Question Video: Finding the Average Rate of Change of Root Functions between Two Points Mathematics • Higher Education

Determine the average rate of change function 𝐴(β„Ž) for 𝑓(π‘₯) = 4π‘₯Β² + 3π‘₯ + 2 at π‘₯ = 1.

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Video Transcript

Determine the average rate of change function 𝐴 of β„Ž for 𝑓 of π‘₯ equals four π‘₯ squared plus three π‘₯ plus two at π‘₯ equals one.

Remember, the average rate of change of a function 𝑓 of π‘₯ between two points defined by π‘Ž, 𝑓 of π‘Ž and π‘Ž plus β„Ž, 𝑓 of π‘Ž plus β„Ž is 𝑓 of π‘Ž plus β„Ž minus 𝑓 of π‘Ž all over β„Ž. We can see in this question that 𝑓 of π‘₯ has been defined for us. It’s four π‘₯ squared plus three π‘₯ plus two. We want to find the average rate of change function for 𝑓 of π‘₯ at π‘₯ equals one. So, we let π‘Ž be equal to one. We don’t actually know what β„Ž is, but that’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„Ž with this function. Let’s break this down and begin by working out what 𝑓 of π‘Ž plus β„Ž is.

We said π‘Ž is equal to one, so we’re actually looking to find 𝑓 of one plus β„Ž. We go back to our function 𝑓 of π‘₯, and each time we see an π‘₯, we replace it with one plus β„Ž. So, 𝑓 of one plus β„Ž is four times one plus β„Ž squared plus three times one plus β„Ž plus two. Let’s distribute our parentheses. One plus β„Ž all squared is one plus two β„Ž plus β„Ž squared, and three times one plus β„Ž is three plus three β„Ž. We can then distribute these parentheses and we get four plus eight β„Ž plus four β„Ž squared. Finally, we collect like terms and we get four β„Ž squared plus 11β„Ž plus nine.

Next, we work out 𝑓 of π‘Ž. Well, of course, we know that that’s 𝑓 of one. This one is slightly more straightforward than 𝑓 of one plus β„Ž. We simply replace π‘₯ with one. And we get four times one squared plus three times one plus two. And that’s equal to nine. We’re now ready to substitute everything into the average rate of change formula. We have 𝑓 of one plus β„Ž minus 𝑓 of one and that’s all over β„Ž. Well, nine take away nine is zero, and then we can divide through by β„Ž. And it simplifies really nicely to four β„Ž plus 11. And so, the average rate of change function 𝐴 of β„Ž for 𝑓 of π‘₯ equals four π‘₯ squared plus three π‘₯ plus two at π‘₯ equals one is four β„Ž plus 11.

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