Study Notes on Derivatives and Rates of Change

Learning Objectives:
- State limit definition of derivative in two forms.
- Compute slope and equation of tangent line using limit definition.
- Explain relationships: secant lines vs. tangent lines, average rate of change vs. instantaneous rate.
- Describe how derivatives relate to tangent lines.
- Identify limit expressions as function derivatives at specified points.
Secant and Tangent Lines:
- Slope of secant line through (a, f(a)) and (x, f(x)) is important for finding tangent line slope.
- Slope of the tangent line at (a, f(a)) involves limits of secant line slopes.
Derivative Definition:
- The derivative of a function y = f(x) at x-value a is defined using limits.
Applications:
- Find derivatives through examples, determining specific functions and values for a.
- Calculate derivatives and tangent line equations at given points (e.g., x = 5).
Interpreting Derivatives and Rates of Change:
- Analyze slopes of secant and tangent lines in real-life contexts (e.g., bike ride distance, coffee temperature).
- Understand units of expressions involving derivatives and rate changes (e.g., temperature over time).
Average vs. Instantaneous Rates of Change:
- Organize expressions by average (left) and instantaneous rates (right).
- Definitions include terms like slope of tangent line, slope of secant line, average velocity, instantaneous velocity, derivative.