WEBVTT
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By finding the linear approximation of the function π of π₯ equals π to the power of π₯ at a suitable value of π₯, estimate the value of π to the power of 0.1.
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Weβre told to use the linear approximation of the function π of π₯ equals π to the power of π₯.
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So we recall the formula.
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If π is differentiable at π₯ equals π, then the equation that can be used to find the linear approximation to the function at π₯ equals π is π of π₯ equals π of π plus π prime of π times π₯ minus π.
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In this example, weβre trying to approximate the value of π to the power of 0.1.
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This is going to be close to the value of π to the power of zero.
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So we let π be equal to zero.
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This means π of π is equal to π of zero.
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And substituting zero into our function π of π₯ equals π to the power of π₯ gives π to the power of zero which is one.
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Next, we find π prime of π.
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First, of course, we need to find an expression for the derivative of our function.
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So we different π to the power of π₯ with respect to π₯.
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The first derivative of π to the power of π₯ is π to the power of π₯.
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So π prime of π becomes π prime of zero which is π to the power of zero.
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And once again, thatβs one.
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Substituting what we know into our formula for our tangent line approximation and we see that π of π₯ is equal to one plus one times π₯ minus zero.
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And that simplifies to π₯ plus one.
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Weβll use this to approximate the value of π to the power of 0.1 by finding π of 0.1.
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Thatβs 0.1 plus one which is 1.1.
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And an estimate to the value of π to the 0.1 is 1.1.
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And if we type this into our calculator, π to the 0.1 is 1.10517 and so on.
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Thatβs very close in value to our estimation.
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And thatβs because 0.1 is fairly close to zero.
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Had we tried the larger value, our number might not have been so accurate.
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Letβs check that.
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For example, π of 0.3 is 0.3 plus one.
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So according to our approximation, π to the 0.3 is approximately 1.3.
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Typing π to the 0.3 into our calculator and we get 1.349858808, still not a bad approximation but not quite as close as π to the power of 0.1.