WEBVTT
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this problem Number four of this to a calculus eight
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edition section two point seven. I find the slope
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of the tangent line to the curve. Why equals
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X minus X cubed or at the point one zero
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, using definition one and using the equation to definition
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one an equation to our shooting here for your reference
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. And we begin with the finish in one,
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the slope can be found by calculating the limit as
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experts, eh? Where in this problem is one
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, since that is the point of interest, that's
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the exploits of the point of interest. One zero
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. The X is a protein one of the function
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f of X, which is X minus X cubed
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, minus your function Evaluated, eh? So at
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AIDS one one minus one cube, well, that's
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just a zero. So the function remains Are the
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function is X minus X cubed minus the function value
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added a one zero on the right abi X minus
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one or mix minutes, eh? If we simplify
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this function in the numerator, we can factor out
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next and we're left with that one minus X squared
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. Well, go ahead and also factored this term
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because it's the difference affairs of squares into one minute
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. Sex one plus six. Um, and the
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numerator denominator is explain this one. We can go
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ahead and cancel this term and this term exhibit They're
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not exactly the same. There's going to be a
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leftover negative one. Um, after it is cancelled
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. Um, and then the next step will be
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to plug in er the value one here, so
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the answer would be a negative one times one,
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which is the sex multiplied by one plus one or
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two. And I can see it's an answer.
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Negative, too, for the soup. And that's
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using definition one. Using equation, too. We
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have a limited h of purchase, a serial of
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the function evaluated a A plus h where is one
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? So we're going to be in putting one plus
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age wherever there's an X. So it's one plus
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each, minus the quantity one plus h cubed.
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And we're going to be subjecting FAA, which we
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saw in this previous part that that term is zero
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and we're dividing by h. Okay, So in
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order to simplify, we're going to expand the generator
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one plus age minus on this term when it is
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when it is expanded or beyond. Plus, each
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times one plus X squared will That's one place to
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age plus eight squared. And if we combined those
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two in the next champ, the numerator will be
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ah, rim plus age minus one plus to age
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plus eight. Squared class, Age plus two h
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squared was h cubed all over h here in the
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numerator the term one and negative one Counsel, Um
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, and simplifying down here and combining like terms.
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We have a tch minus two age. That's negative
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. H ah, minus another age. Think of
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two each. That's the first term. And then
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we're left over with any squared in a two h
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squared. That's three. It's squared with the negative
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in front negative three x squared on and then finally
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left over with negative h cubed over each. Ah
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, now we plug in or we cancel out one
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h her turn here we're left with it's squared and
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then we plug in. H is zero. That
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cancels out these Final two terms, and we're left
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over with an answer negative to which agrees with the
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answer for definition. One for part B. Renee's
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a slope negative too. And we're going to write
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an equation of attention line with the point one zero
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. So we have a point and we have a
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slope. We're in his point slope form. Why
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I'm on this white one equals M times x twenty
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six one where the point is a war memento that
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zero to the point. It's ones here. A
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superstate of two and X is one. So our
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function is actually going to be y equals native to
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x Ah, plus two. And this is the
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equation of the tension line according to the information given
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a part of B finally in part see graft,
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the curve and the tangent line and successively smaller,
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reeling rectangles centered at one zero until the curve and
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the line appear to coincide. So we take the
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function, explain its execute, and we're also going
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to plot the tension line Negative too explicit. Two
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. And here we see an animation ah, zooming
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into the point one zero. So the orange function
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that is exponents x cubed this purple function. The
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linear function is the tension line Negative two x plus
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two and as we see here, it is tangent
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to the curve at one zero And as we successively
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Suman into the point one zero, the no attention
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lane and the curve coincide and become indistinguishable. So
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this is the purpose of this exercise in party.
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We have shown that this the line is indeed tangent
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at one zero and that is reason closer. The
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ah functions are index in distinguishing.