Overview
This lecture covers how to calculate and interpret the average rate of change (slope of a secant line) and the instantaneous rate of change (slope of a tangent line, i.e., the derivative) for functions, using formulas, limits, and examples.
Average Rate of Change
- The average rate of change between points ( a ) and ( b ) is ( \frac{f(b) - f(a)}{b - a} ).
- It represents the slope of the secant line connecting two points on a function’s graph.
- For tables or graphs without explicit functions, use this formula for estimation.
Instantaneous Rate of Change & Derivatives
- The instantaneous rate of change at ( x=a ) is the slope of the tangent line at that point, written as ( f'(a) ).
- Estimate ( f'(a) ) by computing the average rate of change over increasingly smaller intervals around ( a ).
- Derivative as a function: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).
- Alternate form: ( f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} ).
Power Rule & Basic Differentiation
- Power rule: ( \frac{d}{dx}x^n = n x^{n-1} ).
- Coefficients: ( \frac{d}{dx}[k x^n] = k n x^{n-1} ).
- The derivative of a constant is 0; ( \frac{d}{dx}[c] = 0 ).
- The derivative of ( kx ) is ( k ).
Examples & Practice Problems
- For ( f(x) = x^3 ) at ( x=2 ), average rate on [1,3]: 13; instantaneous rate (derivative): 12.
- For ( f(x) = x^2 + 2x - 3 ) on [2,4], average rate: 8; at ( x=3 ), instantaneous rate: 8.
- For ( f(x) = x^4 + 5x ) on [1,4], average rate: 90; at ( x=2 ), instantaneous rate: 37.
- For ( f(x) = x^2 + 3x - 5 ), derivative: ( 2x + 3 ); at ( x=3 ), slope: 9.
Data Table Estimation
- Average rate on [1,3]: ( (11.6-7.1)/2 = 2.25 ).
- Slope between 2 and 4: ( (13.3-8.9)/2 = 2.2 ).
- Estimate instantaneous rate at 4 using [3,5]: ( (16.1-11.6)/2 = 2.25 ).
- Best estimate for instantaneous rate at 1.5 is between closest points (e.g., [1,2]).
Key Terms & Definitions
- Average Rate of Change — The ratio ( \frac{f(b) - f(a)}{b - a} ); slope of the secant line.
- Instantaneous Rate of Change — The derivative at a point; slope of the tangent line.
- Derivative — Describes the instantaneous rate of change; can be found with limits.
- Secant Line — A line connecting two points on a curve.
- Tangent Line — A line touching the curve at one point only.
- Power Rule — A shortcut for differentiating powers of ( x ): ( nx^{n-1} ).
Action Items / Next Steps
- Practice calculating average and instantaneous rates of change for given functions and tables.
- Review the power rule and its application to different functions.
- Complete assigned homework problems on rate of change and differentiation.