Video: Finding the Average Rate of Change of Polynomial Functions at a Point

Find the average rate of change function 𝐴(β„Ž) of 𝑓(π‘₯) = 2π‘₯Β³ + 30 when π‘₯ = π‘₯₁.

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Video Transcript

Find the average rate of change function 𝐴 of β„Ž of 𝑓 of π‘₯ is equal to two π‘₯ cubed plus 30 when π‘₯ is equal to π‘₯ one.

In general, the average rate of change function 𝐴 of β„Ž for a function 𝑓 of π‘₯ at a point π‘₯ is equal to π‘₯ one is given by the expression 𝐴 of β„Ž is equal to 𝑓 of π‘₯ one plus β„Ž minus 𝑓 of π‘₯ one divided by β„Ž, where β„Ž is a small change in π‘₯. We’ve been given the function 𝑓 of π‘₯ is two π‘₯ cubed plus 30. And to find the average rate of change function 𝐴 of β„Ž for our function 𝑓 of π‘₯ when π‘₯ is equal to π‘₯ one, we need to evaluate our function when π‘₯ is equal to π‘₯ one plus β„Ž and when π‘₯ is equal to π‘₯ one. And we can then substitute these into our expression for 𝐴 of β„Ž.

To evaluate 𝑓 at π‘₯ is equal to π‘₯ one plus β„Ž, we replace π‘₯ with π‘₯ one plus β„Ž in our function 𝑓. And this gives us 𝑓 of π‘₯ one plus β„Ž is equal to two times π‘₯ one plus β„Ž cubed plus 30. Expanding π‘₯ one plus β„Ž cubed, we have π‘₯ one plus β„Ž multiplied by π‘₯ one squared plus two β„Žπ‘₯ one plus β„Ž squared, that is, π‘₯ one cubed plus three β„Žπ‘₯ one squared plus three β„Ž squared π‘₯ one plus β„Ž cubed. So that our function 𝑓 evaluated at π‘₯ is equal to π‘₯ one plus β„Ž is two times π‘₯ one cubed plus three β„Žπ‘₯ one squared plus three β„Ž squared π‘₯ one plus β„Ž cubed plus 30.

To make things a little simpler when we use our average rate of change function, we’ve multiplied out the bracket. So we have 𝑓 evaluated at π‘₯ is equal to π‘₯ one plus β„Ž is equal to two π‘₯ one cubed plus six β„Žπ‘₯ one squared plus six β„Ž squared π‘₯ one plus two β„Ž cubed plus 30. To use our average rate of change function, we also need 𝑓 evaluated at π‘₯ is equal to π‘₯ one. And that’s equal to two times π‘₯ one cubed plus 30.

So now if we substitute 𝑓 of π‘₯ one plus β„Ž and 𝑓 of π‘₯ one into our average rate of change function, we have two π‘₯ one cubed plus six β„Žπ‘₯ one squared plus six β„Ž squared π‘₯ one plus two β„Ž cubed plus 30 minus two π‘₯ one cubed plus 30 all divided by β„Ž. And if we remove the brackets, we can see that the final two terms in our numerator are both negative. This negative two π‘₯ one cubed cancels with the positive two π‘₯ one cubed at the beginning. And we can cancel the positive with the negative 30. We’re left then with six β„Žπ‘₯ one squared plus six β„Ž squared π‘₯ one plus two β„Ž cubed all over β„Ž.

Now notice we have a common factor of β„Ž in the numerator, which we can cancel with the β„Ž in the denominator. And this leaves us with our average rate of change function six π‘₯ one squared plus six β„Žπ‘₯ one plus two β„Ž squared. And since 𝐴 is a function of β„Ž, we rearrange with the highest exponent of β„Ž as the lead term.

For 𝑓 of π‘₯ is equal to two π‘₯ cubed plus 30, the average rate of change function when π‘₯ is equal to π‘₯ one is, therefore, 𝐴 of β„Ž is equal to two β„Ž squared plus six β„Žπ‘₯ one plus six π‘₯ one squared.

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