EC-18.1 Functions, and the Language of Change
You can read a function as a rule that turns one quantity into another, state its domain and range, and pick a family of functions that matches the behaviour a measurement actually shows.
Before:nothing requiredUnlocks:EC-3. Differential Equations
Every subject downstream of this one opens by writing a function and assuming you can read it. A circuit course writes a voltage as a function of time, a devices course writes a current as a function of an applied voltage, and a communications course writes a signal as a function of frequency. None of them stops to say what that notation means, so this topic says it once, with the families that keep reappearing and the units that come attached to each.
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The real line, intervals, and what a real number is for here
Everything in this module lives on the real line, and the interval notation for a piece of it is used constantly afterwards without being explained. Getting the open and closed ends right early saves a great deal of confusion about where a result holds.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseA function as a rule, and the domain and range that come with it
A function is a rule that gives one output for each admissible input, and the set of admissible inputs is part of the function rather than an afterthought. Two rules that agree everywhere but disagree about what is admissible are different functions.
NPTEL: Calculus of One Real Variable (IIT Kanpur) · CourseThe families that keep reappearing: linear, polynomial, rational, exponential, logarithmic and trigonometric
Almost every model in this area is built from six families of function, and each has a characteristic shape worth recognising on sight. Knowing which family a measurement resembles is most of the work of choosing a model for it.
NPTEL: Basic Calculus for Engineers, Scientists and Economists (IIT Kanpur) · CourseComposition and the inverse, and when an inverse exists at all
Feeding one function into another is how a chain of physical effects is written down, and undoing it needs the inverse. An inverse exists only when the original never sends two different inputs to the same output, which is a real restriction and not a formality.
Reading a graph: increasing, decreasing, turning, and where the behaviour stops
A graph carries information the formula hides, including where a quantity turns around and where it stops being defined. Reading one fluently is a skill the derivative later makes precise, and it is worth having beforehand.
MIT 18.01SC: Single Variable Calculus · CourseChoosing a model for a measurement, and carrying the units through it
A function that models something physical has units on both sides, and an expression whose units do not balance is wrong before any number is computed. Checking units is the cheapest error detector in engineering mathematics.
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