1. MAT 1221 Survey of Calculus Section 2.1 The Derivative and the Slope of a Graph http://myhome.spu.edu/lauw

2. Expectations Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions doing cancelations Do not skip steps – points are assigned to all essential steps Start your solutions with

3. Reminder WebAssign Homework 2.1 Quiz 02 on Monday Read the next section on the schedule If you do not have WebAssign access because you added the class after the first day of class, talk to Caro or I after class.

4. Recall: What do we care? How fast “things” are going The velocity of a particle The “speed” of formation of chemicals The rate of change of a population

5. Recall: Slope of Tangent Line

6. Preview Definition of Tangent Lines Definition of Derivatives The limit of Difference Quotients are the Derivatives

7. Example 1 The Tangent Problem Slope=?

8. Example 1 The Tangent Problem We are going to use an “limiting” process to “guess” the slope of the tangent line at x=1. Slope=?

9. Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

10. Example 1 The Tangent Problem Then we compute the slope of the secant line between x=1 and x=2. Slope=?

11. Example 1 The Tangent Problem As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1. Slope=?

12. Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

13. Observation… Let h be the distance between the two points.

14. Example 1 The Tangent Problem Let us record the results in a table. hslope 22 1 0.1 0.01

15. Example 1 The Tangent Problem We “see” from the table that the slope of the tangent line at x=1 should be _________.

16. Use of Limit Notations When h is approaching 0, is approaching 1. We say as h  0, Or,

17. Definition (Geometric Property)

18. Definition (Function Property)

19. Example 2

22. Example 2 Step 1

23. Example 2 Step 2

24. Example 2 Step 3

25. Example 2

26. Example 3

27. Recall: Point-Slope Form

28. Example 3

30. Transitional Statement...

31. Classwork

32. Expectations Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions when doing cancelations If you choose not to follow the expectations, you paper will not be counted