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Applications of Derivatives

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  1. Applications of Derivatives Section 4.1 Section 5.2

  2. Applications of Derivatives • Derivatives allow you to sketch the shape of functions

  3. Applications of Derivatives • Ex: Amount of cargo unloaded at a port related to the number of trucks

  4. Applications of Derivatives

  5. Applications of Derivatives

  6. Applications of Derivatives

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  10. Applications of Derivatives

  11. Applications of Derivatives • Sketch the function c(w) based on the following: c(0) = 200 c(5) = 176 c(20) = 121 c’(0) = -50 c’(5) = -44 c’(20) = -30

  12. Applications of Derivatives • Derivatives allow you to approximate functions

  13. Applications of Derivatives

  14. Applications of Derivatives

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  17. Applications of Derivatives

  18. Applications of Derivatives

  19. Applications of Derivatives

  20. Applications of Derivatives • Suppose that for the function c(w), c(10) = 155 and c’(10) = -39. What is the approximate value of c(20)?

  21. Extreme Points 0 + slope - slope

  22. Extreme Points Population of Cleveland

  23. Extreme Points

  24. Extreme Points

  25. Extreme Points

  26. Extreme Points • Conclusions • At the minimum/maximum values of a function, the value of the derivative is 0. • At the inflection points of a function, the value of the derivative reaches a minimum/maximum.

  27. Extreme Points • Finding roots • Easy for linear, quadratic • Hard for higher order polynomials, other function Y= GRAPH CALC 2: zero

  28. Extreme Points • In-Class • Find the maxima and minima for the following functions • 0.04x3 - 0.88x2 + 4.81x +12.11 • 0.0004x4 – 0.007x3 + 0.03x2 – 0.035x + 10

  29. Extreme Points • Cost of production • How many machines are needed to minimize the cost per unit?

  30. Extreme Points

  31. Extreme Points

  32. Extreme Points • Fit a quadratic model to the data

  33. Extreme Points • How many machines are needed to minimize the cost per unit?

  34. Extreme Points

  35. Extreme Points • How many machines are needed to minimize the cost per unit? • The number that sets c’(m) = 0 (root)

  36. Extreme Points • Revenue over time • In what month was revenue maximized?

  37. Extreme Points

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  39. Extreme Points • Fit a quartic model to the data

  40. Extreme Points • In what month was revenue maximized?

  41. Extreme Points

  42. Extreme Points • In what month was revenue maximized? • Find the 3 numbers that set r’(t) = 0 Y= GRAPH CALC 2: zero

  43. Extreme Points • In what month was revenue maximized? • Find the 3 numbers that set r’(t) = 0

  44. Extreme Points

  45. Extreme Points • In-Class