1. 1 2.1 - The Tangent and Velocity Problem © John Seims (2008)

2. 2 Definition: Secant Line f(x2)f(x2) f(x1)f(x1) x1x1 x2x2 P Q Secant Line – A line passing thorough two points on a graph of a function. The slope of the secant line is called the average rate of change. Average Rate of Change

3. 3 Definition: Tangent Line f(x1)f(x1) x1x1 P Tangent Line – A line touching the graph of a function at a point P. The line may touch the graph elsewhere and still be considered a tangent line at x = x 1.

4. 4 Instantaneous Rate of Change f(x1)f(x1) x1x1 P The slope of the tangent line to the graph of a function at a point is called the instantaneous rate of change of the function at x 1. How can we determine the slope of the tangent line using a secant line?

5. 5 The Slope of a Tangent Line http://www.scottsarra.org/applets/calculus/ SecantTangent.html To estimate the slope of a tangent line at a point x 1, choose a point _____ that is very ________ to x 1 and determine the ___________ of the line between the coordinates of these two points.