Instantaneous Rate of Change and Derivatives Summary

Introduction to Derivatives

  • Focus: Understanding derivatives as instantaneous rates of change.
  • Importance: Differentiating helps us understand how quantities change, especially in motion.

Average vs. Instantaneous Rate of Change

  • Average Rate of Change:
    • Measures the change over an interval.
    • Calculated as:
      [ \text{Average Rate} = \frac{f(c + h) - f(c)}{h} ]
      where ( h ) is the length of the interval.

Defining Instantaneous Rate of Change

  • Forward Instantaneous Rate of Change:
    • Defined using limits when ( h \to 0^+ ) (approaches zero from the right).
    • Formula:
      [ R{f}(c) = \lim{h \to 0^+} \frac{f(c + h) - f(c)}{h} ]
  • Backward Instantaneous Rate of Change:
    • Defined using limits when ( h \to 0^- ) (approaches zero from the left).
    • Formula:
      [ R{b}(c) = \lim{h \to 0^-} \frac{f(c) - f(c - h)}{h} ]

Key Concept: Derivative

  • The derivative ( f'(c) ) at point ( c ) exists only if both forward and backward limits are equal and finite.
  • Formula for derivative:
    [ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} ]

Example: Finding the Derivative of a Quadratic Function

  • Given function: ( f(t) = 3t^2 - 60t ) on ( t \in [0, 20] ).
  • Find ( f'(5) ):
    1. Start with the limit definition:
      [ f'(5) = \lim_{h \to 0} \frac{f(5 + h) - f(5)}{h} ]
    2. Substitute ( f(5+h) ) and ( f(5) ) into the limit:
    • Compute ( f(5 + h) ) and ( f(5) ).
    1. Conduct algebra to simplify the expression to isolate ( h ).
    2. Find the limit as ( h \to 0 ). Results in ( f'(5) = -30 ).

Definition of Differentiability

  • A function ( f ) is differentiable at ( c ) if ( f'(c) ) exists and is finite.

Example of Non-Differentiability: Absolute Value Function

  • Consider the piecewise function defined at a point where behavior changes (like a corner):
    • We analyze both forward and backward instantaneous rates of change to check for differentiability.
    • If two limits (from either direction) don’t match, the function is not differentiable at that point.

Piecewise Functions and Differentiability

  • Not all piecewise functions are non-differentiable at endpoints; some may be valid if they are continuous without sharp corners or breaks.
  • Identifying:
    • Continuity does not guarantee differentiability.
    • Sharp corners often indicate non-differentiability.

Conclusion

  • Importantly, we use derivatives to find instantaneous rates of change and comprehend motion.
  • Future study will include advanced methods to compute derivatives more efficiently, alongside continuous practice applying limit definitions.