Lecture Notes: Understanding Functions

Introduction to Functions

• Definition of a Function:
  • A function is a rule or process that assigns a corresponding output to each input.
  • There is a maximum of one output per input.
  • Some inputs may not have an output (e.g., square root of negative one).

Exploring Functions Through an Example

• Example: Wheel with a radius of 1, rotating counterclockwise by angle ( \theta ).
  • Input: Angle ( \theta ).
  • Output: Height of point (( h(\theta) )).
• Graphing the Relationship:
  • Axes: ( \theta ) (input) and ( h(\theta) ) (output).
  • Initial conditions: ( \theta = 0 ) gives height 0.
  • Example calculations:
    • ( \theta = \frac{\pi}{4} ) results in height slightly less than 1.
    • ( \theta = \frac{\pi}{2} ) results in height 1.
  • Graph forms sinusoidal pattern as the wheel rotates.

Trigonometric Representation

• Function Relation:
  • Height is defined by ( \sin(\theta) ).
  • Calculations for angles:
    • ( \sin(0) = 0 ).
    • ( \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} ).
    • ( \sin(\frac{\pi}{2}) = 1 ).

Describing Functions in Various Forms

• Verbal Descriptions: Starting point and problem definition.
• Graphs & Sketches: Visual representation aids understanding.
• Formulas: Often used in mathematics but not the only method.
• Tables of Data: Common in engineering and sensor data.

General Function Model

• Input and Output:
  • Input called independent variable.
  • Output called dependent variable.
• Terminology:
  • While less frequent, dependent and independent terms may appear in experiments.

Familiar Functions and Single Input-Output

• Examples:
  • ( y = x^2 ): Single x input gives single y output.
  • Exponentials ( x = e^t ) and logarithms ( g(x) = \ln(x) ) follow similar patterns.

Expanding the Concept of Functions

• Multi-dimensional Inputs & Outputs:
  • Potential for functions with multiple inputs (e.g., ( x ) & ( y )) and outputs (e.g., ( z )).
  • Functions involving time and multiple outputs will be explored.

Conclusion

• Functions can be expressed and explored in a variety of ways beyond just formulas.
• Understanding the breadth of function descriptions enhances mathematical and engineering problem-solving.
• The next topics will explore more complex function formulations and their calculus implications.