Overview
This lesson covers how to find and interpret rates of change for polynomial functions, focusing on average and instantaneous rates of change using secants and tangents.
Rates of Change in Polynomial Functions
- The rate of change in a polynomial function varies at different points on its curve.
- Average rate of change is the slope of a secant line between two points.
- Instantaneous rate of change is the slope of a tangent line at a single point.
- Areas where the function changes direction (maxima/minima) have a slope of zero.
Example: Average Rate of Change
- To find average rate of change between ( x = 2 ) and ( x = 6 ) for a given quadratic, first find ( f(2) ) and ( f(6) ).
- ( f(2) = -2 ), ( f(6) = 46 ); coordinates are (2, -2) and (6, 46).
- Average rate of change = ( (46 - (-2)) / (6 - 2) = 48 / 4 = 12 ).
- Steps: find y-values at both x-values, then compute the slope.
Slope Behavior on Polynomial Graphs
- Tangents on the left side of the vertex are negative (decreasing).
- Tangents at the vertex are zero (max/min point).
- Tangents on the right side of the vertex are positive (increasing).
Example: Estimating Instantaneous Rate of Change
- To estimate slope at ( x = 1 ) for ( f(x) = 3x^2 - 4x - 1 ), use ( [f(1.001) - f(1)] / (1.001 - 1) ).
- Compute ( f(1) = -2 ), estimate yields slope ≈ 2.
- Using calculus, the derivative (slope function) is ( f'(x) = 6x - 4 ), so at ( x = 1 ), slope = 2.
Tangent Line Equation at a Point
- Point of tangency: ( (1, -2) ), slope: 2.
- Equation: ( y = mx + b ) with ( m = 2 ) and point ( (1, -2) ).
- Substitute: ( -2 = 2(1) + b ), thus ( b = -4 ).
- Tangent equation: ( y = 2x - 4 ).
Key Terms & Definitions
- Polynomial Function — a function involving terms with variables raised to whole number powers.
- Secant Line — a line joining two points on a graph, used to find average rate of change.
- Tangent Line — a line touching the graph at one point, used to find instantaneous rate of change.
- Average Rate of Change — change in y over change in x between two points ((f(b)-f(a))/(b-a)).
- Instantaneous Rate of Change — slope of the tangent at a single point; limit of the average rate as interval approaches zero.
- Derivative — function that gives the slope of another function at any point.
Action Items / Next Steps
- Practice finding average and instantaneous rates of change for given polynomial functions.
- Review textbook examples and attempt similar questions for homework.
- Prepare for future lessons by revisiting Chapter 2 on rates of change.