Question Video: Finding the Average Rate of Change of Root Functions between Two Points | Nagwa Question Video: Finding the Average Rate of Change of Root Functions between Two Points | Nagwa

# Question Video: Finding the Average Rate of Change of Root Functions between Two Points Mathematics • Second Year of Secondary School

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Determine the average rate of change function π΄(β) for π(π₯) = 4π₯Β² + 3π₯ + 2 at π₯ = 1.

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### Video Transcript

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.

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. We can see in this question that π of π₯ has been defined for us. Itβs four π₯ squared plus three π₯ plus two. We want to find the average rate of change function for π of π₯ at π₯ equals one. So, we let π be equal to one. We donβt actually know what β is, but thatβs fine. 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. Letβs break this down and begin by working out what π of π plus β is.

We said π is equal to one, so weβre actually looking to find π of one plus β. We go back to our function π of π₯, and each time we see an π₯, we replace it with one plus β. So, π of one plus β is four times one plus β squared plus three times one plus β plus two. Letβs distribute our parentheses. One plus β all squared is one plus two β plus β squared, and three times one plus β is three plus three β. We can then distribute these parentheses and we get four plus eight β plus four β squared. Finally, we collect like terms and we get four β squared plus 11β plus nine.

Next, we work out π of π. Well, of course, we know that thatβs π of one. This one is slightly more straightforward than π of one plus β. We simply replace π₯ with one. And we get four times one squared plus three times one plus two. And thatβs equal to nine. Weβre now ready to substitute everything into the average rate of change formula. We have π of one plus β minus π of one and thatβs all over β. Well, nine take away nine is zero, and then we can divide through by β. And it simplifies really nicely to four β plus 11. 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.

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