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This is problem number thirty three of this tour.
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Calculus. Eighth edition, Section two point eight.
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Party if ever Vex equals X to the fourth plus
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two. Ex find every crime of X king.
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Let's use the Internet derivative definition. Find the paramedics
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limit as h approaches zero of this function of X
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. You know, you wanted a explosive JJ to
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That means explosive quantity to the fourth power. Plus
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two times actress H they were gonna subtract the function
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of X X to the fourth US to X is
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all divided by just each. Okay, good.
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Next step is to expand every term. Here we
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have the binomial to the fourth power. Um and
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then she read some other numbers as well. So
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this binomial to the fourth power will be X to
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the fourth. Because for X cubed H plus six
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x squared each squared plus for X h cute plus
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each to the fourth. That's just this final meal
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to the fourth power. And we're going to distribute
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the to here to the X and age plus two
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x plus to age. And they were in a
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subject exit. The fourth and two exits one okay
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, and this is all over H I'LL take a
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look at the numerator and see what we can cancel
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out a positive excellent forth and a negative x a
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fourth to those go away a positive to X and
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negative tricks and then an agent A dominator will cancel
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with an ancient each of the terms in the numerator
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since each of the terms of numerator have at least
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one each king. So what we're left with is
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the limit as h purchase zero of for X cubed
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less six ax squared each plus for eggs each square
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pas age cubed plus two Now, as a chair
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purchase zero each of these age terms a purchase zeros
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for those No way and we should be love with
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for X cubed plus two And this will indeed be
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our derivative of paramedics. Now we will verify and
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party by chicken or answer with party um to make
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sure that it's reasonable comparing the graph of f end
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of crimes were going plot both FX excellent forthwith to
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X And if Prem of X for Cuba for X
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cube two plus two and see if it is,
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this answer is reasonable. So we have pulled up
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at the next, which is in blue, and
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the derivative, which is in green here with their
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functions that we were given in which we found.
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And let's discuss whether this makes it, since the
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function F of X is a something similar to a
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problem. So on the left side it's decreasing,
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and when it's decreasing, its slope is negative,
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so as a very large negative slope. But then
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it gets less negative as it gets close to this
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minimum point. So that's what we see here.
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The function, the green function, the dirt of
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function is mostly negative. Insolent reaches is until it
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keeps approaching Weikel zero. So purchase of slope of
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Zero exactly where the minimum is. So this is
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consistent, the slope of zero at the minimum,
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so that is correct afterward, of the soap increases
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, the slip is positive for the remaining part,
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and this is shown on the slope craft, the
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derivative crafting green and what we see here is that
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the slope increases and then stays the same for a
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little bit and then increases again. And that's what
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explains his behavior here. This soap is positive initially
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and increases and becomes more positive until a point here
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. Actually, the slip seems a little constant exactly
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equal to around two. So the slopes ese constant
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for a little bit. But then it starts increasing
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again. Until this derivative that we found Brinkley is
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consistent with our two cafs shown here.