WEBVTT
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So here is the power series for the tangent inverse
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of X function. So tangent inverse of ex uh
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is equal to x minus X cubed over three plus
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. Extra fifth over five minus x to the seventh
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over seven plus. An infinite more number of terms
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. Uh And this is the power series representation for
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tangent inverse affects as long as X is in the
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interval from negative 1 to 1. We want to
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find the power series representation for the uh Inverse tension
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of two x. And this is easily done.
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If tangent inverse of X equals this series, then
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tangent inverse of two X. Uh to find a
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series for this function simply substitute in two X everywhere
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you see X. So tangent inverse of two x
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Will be two x minus and then substitute two X
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and three X or two X. Two third over
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three plus two X to the 5th over five uh
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minus noticed plus uh And subtraction science keep alternating.
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So tangent inverse of two X uh minus two X
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to the seventh over seven plus. And of course
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the series goes on uh indefinitely. So now we
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just got to simplify uh two extra third, two
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extra fifth two x to the seventh. So we
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have two x minus law, two X to the
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third is really to to the third time's extra 32
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to 38. So we have eight over three times
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execute plus and then two X to the fifth,
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two to the fifth is 32. So we'll have
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32 X to the fifth over five. 32 X
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to the 5th Over five. And we can just
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put the 30 to 35 Uh-2 to the 7th
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. Double check in the calculator. Two to the
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seventh. Power is 1 28. So we're going
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to have 128 Times X to the 7th over seven
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plus. Obviously an infinite more number of terms.
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So this is the power series representation for detention inverse
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of two X. Now uh We can take partial
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sums of this series to approximate the tangent inverse of
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two x function. So as some with just one
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term would be two X. The some with two
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terms would be two x minus eight thirds. Execute
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if you wanted to use a some with three terms
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to approximate the function, then you just use the
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first three terms in the series. Um And likewise
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, if you want to use a some with four
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terms to approximate the tangent inverse of two X function
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. Use all four of these terms. So what
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we're going to do next is we're going to graft
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the tangent inverse of two X. Function on the
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graphing calculator. Using dez mo's. Then we're gonna
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grab some partial sums. Uh So let's use a
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some well graph uh Detainment inverse of two x function
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and then we'll graph partial stumps. So as to
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of ex uh simply means the partial some of this
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part of this power series using the first two terms
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. So we will graph two x minus eight thirds
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X cubed. And then uh we'll also graph Uh
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as three which is the partial sum using the first
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three terms of the series. So s three of
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x will be two X Uh-8/3 execute plus 32/5
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extra to 5th. So we're gonna graph the actual
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function tangent inverse of two X. And then we're
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gonna graph uh the partial sum from its power series
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partial sum using the first two terms and a partial
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sum using the first three terms of the power series
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. And our goal is to compare or to see
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how well these partial sums. Uh These partial some
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functions approximate uh the tangent inverse function more specifically as
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we use more terms in from the power series.
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Uh Do we get a better approximation uh to the
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actual function? Okay, so here using dez most
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currently you see the graph of the red graph is
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the graph of the function tangent inverse of two.
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Xd actual function uh defined for excess between negative one
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and positive one. Now, since our power series
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that we derived for the tangent inverse of two X
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function uh really is only defined for X values between
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negative one and one. Uh Here is the first
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two terms from the power series. And when we
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graph it we're going to make sure that we keep
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X restricted to the interval from negative 1 to 1
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and the same thing. When we take uh the
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third partial sum uh as three of X. Basically
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the power series, Using the first three terms.
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Once again, when we graph these three terms the
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first three terms of the power series, we're going
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to make sure that we keep X restricted between negative
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one And a positive one. So you're currently looking
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at this red graph is a graph of tangent inverse
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of two. Acts next I'm going to grab along
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with it. Uh Just um the partial some consisting
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of the first two terms of the power series,
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and I was actually actually pretty impressed just how close
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using just two terms from the power series approximates the
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actual graph. So take a look at this.
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Okay, um So the blue graph does a really
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good job of approximating the actual uh graph of the
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function in red. So the blue graph comes really
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close to the red graph at least for x values
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between negative one half and one half, it really
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kind of diverges away from it. Uh For excess
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between negative one and negative one half and one half
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Uh positive one. but for ex equal negative one
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half up to X equals positive one half. Using
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the partial some consisting of just two terms from the
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power series. Does a really good job of approximating
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the actual uh inverse tangent function at least on a
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portion of this interval. So now let's see how
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using the partial sum as three of ex uh some
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of the first three terms of the power series.
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Let's see how close uh three terms from the power
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series approximates. Uh The actual tangent universe function.
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Alright, this one does even a little bit better
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. Um So this green uh graph that you see
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here is using three terms from the power series.
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So you can see that three terms from the power
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series. Uh does a really nice job of approximating
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uh the inverse tangent function represented by the red graph
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. Once again, it starts to divert away from
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it a little bit when X is less than negative
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one half and when X is greater than one half
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, but it stays closer to it using three terms
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um stays closer to the actual inverse tangent function than
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to to term sticks. Now, let's look at
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all three of them together. All right, so
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the red graph is the actual inverse tangent function.
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Do Blue graph is when we use two terms from
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the power series, and the green graph is when
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we used three terms from the power series. So
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, the more terms you use from the power series
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, the closer you get to the actual function or
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the actual graph of the actual function. So the
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green graph does a lot better job than the blue
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graph does of staying close uh to the inverse tangent
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function