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maxima and minima

maxima and minima. SKETCHING THE GRAPH USING THE FIRST DERIVATIVE TEST. Standard of Competence : To use The concept of Function Limit and Function deferential in problem solving. Basic Competenc e : To use The derived to find the caracteristic of functions and to solve the problems.

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maxima and minima

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  1. maxima and minima

  2. SKETCHING THE GRAPH USINGTHE FIRST DERIVATIVE TEST

  3. Standard of Competence:To use The concept of Function Limit and Function deferential in problem solving Basic Competence: To use The derived to find the caracteristic of functions and to solve the problems • Indicator: • To find the function increases and the functiondecreases by first derivative concept • To sketch the function graph by the propertis of the Derived Functions • To find endpoints of function graph

  4. Definitions of Increasing and Decreasing Functions

  5. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. dec inc inc

  6. The increasing/decreasing concept can be associated with the slope of the tangent line. The slope of the tangent line is positive when the function is increasing and negative when decreasing

  7. Test for Increasing and Decreasing Functions

  8. Theorem 3.6 The First Derivative Test

  9. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1: Graph the function f given by • and find the relative extremes. • Suppose that we are trying to graph this function but • do not know any calculus. What can we do? We can • plot a few points to determine in which direction the • graph seems to be turning. Let’s pick some x-values • and see what happens.

  10. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued):

  11. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued): • We can see some features of the graph from the sketch. • Now we will calculate the coordinates of these features • precisely. • 1st find a general expression for the derivative. • 2nd determine where f (x) does not exist or where • f (x) = 0. (Since f (x) is a polynomial, there is no • value where f (x) does not exist. So, the only • possibilities for critical values are where f (x) = 0.)

  12. Optimizing an Open Box • An open box with a square base is to be constructed from 108 square inches of material. • What dimensions will produce a box that yields the maximum possible volume?

  13. Which basic shape would yield the maximum volume? • Should it be tall? • Should it be square? • Should it be more cubical? • Perhaps we could try calculating a few volumes and get lucky.

  14. Is this the maximum volume?

  15. Is this the maximum volume?

  16. Is this the maximum volume?

  17. Is this the maximum volume?

  18. How are we doing? • Guess and Check really is not a very efficient way to approach this problem. • Lets use Calculus and get directly to the solution of this problem. • We can apply the maxima theory for a derivative to resolve this problem.

  19. Working rule for finding points of maxima and minima • Let f be a function such that f’ (x) exists. • 1) if f ’ (C) = 0 and f(C) ’’ > 0 then f has local minima • 2) if f ’(C) = 0 and f ’’(C) < 0 then f has local maxima

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