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GISAT 151 Applications of the Derivative

Lecture 12. GISAT 151 Applications of the Derivative. Maxima and Minima. How many production runs to make??. You are planning this year’s printing schedule for your publishing company’s latest best-seller and your job is to minimize cost while ensuring that sales will be met!.

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GISAT 151 Applications of the Derivative

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  1. Lecture 12 GISAT 151Applications of the Derivative Maxima and Minima

  2. How many production runs to make?? You are planning this year’s printing schedule for your publishing company’s latest best-seller and your job is to minimize cost while ensuring that sales will be met!

  3. How many production runs to make?? Here’s what you know… • ·        Making predicts that it will sell 100,000 copies per month over the coming year. • ·        It costs $5,000 to set up the printing presses. • ·        Once the presses are set up, it costs $1 to print each book. • ·        It costs $0.01 to store each book for one month. • ·        You may make as many printing runs as you like, however to achieve efficient printing operations you must print the same number of books for each run and the runs must be evenly spaced throughout the year.

  4. A function f has a relative (local) maximum at x = c if for all values of x close to c. A function f has a relative (local) minimum at x = c if for all values of x close to c. Relative Maximums Relative Minimums Relative Extrema

  5. Absolute Extrema A function f has an absolute (global) maximum at x = c if for all x in the domain of f. A function f has a absolute (global) minimum at x = c if for all x in the domain of f. Absolute Maximum Absolute Minimum

  6. Critical Points of f A critical point of a function f is a point in the domain of f where (stationary point) (singular point)

  7. Candidates for Relative Extrema • Stationary points: x such that x is in the domain of f and • Singular points: x such that x is in the domain of f and • Endpoints: endpoints of the domain (if any). Closed intervals have endpoints, open intervals do not.

  8. Relative Extrema Ex. Find all the relative extrema of Stationary points: Singular points: None Endpoints: None Relative max. f (0) = 1 4 Relative min. f (4) = –31

  9. Relative Extrema Ex. Find all the relative extrema of Stationary points: Singular points: Relative max. Relative min.

  10. Increasing/Decreasing/Constant

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

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

  13. Absolute Extrema If a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and minimum on [a, b]. Each extremum must occur at a critical point or an endpoint. a b a b a b Attains absolute max. and min. Attains absolute min. but not max. No absolute min. or max. Interval open Not continuous

  14. Example Find the absolute extrema of Critical values at x = 0, 2 Absolute Min. Evaluate Absolute Max.

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