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Determine the average rate of change function π΄ of β for π of π₯ equals four π₯ squared plus three π₯ plus two at π₯ equals one.
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Remember, the average rate of change of a function π of π₯ between two points defined by π, π of π and π plus β, π of π plus β is π of π plus β minus π of π all over β.
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We can see in this question that π of π₯ has been defined for us.
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Itβs four π₯ squared plus three π₯ plus two.
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We want to find the average rate of change function for π of π₯ at π₯ equals one.
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So, we let π be equal to one.
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We donβt actually know what β is, but thatβs fine.
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This question is essentially asking us to derive a function that will allow us to find the average rate of change for any value of β with this function.
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Letβs break this down and begin by working out what π of π plus β is.
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We said π is equal to one, so weβre actually looking to find π of one plus β.
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We go back to our function π of π₯, and each time we see an π₯, we replace it with one plus β.
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So, π of one plus β is four times one plus β squared plus three times one plus β plus two.
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Letβs distribute our parentheses.
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One plus β all squared is one plus two β plus β squared, and three times one plus β is three plus three β.
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We can then distribute these parentheses and we get four plus eight β plus four β squared.
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Finally, we collect like terms and we get four β squared plus 11β plus nine.
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Next, we work out π of π.
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Well, of course, we know that thatβs π of one.
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This one is slightly more straightforward than π of one plus β.
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We simply replace π₯ with one.
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And we get four times one squared plus three times one plus two.
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And thatβs equal to nine.
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Weβre now ready to substitute everything into the average rate of change formula.
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We have π of one plus β minus π of one and thatβs all over β.
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Well, nine take away nine is zero, and then we can divide through by β.
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And it simplifies really nicely to four β plus 11.
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And so, the average rate of change function π΄ of β for π of π₯ equals four π₯ squared plus three π₯ plus two at π₯ equals one is four β plus 11.