient Greeks studied problems that today are routinely solved using calculus
• Method of Exhaustion


  Inscribe polygons of increasing number of sides, starting with a square, then a pentagon, then a hexagon, and so on, each succeeding figure would have an area that is closer to the actual area of the circle, taking the limit as n approaches infinity
• Differential calculus


  Concerned with the study of the rates at which quantities change
• Derivative of a function


  Describes the rate of change of the function near a chosen input value, the slope of the tangent line to the graph of the function at that point

  
• Differentiation is the reverse process to integration, as stated by the Fundamental Theorem of Calculus
• Differentiation has applications to nearly all quantitative disciplines, such as physics, chemistry, and more
• Derivatives are frequently used to find the maxima and minima of a function, and equations involving derivatives are called differential equations and are fundamental in describing natural phenomena
• Speed Camera System


  Cameras installed on bridges above traffic lanes, in two different locations along a roadway, each camera automatically captures an image of the vehicle and a computer identifies and recognizes the license plate of the passing vehicle, the pair of images taken of a single vehicle are matched by the license plate and speed is calculated