Instantaneous Rate of Change

1. Instantaneous Rate of Change Sections 3.1-3.3 Section 4.1

2. Average Change • Defined equations in terms of their changes • e.g., exponential  constant percentage change • Will use this concept motivate derivatives

3. Average Change • Rate of change • Difference between two values • Percentage change • Difference between two values as a percentage of the original value • Average change • Change per unit of time

4. Average Change • Ex: r(t) = pool sales

5. Average Change

6. Average Change • Rate of change • Percent change • Average change

7. Average Change Rate of change from April to August Rate of change

8. Average Change Rate of change from April to August r(8) - r(4)

9. Average Change Average rate of change from April to August r(8) - r(4)

10. Average Change Average rate of change from April to August r(8) - r(4) 8-4

11. Average Change Average rate of change from April to August r(8) - r(4) 8-4

12. Average Change • In-Class • Pg 167: 1, 2, 4, 6, 7, 8, 10

13. Average Change Average rate of change from April to August r(8) - r(4) 8-4

14. Average Change Average rate of change from April to August rise run

15. Average Change Average change between two points Slope of the secant line between the two points =

16. Instantaneous Rate of Change • Rate of change at this instant • Average rate of change over an infinitesimally small range

17. Instantaneous Rate of Change

18. Instantaneous Rate of Change

19. Instantaneous Rate of Change

20. Instantaneous Rate of Change

21. Instantaneous Rate of Change

22. Instantaneous Rates of Change • Tangent line • Secant line that touches the graph at the point evaluated The instantaneous rate of change is the slope of the tangent line at the point evaluated

23. Instantaneous Rates of Change • Local linearity • Zoom in enough and anything looks like a line.

24. Instantaneous Rate of Change

25. Instantaneous Rate of Change

26. Instantaneous Rate of Change

27. Instantaneous Rates of Change • Tangent lines don’t intersect graph at the point of tangency, but • Tangent lines can intersect graph at other points

28. Instantaneous Rates of Change

29. Instantaneous Rates of Change

30. Instantaneous Rates of Change

31. Instantaneous Rates of Change • Concave Down

32. Instantaneous Rates of Change • Concave Up

33. Instantaneous Rates of Change • Exists only where you have a continuous function • Does not exist at breakpoints

34. Instantaneous Rates of Change

35. Instantaneous Rates of Change

36. Instantaneous Rates of Change

37. Instantaneous Rates of Change

38. Instantaneous Rates of Change • In-Class • Pg 185: 7, 8, 9, 10

39. Derivatives Section 3.3 Section 4.1

40. Derivatives

41. Derivatives

42. Derivatives

43. Derivatives

44. Derivatives

45. Derivatives • Another phrase for instantaneous rate of change Instantaneous rate of change Rate of change = = Slope of curve Slope of tangent line = = Derivative

46. Derivatives • Notation “Derivative of f with respect to x”

47. Derivatives • Ex: Profit versus number of employees • p(t) = 20*ln(t)

48. Derivatives

49. Derivatives • Notation • Interpretation

50. Derivatives • In-Class • Pg 203: 1, 2, 4, 6, 10