Question Video: Finding the Average Rate of Change of Polynomial Functions at a Point Mathematics • Higher Education

Determine the average rate of change function 𝐴(β„Ž) for 𝑓(π‘₯) = 6π‘₯Β² βˆ’ 3 at π‘₯ = 1.

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Video Transcript

Determine the average rate of change function 𝐴 of β„Ž for 𝑓 of π‘₯ is equal to six π‘₯ squared minus three at π‘₯ is equal to one.

The average rate of change function 𝐴 of β„Ž for a function 𝑓 of π‘₯ at π‘₯ is equal to π‘₯ one is given by 𝐴 of β„Ž is equal to 𝑓 at π‘₯ is equal to π‘₯ one plus β„Ž minus 𝑓 at π‘₯ is equal to π‘₯ one over β„Ž, where β„Ž is a small change in π‘₯. We’ve been given the function 𝑓 of π‘₯ is six π‘₯ squared minus three and asked to determine the rate of change function 𝐴 of β„Ž at π‘₯ is equal to one. This means that in our average rate of change function 𝐴 of β„Ž, π‘₯ one is equal to one. So, we have the function 𝑓 of π‘₯ is six π‘₯ squared minus three and π‘₯ one is equal to one. And our average rate of change function 𝐴 of β„Ž is equal to 𝑓 at π‘₯ is equal to one plus β„Ž minus 𝑓 at π‘₯ is equal to one all over β„Ž.

Of course, we haven’t finished yet. We need to evaluate our function 𝑓 at π‘₯ is equal to one plus β„Ž and 𝑓 at π‘₯ is equal to one. So, let’s do this. Replacing π‘₯ with one plus β„Ž in our function 𝑓 gives us six times one plus β„Ž squared minus three. That is six times one plus two β„Ž plus β„Ž squared all minus three, which is six plus 12β„Ž plus six β„Ž squared minus three. And collecting like terms and rearranging gives us six β„Ž squared plus 12β„Ž plus three.

Now, we evaluate 𝑓 at π‘₯ is equal to one. We have six times one squared minus three, which is six minus three, and that’s equal to three. We have 𝑓 at π‘₯ is equal to one plus β„Ž is six β„Ž squared plus 12β„Ž plus three and 𝑓 at π‘₯ is equal to one is equal to three, which we can now substitute into our average rate of change function 𝐴 of β„Ž. This gives us 𝐴 of β„Ž is equal to six β„Ž squared plus 12β„Ž plus three minus three all over β„Ž.

The positive three cancels with the negative three so that we have six β„Ž squared plus 12β„Ž over β„Ž. We have a common factor of β„Ž in the numerator. If we rewrite this as β„Ž times six β„Ž plus 12 over β„Ž, we can cancel this β„Ž in the numerator with the β„Ž in the denominator, and we’re left with six β„Ž plus 12.

The average rate of change function 𝐴 of β„Ž for 𝑓 of π‘₯ is equal to six π‘₯ squared minus three at π‘₯ is equal to one is six β„Ž plus 12.

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