Lecture Notes on Derivatives and Discontinuities
Summary of the Lecture
This lecture focused on exploring different perspectives on what a derivative represents, particularly focusing on its interpretation as a rate of change. Examples from physics were used, such as the current being the derivative of charge with respect to time, and speed as the derivative of distance. The lecture also encompassed a broad discussion on continuity, limits, and different types of discontinuities in functions.
Key Points Discussed in the Lecture
Derivatives
- Definition Revisited: Derivative as the slope of a tangent line to a function at a point.
- Computational Examples:
- Derivative of ( \frac{1}{x} ) is ( -\frac{1}{x^2} ).
- Derivative of ( x^n ) is ( nx^{n-1} ).
Interpretation as Rate of Change
- General Formula: Rate of change is represented as ( \frac{\Delta y}{\Delta x} ) and in the limit, as the derivative ( \frac{dy}{dx} ).
- Physical Examples:
- Electric current represented as ( \frac{dQ}{dt} ).
- Speed as the rate of change of distance ( \frac{ds}{dt} ).
Example: The Pumpkin Drop
- Event Description: Pumpkins are dropped from a building, involving calculations of speed and impact using derivatives.
- Formulas Used:
- ( H(t) = 80 - 5t^2 ) to model the pumpkin's height over time.
- Average speed ( \frac{\Delta H}{\Delta t} ).
- Instantaneous speed at impact ( H'(t) ) derived from ( H(t) ).
Discontinuities and Limits
- Easy Limits: Direct substitution if function is continuous at the point.
- Continuity:
- Defined as the left and right-hand limits at a point being equal to the function’s value at that point.
- Different types of discontinuities discussed:
- Jump Discontinuity: Left and right limits exist but are not equal.
- Removable Discontinuity: Limits from left and right are equal but not to the function’s value; a "hole" in the graph.
- Infinite Discontinuity: Function approaches infinity, e.g., ( \frac{1}{x} ) as ( x ) approaches zero.
- Other Types: Oscillating functions like ( \sin(\frac{1}{x}) ) as ( x ) approaches zero.
Differentiability Implies Continuity
- Theorem and Proof:
- If a function is differentiable at a point, it is also continuous at that point.
- Proof involves showing that the limit of the difference quotient equals zero.
Conclusion
The lecture closed by illustrating the practical application of derivative concepts using physical examples and transitioning into a detailed analysis of continuity and types of discontinuities. This set a foundation for deeper exploration into calculus and its real-world applications, emphasizing the dual perspectives of theoretical concepts and tangible examples.