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Understanding Maxima and Minima in Optimization

In this activity, teams will investigate the relationship between regions of increase and decrease in functions to find maxima and minima. Students will learn how to determine critical points by taking the derivative of a function and setting it equal to zero. Definitions regarding local and global maxima and minima will be discussed. Practice with graphing functions and identifying critical points will reinforce these concepts. Homework includes exercises and optional review resources for further understanding.

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Understanding Maxima and Minima in Optimization

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  1. Questions?

  2. Maxima and Minima Friday, Feb 28th

  3. When we’re not that good. • Not that good.

  4. Investigative activity Review yesterday: In your teams, investigate the relation between regions of increase and decrease.

  5. optimization How do you find the maximum or minimum of a function? • Take the function’s derivative, f’(x) • Set the derivative equal to zero. • Solve for x.

  6. Maxima and minima A few definitions: • Local maximum point is the point (a, f(a)) where the slope is zero, changing from positive to negative. • Local maximum value is the y-coordinate, f(a) • Local minimum point is the point (a, f(a)) where the slope is zero, changing from negative to positive. • Local minimum value is the y-coordinate, f(a) • Critical number is the x-coordinate, a, where f’(a) = 0 or doesn’t exist.

  7. Maxima and minima A few more definitions: • Global maximum (aka “absolute maximum”) point is the highest point of f(x) in the whole domain. • Global minimum (aka “absolute minimum”) point is the lowest point of f(x) in the whole domain.

  8. Practice with a friend • Draw up a graph of anything you like (preferably one with lots of dips and bumps). Suggestions: graph position vs time of crazed rabbit, happiness vs class time of a calculus student, temperature vs location in CCHS, etc. • Give your graph a clearly labelled axis. • Give your graph to a friend. • Label your friend’s graph with the local maximum values, f(a1), f(a2), etc., the local minimum values, f(a3), f(a4), etc., and the critical values, a1, a2, etc.

  9. Maxima and minima Example: • Find the critical points of f(x) = x3 – 12x – 3. • Determine local maximum and minimum values • Determine the global maximum and minimum values of this function between -3 < x < 4.

  10. In your teams! The height of Lorin’s epic Olympic-style ski jump that resulted in battle wounds is given by the function: h(t) = -4.9t2 + 8.1t + 1 What is the maximum height that Lorin reaches (aka. global maximum value)? At what time does he reach this height (aka. critical value)?

  11. Homework • Page 163 #2, 3, 5, 9, 13, 15, 22 • Optional: Review for your test via Khan academy. These tutorials take a slightly different approach to learning the same concepts that we learned, which can be helpful to broaden your understanding.

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