Idea: Every polynomial looks like a line, if you zoom close enough.[br][br]Earlier lessons. What is change? What is the the rate of change for a constant function? What is the rate of change for a linear function?[br][br]Earlier result: rate of change for a linear function is constant, (instantaneous rate of change is also a constant).[br][b][br][/b]Tasks for students to solve in small groups, 2-3 in each.[b][br][/b]Problem 1. [br][list=1][*]With Geogebra, draw the graph of a function [math]f(x)=x^3-2x[/math]. Zoom in close enough at any point. How does the graph look like?[br][/*][*]When zoomed in, add two points to graph and draw a line through those points. [br][/*][*]Zoom out. What is the connection between the slope of the line and instantaneous rate of change of the function [math]f(x)[/math]at the point you zoomed in?[br][/*][/list][br]Problem 2. Study the graph of the function [math]g(x)=|x^3-2x|[/math]. Does it look like a line, if you zoom close? [br][br]Problem 3. Use the tangent tool in Geogebra and find out the instantaneous rate of change for a function [math]f(x)=x^2[/math]when [math]x=1[/math], [math]x=2[/math] and [math]x=3[/math].[br][br]What is the general rule? Check, if your general rule holds, when [math]x=-1[/math].