WEBVTT
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for this problem were given the function y equals 2/1
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plus e to the negative X, and we want
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to find the equation of the tangent line at the
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0.1 So remember to get the slope of the tangent
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line, you find the derivative. So we're going
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to use the quotient rule to find the derivative of
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the function. So what we have here is the
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bottom times, the derivative of the top, minus
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the top times, a derivative of the bottom.
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And we use the chain rule to find the derivative
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of E to the negative X. And then that's
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over the bottom squared. Next, we want to
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simplify that, and we want to plug in the
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number zero for X because we're finding the derivative at
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the 0.1 So a substitute in zero everywhere we have
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an X and remember that each of the zero is
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one, so that simplifies things quite a bit.
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And that gives us negative two times one times negative
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, 1/2 squared that simplifies to be 2/4. So
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that's 1/2. And remember, that is the slope
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of the tangent line. So now that we have
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the slope and we have the 0.1 We can use
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our point slope form of the equation of a line
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. Why minus y one equals M times X minus
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X one. We can substitute our slope in there
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and our 0.1 and that will give us the equation
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of the tangent line, distribute the 1/2 and then
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add one to both sides and we have y equals
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1/2 X plus one. So that takes care of
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part A and then for part B. What we
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want to do is illustrate by graphing the curve and
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the Tanja line on the same screen. So we
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grab a calculator, we go to y equals and
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we type in. Our function is why one and
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we type in our tangent line is why too.
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All right now we graph and I'm using zoom decimal
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number four and the blue one is the curve,
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and the red one is a tangent line. And
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here we have the cursor at the 10.1 so we
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can see that that line does appear to be tangent
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to the curve. At that point