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Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4. Applications of the Derivative. Maxima and Minima Applications of Maxima and Minima The Second Derivative - Analyzing Graphs. Absolute Extrema. Let f be a function defined on a domain D. Absolute Maximum.

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Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4

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  1. Chapter 5Applications of the DerivativeSections 5.1, 5.2, 5.3, and 5.4

  2. Applications of the Derivative • Maxima and Minima • Applications of Maxima and Minima • The Second Derivative - Analyzing Graphs

  3. Absolute Extrema Let f be a functiondefined on a domain D Absolute Maximum Absolute Minimum

  4. Absolute Extrema A function f has an absolute (global) maximum atx = c if f (x) f (c)for allx in the domain D of f. The number f (c) is called the absolute maximumvalue of f in D Absolute Maximum

  5. Absolute Extrema A function f has an absolute (global) minimum atx = c if f (c) f (x)for allx in the domain D of f. The number f (c) is called the absolute minimumvalue of f in D Absolute Minimum

  6. Generic Example

  7. Generic Example

  8. Generic Example

  9. Relative Extrema A function f has a relative (local) maximum at xc if there exists an open interval (r, s) containing c suchthat f (x) f (c) for all r  x  s. Relative Maxima

  10. Relative Extrema A function f has a relative (local) minimum at xc if there exists an open interval (r, s) containing c suchthat f (c) f (x) for all r  x  s. Relative Minima

  11. Generic Example The corresponding values of x are called Critical Points of f

  12. Critical Points of f A critical number of a function f is a number cin the domain off such that (stationary point) (singular point)

  13. Candidates for Relative Extrema • Stationary points: any x such that xis in the domain of f and f'(x)  0. • Singular points: any x such that xis in the domain of f and f'(x)  undefined • Remark:notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

  14. Fermat’s Theorem If a function f has a local maximum or minimum at c, then c is a critical number of f Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum

  15. Generic Example Two critical points of f that do not correspond to local extrema

  16. Example Find all the critical numbers of Stationary points: Singular points:

  17. Local min. Local max. Graph of

  18. Extreme Value Theorem If a function f is continuous on a closed interval [a,b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint. a b a b a b Attains max. and min. Attains min. but no max. No min. and no max. Open Interval Not continuous

  19. Finding absolute extrema on [a,b] • Find all critical numbers for f (x) in (a,b). • Evaluate f (x) for all critical numbers in (a,b). • Evaluate f (x) for the endpoints a and b of the interval [a,b]. • The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a,b].

  20. Absolute Max. Absolute Min. Absolute Max. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Evaluate

  21. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max. Absolute Min.

  22. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Evaluate Absolute Min.

  23. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Absolute Min.

  24. Increasing/Decreasing/Constant

  25. Increasing/Decreasing/Constant

  26. Increasing/Decreasing/Constant

  27. The First Derivative Test

  28. A similar Observation Applies at a Local Max. Generic Example

  29. The First Derivative Test Determine the sign of the derivative of f to the left and right of the critical point. left right conclusion f (c) is a relative maximum f (c) is a relative minimum No change No relative extremum

  30. Relative Extrema Example:Find all the relative extrema of Stationary points: Singular points: None

  31. Relative max. f (0) = 1 Relative min. f (4) = -31 The First Derivative Test Find all the relative extrema of + 0 - 0 + 0 4

  32. The First Derivative Test

  33. The First Derivative Test

  34. Another Example Find all the relative extrema of Stationary points: Singular points:

  35. Stationary points: Singular points: Relative max. Relative min. + ND + 0 - ND - 0 + ND + -1 0 1

  36. Local min. Local max. Graph of

  37. Domain Not a Closed Interval Example: Find the absolute extrema of Notice that the interval is not closed. Look graphically: Absolute Max. (3, 1)

  38. Optimization Problems • Identify the unknown(s). Draw and label a diagram as needed. • Identify the objective function. The quantity to be minimized or maximized. • Identify the constraints. 4. State the optimization problem. 5. Eliminate extra variables. 6. Find the absolute maximum (minimum) of the objective function.

  39. Optimization - Examples An open box is formed by cutting identical squares from each corner of a 4 in. by 4 in. sheet of paper. Find the dimensions of the box that will yield the maximum volume. x 4 – 2x x x x 4 – 2x

  40. Critical points: The dimensions are 8/3 in. by 8/3 in. by 2/3 in. giving a maximum box volume of V 4.74 in3.

  41. Optimization - Examples An metal can with volume 60 in3 is to be constructed in the shape of a right circular cylinder. If the cost of the material for the side is $0.05/in.2 and the cost of the material for the top and bottom is $0.03/in.2 Find the dimensions of the can that will minimize the cost. top and bottom cost side

  42. So Sub. in for h

  43. Graph of cost function to verify absolute minimum: 2.5 So with a radius ≈ 2.52 in. and height ≈ 3.02 in. the cost is minimized at ≈ $3.58.

  44. Second Derivative

  45. Second Derivative - Example

  46. Second Derivative

  47. Second Derivative

  48. Concavity Let f be a differentiable function on (a, b). 1.f is concave upward on (a, b) if f' is increasing on aa(a, b). That is f ''(x)0 for each value of x in (a, b). 2.f is concave downward on (a, b) if f' is decreasing on (a, b). That is f ''(x)0 for each value of x in (a, b). concave upward concave downward

  49. Inflection Point A point on the graph of f at which fis continuousandconcavity changes is called an inflection point. To search for inflection points, find any point, c in the domain where f ''(x)0 or f ''(x)is undefined. If f ''changes sign from the left to the right of c, then (c,f (c))is an inflection point of f.

  50. Example: Inflection Points Find all inflection points of

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