WEBVTT
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we're told a resistor of big are owns is connected
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across the battery of big ive bolts with internal resistance
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. Little are owns and the power in Watts and
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the external resistor is given by the equation. Big
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P equals big e squared times big are over big
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r plus Little are squared. Mhm. Now if
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Big E and little are are fixed but big are
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varies were asked what the maximum value of our power
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is. First we'll find the derivative treating biggie and
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little ours constants and bigger is the variable. So
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we have a ratio. You have that p prime
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of big are is the bottom big R plus little
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r squared times the derivative the top e squared minus
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the top e squared big are times that are at
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the bottom which is to bigger Plus little are all
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over the bottom Bigger plus little r squared squared,
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which is big are possible Large Fourth, this simplifies
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Eventually Two big e squared little ar minus big e
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squared big are over big r plus little are cute
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So I skipped a few steps here. You might
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want to do these on your own. We want
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to find when this is equal to zero to find
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our critical values. Well, clearly, this is
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only equal to zero when the numerator is zero.
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And so when big R is equal to little are
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now we find that when bigger is less than little
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are Well, then we have that are derivative keep
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prime of big are well, this is greater than
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zero And when big R is greater than little are
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p prime of big art is less than zero.
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Therefore, it follows that at the first derivative test
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he has a maximum value at a big are equals
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Little are plugging in big are equals Little are we
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find the maximum power p of little are this is
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e squared times little are over Yeah, little R
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plus little r squared, which is the same as
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you squared over four Little are