# The y value at such a point is the actual relative

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The y-value at such a point is the actual relative extreme value, while the x-value is the location at which the extreme value occurs For parabolas, the relative extremum is the coordinate of the vertex and can be found algebraically. For most other functions, these extreme values require the use of calculus or a graphing calculator. Example 3 Find the coordinates of all relative extrema for the following functions using a graphing calculator. (a) 4 2 ( ) 7 6 f x x x x ± ² (b) ³ ´ 2/3 ( ) 3 4 2 U x x ± ± (c) 2 ( ) g x x x x ± Functions like the ones in the previous example have graphs that are curvy, meaning they increase or decrease at different rates. We are all familiar with speed. For instance, if you drive a total distance of 130 miles in 2 hours, your average speed, or average rate of change (AROC), would be ࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵? = ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵? ࠵?࠵?࠵?࠵? = 130 ࠵?࠵? 2 ℎ࠵? = 65 ࠵?/ℎ This doesn’t mean, necessarily, that you had to be traveling at 65 mph the entire time, for you likely went slower at times, and therefore, faster than 65 mph at other times. We can do the same calculation for functions over x intervals, talking about how “fast,” on average, the y - values change on the interval. In the above example, if we let the function d(t) represent the distance you traveled, in miles, and let t represent the time spent on your trip, in hours, the calculation looks like this: ࠵?࠵?࠵? ࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?ℎ࠵?࠵?࠵?࠵? ࠵?࠵? ࠵?࠵?࠵?࠵?࠵?࠵?࠵?࠵? = ࠵?(2) − ࠵?(0) 2 − 0 = 130 ࠵?࠵? 2 ℎ࠵? = 65 ࠵?/ℎ

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Average Rate of Change on a Closed Interval Example 4 Sketch the function (without a calculator) ³ ´ 2 ( ) 4 f x x ± , then find the average rate of change on the following closed intervals.

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