WEBVTT
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Given that π¦ equals π of π₯ is a function for three known values where π of one is 1.5, π of two is 2.75, and π of three is 3.25, estimate π prime of two.
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So we donβt know the function.
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But weβve been given some π₯ values with the corresponding values of π of π₯.
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Letβs sketch these points on a graph to help us visualize what weβve got.
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So weβve drawn some axes and marked on the points one, 1.5; two, 2.75; and three, 3.25.
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So our function may look something like this.
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It could of course have an entirely different shape that still go through these points.
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However, this would be the most obvious graph.
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And we want to estimate the value of π prime of two, so the value of the derivative, the value of the slope at π₯ equals two.
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So how do we do this?
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Itβs gonna be hard to guess the slope of this tangent line.
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So instead, we connect the two points either side.
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This is called a secant line.
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And weβll estimate the slope at π₯ equals two by working out the slope of this line instead.
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We remember that the slope is the changing π¦-coordinates over the change in π₯-coordinates.
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So for our points β β one, 1.5 and three, 3.25 β β the slope of the line joining those points together is a 3.25 minus 1.5 over three minus one which is 1.75 over 2 or 0.875.
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And we must remember that this is just an estimate of course.