1. 2.4 Rates of Change and Tangent Lines Calculus

2. Finding average rate of change • Find the average rate of change of over the interval [1, 3]. • 12

3. Slope of a secant line • Use points P(23, 150) and Q(45, 340) to compute the average rate of change and the slope of the secant line PQ. • 8.6 flies/day • We can always think about average rate of change as the slope of a secant line.

4. Instantaneous rate of change • What about the growth of the population on day 23? We move point Q closer to point P to get a better estimate. • Notice the secant line appears to be approaching the tangent line. • So we could use the slope of the tangent line as the instantaneous rate of change at

5. Steps for finding the slope of the tangent • Start with what we can calculate- the slope of the secant through a point P and a point nearby (Q) on the curve. • Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. • Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.

6. Definition: Slope of a curve at a point • The expression is the difference quotient of f at a.

7. Example: Finding slope and tangent line • Find the slope of the parabola at the point P(2, 4). Write an equation for the tangent to the parabola at this point.

8. Example: • Find the slope of the curve at . • Where does the slope equal -1/4?

9. Lines normal to a curve • The normal lineto a curve at a point is the line perpendicular to the tangent at that point. • Write an equation for the normal to the curve at

10. Free fall…again • Find the speed of the falling rock (discussed earlier in this chapter) at sec. • Remember: • 32 ft/sec