L09 - Lecture 9 Tangent Line Slope as Rate of Change...

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Lecture 9 — Tangent Line Slope as Rate of Change Suppose that at the snowline of a mountain range, the diversity of an insect genus (measured in average number of species N per acre) depends linearly upon one’s descent x from that point, in hundreds of feet, according to the equation N = 10 + 1 . 2 x . The slope 1 . 2 tells us that every 100 feet of descent will increase the average number of species in the area by EXACTLY 1 . 2 . That is, it measures the rate at which N changes as x changes by one unit. The linear relationship scales this rate in perfect proportion (similar triangles) and is independent of the elevation. How much does N change if one is: 200 feet below the snowline; descends 100 feet 500 feet below the snowline; descends 100 feet 500 feet below the snowline; descends 50 feet
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Most functions are not linear. How do we measure the rate at which they change? Suppose now that diversity N is a function of the descent x given by: N ( x ) = 10 + 0 . 2 x 2 . The steep- ness of the curve varies from point to point, so the slope must be measured a single point at a time. At what rate is N changing with respect to x at 100 feet below the snowline ( x = 1 )? If we measure the slope from our point at x = 1 to another at x = ¯ x on a small interval, then we also measure the average rate of change on the interval: Slope from ( 1 ,N (1) ) to ( ¯ x,N x ) ) = = change in avg # of species change in elevation in 100s
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This note was uploaded on 05/27/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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L09 - Lecture 9 Tangent Line Slope as Rate of Change...

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