1. MAT 1234 Calculus I Section 1.4 The Tangent and Velocity Problems http://myhome.spu.edu/lauw

2. WebAssign Homework 1.4 Good News…Only 2 problems… If you need help on the first assignment that due tomorrow, tutoring is open after class! Happy Friday!

3. The Winner of the Contest is…

4. Preview Big Picture: Why do we study differential calculus this quarter?

5. Two Worlds and Two Problems ?

6. 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

7. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

8. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

9. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

10. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

11. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

12. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

13. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

14. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

15. Tesla Model S on Fire Tesla Stock Plummets After Model S Catches On Fire

16. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

17. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

18. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

19. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

20. What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40sA moving vehicle The distance traveled is  30 ml/s at t=5sOil leaking from a tank The oil leaked is  -30ml/s at t=5sOil in the tank The oil in the tank is  -$5/min at t=8:05amThe value of a stock The value is 

21. What is “Rate of Change”? We are going to look at how to understand and how to find the “ rate of change ” in terms of functions.

22. The Problems The Tangent Problem The Velocity Problem

23. Example 1 The Tangent Problem Slope=?

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

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

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

27. 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=?

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

29. Example 1 The Tangent Problem We want a formula to compute this over and over again with other points in terms of their distance. Observation….

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

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

32. Example 1 The Tangent Problem We “see” from the table that the slope of the tangent line at x=1 should be _________. hslope 22 11.5 0.11.05 0.011.005

33. Limit Notations When h is approaching 0, is approaching 1. We say as h  0, Or,

34. Definition For the graph of, the slope of the tangent line at is if it exists.

35. Two Worlds and Two Problems

37. Example 2 The Velocity Problem y = distance dropped (ft) t = time (s) Displacement Function (Positive Downward) Find the velocity of the ball at t=2.

38. Example 2 The Velocity Problem Again, we are going to use the same limiting process. Find the average velocity of the ball from t=2 to t=2+h by the formula

39. Example 2 The Velocity Problem thAverage Velocity (ft/s) 2 to 31 2 to 2.10.1 2 to 2.010.01 2 to 2.0010.001

40. Example 2 The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be _________ft/s.

41. Example 2 The Velocity Problem We “see” from the table that velocity of the ball at t=2 should be 64ft/s. The instantaneous velocity at t=2 is 64 ft/s. (The ball is traveling at 64 ft/s 2 seconds after it dropped.)

42. Limit Notations When h is approaching 0, is approaching 64. We say as h  0, Or,

43. Definition For the displacement function, the instantaneous velocity at is if it exists.

44. Two Worlds and Two Problems

45. Review and Preview Example 1 and 2 show that in order to solve the tangent and velocity problems we must be able to find limits. In the next few sections, we will study the methods of computing limits without guessing from tables.

46. Review and Preview Example 1 and 2 show that in order to solve the tangent and velocity problems we must be able to find limits. In the next few sections, we will study the methods of computing limits without guessing from tables. (BTW, why guessing is not enough?)