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Use the table to estimate π prime of six.
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The question wants us to use this table to approximate the value of π prime of six.
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And we know that π prime of six is the slope of our function when π₯ is equal to six.
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From our table, we can see that when π₯ is equal to six, π of π₯ is equal to three.
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We can also see from our table weβre given values of π of π₯ for π₯ less than six and values of π of π₯ for π₯ is greater than six.
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This means we could choose to approximate our slope by using a left derivative or a right derivative.
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To approximate the slope of our curve when π₯ is equal to six by using a left derivative, weβll want to find the slope of the line connecting the point on our curve when π₯ is equal to six to a point to the left on our curve.
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And the closer this point is, the more accurate our estimate will be.
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And the closest value of π₯ to six in our table where π₯ is less than six is π₯ is equal to five.
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We can also do the same on the right-hand side.
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We see the closest value of π₯ to six where π₯ is greater than six in our table is when π₯ is equal to eight.
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So this means we could approximate π prime of six by taking the slope of the line between the point five, two and the point six, three.
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The slope of this line will be the change in π¦ divided by the change in π₯.
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Thatβs three minus two divided by six minus five.
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And we can calculate this to give us one.
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However, we can also approximate π prime of six by the slope of the line between the point six, three and the point eight, six.
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And the slope of this line will be six minus three divided by eight minus six, which we can evaluate to give us three over two.
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And these are both fairly reasonable estimates of π prime of six.
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However, we can get a more accurate estimate by taking the average of these two values.
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We call this the average of the left and right approximations for the derivative.
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This gives us that π prime of six is approximately equal to one plus three over two all divided by two.
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And if we calculate this, we get five divided by four, which weβll write as 1.25.
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Therefore, by taking an average of the left and right approximations for the derivative of the function π of π₯ given to us in the table, we were able to show that π prime of six was approximately equal to 1.25.