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RELATIVE & ABSOLUTE EXTREMA
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IDENTIFYING EXTREMA ON A GRAPH
local or relative maximum local or relative minimum
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IDENTIFYING EXTREMA ON A GRAPH
The vertex of the parabola would be considered both a relative minumum and an absolute minimum
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FUNCTIONS WITH RESTRICTED DOMAINS
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HOW CAN WE FIND THESE HIGH AND LOW POINTS ON A GRAPH MATHEMATICALLY?
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WHAT ARE “CRITICAL VALUES”?
A CRITICAL VALUE is any value of 'x' where f'(x) = 0 or where f'(x) is undefined. MAXIMUMS AND MINIMUMS CAN ONLY OCCUR AT 1) CLOSED ENDPOINTS or 2) AT CRITICAL VALUES.
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FINDING CRITICAL VALUES
FIND THE CRITICAL VALUES FOR EACH FUNCTION BELOW. THEN, USE A GRAPHING UTILITY TO DETERMINE IF THIS CRITICAL VALUE WILL RESULT IN A RELATIVE MAXIMUM, RELATIVE MINIMUM OR NEITHER Example 1: Pierre de Fermat Prediction: relative minimum
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Let’s look at the graph of the derivative
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FOR FINDING RELATIVE EXTREMA
THE 1ST DERIVATIVE TEST
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The 1st Derivative Test Let x = c be a critical value for a function f(x) where f(c) exists. Then... If f' changes from + to - at x = c, then f has a relative maximum at x = c b) If f' changes from - to + at x = c, then f has a relative minimum at x = c
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Let’s look at the graph of the derivative
y’ < 0 to the left of x = 1.5 And y’ > 0 to the right of x = 1.5 So, by the 1st derivative test there must exist a relative minimum at x = 1.5
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FINDING RELATIVE EXTREMA – NO CALCULATOR
Example 2: Find all critical values and then determine if each critical value results in a maximum or minimum. Example 3: Find all critical values and then determine if each critical value results in a maximum or minimum. no maximums or minimums y’ > 0 to the left of x = -2 and y’ < 0 to the right of x = -2 So, by the 1st D.T. there must exist a relative maximum at x = -2 y’ < 0 to the left of x = 4/3 and y’ > 0 to the right of x = 4/3 So, by the 1st D.T. there must exist a relative minimum at x = 4/3
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FINDING RELATIVE EXTREMA – NO CALCULATOR
Example 4: Find all critical values and then determine if each critical value results in a maximum or minimum. y’ > 0 to the left and y’ < 0 to the right of x = /2 So, by the 1st D.T. there must exist a relative maximum at x = /2 y’ < 0 to the left and y’ > 0 to the right of x = 3/2 So, by the 1st D.T. there must exist a relative minimum at x = 3/2
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Example 5: The function below represents the derivative of a function f that has a domain of all real numbers. Use f’ to determine the values of ‘x’ where function f has relative extrema. y’ > 0 to the left and y’ < 0 to the right of x = .2301 So, by the 1st D.T. there must exist a relative maximum at x = .2301 y’ < 0 to the left and y’ > 0 to the right of x = So, by the 1st D.T. there must exist a relative minimum at x =
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FINDING RELATIVE EXTREMA – CALCULATOR
Example 6: Find all critical values and then determine if each critical value results in a maximum or minimum y’ < 0 to the left and y’ > 0 to the right of x = 0 So, by the 1st D.T. there must exist a relative minimum at x = 0
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FINDING RELATIVE EXTREMA –CALCULATOR
Example 7:Find all critical values and then determine if each critical value results in a maximum or minimum y’ > 0 to the left and y’ < 0 to the right of x = 0 So, by the 1st D.T. there must exist a relative maximum at x = 0 y’ > 0 to the right of x = -2 and x = -2 is a left endpoint So, there must exist a relative minimum at x = -2 y’ < 0 to the left of x = 2 and x = 2 is a right endpoint So, there must exist a relative maximum at x = 2
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INCREASING & DECREASING
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WHAT DO YOU REMEMBER ABOUT INCREASING & DECREASING FROM PRE-CALCULUS?
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CALCULUS DEFINITIONS:
Increasing: If 'f' is increasing on an interval of its domain then it is true that f'(x) ≥ 0 for all x on that interval. Decreasing: If 'f' is decreasing on an interval of its domain then it is true that f'(x) 0 for all x on that interval. Question: At what point(s) would a function change from increasing to decreasing?
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INCREASING/DECREASING
Example 1 – non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing:
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INCREASING/DECREASING
Example 2 non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing: none
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INCREASING/DECREASING
Example 3 non-calculator: Use the derivative to find the exact intervals of increasing and decreasing. Increasing: Decreasing:
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INCREASING/DECREASING
Example 4 non-calculator: Graph the following using a graphing utility. Predict the intervals of increasing and decreasing. Then use the derivative to find the exact intervals of increasing and decreasing. Decreasing:
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Putting it all Together non-calculator
Example 1: Find all relative maximum and minimums for the function below and find all intervals of increasing and decreasing.
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Putting it all Together non-calculator
Example 2: Find all relative maximum and minimums for the function below. Determine if any of these are absolute maximums or minimums.
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Putting it all Together - calculator
Example 3: Find all relative maximum and minimums for the function below and find all intervals of increasing and decreasing.
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