WEBVTT
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Find the average rate of change function π΄ of β of π of π₯ is equal to two π₯ cubed plus 30 when π₯ is equal to π₯ one.
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In general, the average rate of change function π΄ of β for a function π of π₯ at a point π₯ is equal to π₯ one is given by the expression π΄ of β is equal to π of π₯ one plus β minus π of π₯ one divided by β, where β is a small change in π₯.
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Weβve been given the function π of π₯ is two π₯ cubed plus 30.
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And to find the average rate of change function π΄ of β for our function π of π₯ when π₯ is equal to π₯ one, we need to evaluate our function when π₯ is equal to π₯ one plus β and when π₯ is equal to π₯ one.
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And we can then substitute these into our expression for π΄ of β.
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To evaluate π at π₯ is equal to π₯ one plus β, we replace π₯ with π₯ one plus β in our function π.
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And this gives us π of π₯ one plus β is equal to two times π₯ one plus β cubed plus 30.
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Expanding π₯ one plus β cubed, we have π₯ one plus β multiplied by π₯ one squared plus two βπ₯ one plus β squared, that is, π₯ one cubed plus three βπ₯ one squared plus three β squared π₯ one plus β cubed.
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So that our function π evaluated at π₯ is equal to π₯ one plus β is two times π₯ one cubed plus three βπ₯ one squared plus three β squared π₯ one plus β cubed plus 30.
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To make things a little simpler when we use our average rate of change function, weβve multiplied out the bracket.
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So we have π evaluated at π₯ is equal to π₯ one plus β is equal to two π₯ one cubed plus six βπ₯ one squared plus six β squared π₯ one plus two β cubed plus 30.
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To use our average rate of change function, we also need π evaluated at π₯ is equal to π₯ one.
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And thatβs equal to two times π₯ one cubed plus 30.
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So now if we substitute π of π₯ one plus β and π of π₯ one into our average rate of change function, we have two π₯ one cubed plus six βπ₯ one squared plus six β squared π₯ one plus two β cubed plus 30 minus two π₯ one cubed plus 30 all divided by β.
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And if we remove the brackets, we can see that the final two terms in our numerator are both negative.
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This negative two π₯ one cubed cancels with the positive two π₯ one cubed at the beginning.
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And we can cancel the positive with the negative 30.
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Weβre left then with six βπ₯ one squared plus six β squared π₯ one plus two β cubed all over β.
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Now notice we have a common factor of β in the numerator, which we can cancel with the β in the denominator.
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And this leaves us with our average rate of change function six π₯ one squared plus six βπ₯ one plus two β squared.
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And since π΄ is a function of β, we rearrange with the highest exponent of β as the lead term.
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For π 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, π΄ of β is equal to two β squared plus six βπ₯ one plus six π₯ one squared.