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3. Increasing, Decreasing, and the 1st derivative test
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Graphs We will be spending the next several days talking about curve sketching We’ll use information about the function, the 1st derivative, and the 2nd derivative in order to make a sketch of the function
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If a graph exists on an interval, it is doing one of three things
Increasing (y-values increase as x increases) Decreasing (y-values decrease as x increases) Staying constant (y values stay the same as x increases) The derivatives tell us this information because it tells us the slope.
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Where does a graph change its increasing/decreasing/constant status??
It only changes at a critical value or at a discontinuity!! Without a graph, we must find any critical values or discontinuities, then test the intervals in between them.
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1st derivative test for local extrema
At a critical point c If f’ changes sign from positive to negative, then f has a local max If f’ changes sign from negative to positive, then f has a local min If f’ doesn’t change sign, then no local extrema When using the 1st derivative test to justify relative extrema, you MUST write a concluding statement clearly communicating the type of sign change of f’ at each x = c. You may not use the word “IT”
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Example 1 Inc Dec Inc -2 2 Plug in values in each interval to f’
Local max Local min
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Example 2
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Example 3 Find the x-coordinates of the local extrema of the function on the interval
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Example 4 Find the relative extrema of the function Justify.
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Example 5 Find the x-coordinates of the relative extrema of the function Justify.
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