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Eulers Number - Differential Equations in Action

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    So now that we now that the slope of two to the power of X at X equals zero is slightly less than one,
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    and the slope of three to the power of X at X equals zero is slightly larger than one,
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    so it's not hard to imagine that somewhere in between two and three, there is a number called E--
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    Euler's number--so that E to the power of X has a slope of exactly one at X equals zero.
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    This function is called The Exponential Function. Whereas two to the power of X and three to the power of X are just exponential functions.
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    As this exponential function, The Exponential Function is so important, it gets a special name;
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    It's not just E to the power of X. It's called exp of X. This is the natural exponential function.
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    And we already know that its basis, E, is a number between two and three.
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    Now we can start from this idea and compute Euler's number.
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    E is nothing but E to the first power. Great. Let me write this one as 10,000 divided by 10,000.
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    Up to now, this looks completely crazy. Where am I going? I'm going to apply the laws for powers.
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    This is E to the one over 10,000 to the power of 10,000. Remember that rule--if you take a power of a power,
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    you multiply the exponents. This is what you get. But E to the power of one over 10,000,
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    that's something pretty close to X equals zero on this blue curve.
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    So rather than looking up the value on that blue curve, I'm using the red line, the tangent line, to compute that value,
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    to approximate that value, that is. The slope is one, so we can just put X in here, plus the Y intercept is one as well.
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    And now I'm using this equation for the tangent line to approximate this value.
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    The value of E to the X at X equals one over 10,000. So this is approximately equal to one over 10,000 plus one.
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    We simply looked up that value using the tangent line. Let me plug that in. Then we have E is approximately equal to--
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    I changed the order here to make it look nicer--one plus one over 10,000, to the power of 10,000.
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    Guess what? We have an equation, well, an approximate equation, for Euler's number.
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    The official definition for Euler's number is the following: E equals the limit of N goes to infinity of one plus one over N
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    to the power of N. I hope that you can now easily guess where this comes from.
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    The larger this number gets, the better these approximations get. And the result is 2.7 something.
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    Footnote: This equation is pretty elegant when it comes to deriving E, but when it comes to computing the actual value of E,
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    this equation is pretty inefficient.
Eulers Number - Differential Equations in Action
Video Language:
CS222 - Differential Equations
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