1. Warm Up Determine a) ∞ b) 0 c) ½ d) 3/10 e) 1

2. 2.4 – Rates of Change and Tangent Lines

3. GOAL I will be able to calculate the slope of a line tangent to a curve through the definition of the slope of a curve

4. Average rate of change

5. Example

6. You try

7. Remember Geometry? Secant line is a line which passes through at least 2 points on a curve Tangent line is a line which passes through exactly one point on a curve.

8. Average rate of change=slope of the secant line Instantaneous rate of change = slope of tangent line

9. Example

10. Solution

11. Continued

13. In general: The slope of the curve f(x) at the point (a, f(a)) is Provided the limit exists.

14. You try Find the slope of the curve f(x) = x 2 + x at x = 5. When you are asked to find the slope at a point a, use the interval [a, a+h]. So in this case, use [5, 5+h] Find f(5) and f(5+h). Then plug those into the slope equation.

15. Solution

16. Example If the function f given by f (x) = x 2 + x has an average rate of change of 7 on the interval [0, k], then k = ? (a) -8 (b) 2 (c) 6 (d) 30 (e) k cannot be determined from the information given.

17. Definition The normal line to a curve at a point is the line perpendicular to the tangent line at that point. So: to find the equation of the normal line, follow the process of finding slope, then use the opposite reciprocal.

18. Time check!

19. Example

20. Continued When is the slope equal to 9? 9= 2a+1 At a=4

21. Continued

23. Homework 2.4