Stewart6e4_3

# Stewart6e4_3 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 4.3: Derivatives and the shape of things We’ve already seen ways in which we can tell stuff about f by examining its derivative f . For instance, if f ( x ) = 0 everywhere on an interval, then f is constant on that interval, as a consequence of the . More generally, if f ( x ) > 0 for all x on an interval, we expect that the function should be increasing, since the tangent lines “guide” it upwards; an analogous statement holds in case f ( x ) < 0 on an interval: The Increasing/Decreasing (I/D) Test. 1. If f ( x ) > 0 on an interval, then f is increasing on that interval. 2. If f ( x ) < 0 on an interval, then f is decreasing on that interval. Proof. This is easy! Let’s suppose that f ( x ) > 0 everywhere on the interval [ a, b ], and let’s take x 1 < x 2 inside of that interval. What can you say, by the MVT? Example. If f ( x ) = 3 x 4 - 4 x 3 - 12 x 2 + 5, where if f increasing? Where is it decreasing? Use the I/D test to find out!

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The I/D Test gives us a way to check whether numbers give us local extrema: The First Derivative Test.
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