maxima and minima
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Transcript of maxima and minima
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SKETCHING THE GRAPH USINGTHE FIRST DERIVATIVE TEST
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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 function decreases by first derivative concept•To sketch the function graph by the propertis of the Derived Functions•To find end points of function graph
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Definitions of Increasing and Decreasing Functions
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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. inc inc
dec
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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
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Test for Increasing and Decreasing Functions
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Theorem 3.6 The First Derivative Test
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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.
f (x)2x3 3x2 12x 12.
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Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
• Example 1 (continued):
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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.)
f (x)6x2 6x 12
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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?
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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.
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Is this the maximum volume?
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Is this the maximum volume?
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Is this the maximum volume?
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Is this the maximum volume?
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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.
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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