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- The Subtle Reason Taylor Series Work | Smooth vs. Analytic Functions
The Subtle Reason Taylor Series Work | Smooth vs. Analytic Functions
Learn about the subtle nuances of Taylor series and how they are used to approximate functions like e^x, sin(x), and cos(x) near specific input values in calculus. Discover the differences between smooth and analytic functions in this insightful video.
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thanks to Surf shark VPN for supporting
this video more on them later how do you
calculate functions like e to the X sin
of X or cosine of x at any given x value
like how might your calculator do it
this is a very old and classic question
and if you taking a calculus course you
might already know the very common
technique for doing this that I'm about
to show but even if you do please stick
with me because there's a Nuance to it
we're going to explore that you may not
have thought about the basic idea is to
approximate the function near an input
value where the function and its
derivatives are easy to calculate let's
take e to X as our example function e to
the x is easy to calculate at xal 0
since anything to the0 power is 1 and if
you know a little calculus you might
know that all the derivatives of e to
the X are just copies of itself so all
its derivatives are also easy to
evaluate at x equal 0 and they're all
equal to 1 using this information about
e to the X near xal 0
we can approximate the value of e to the
X near there we can start with a linear
approximation approximating e to the X
near xal 0 with a linear function this
amounts to finding the formula for the
tangent line of the curve at xal 0 since
a tangent line gives the best possible
linear approximation for a function
around the point of tangency since it
matches two properties of the curve at
that point its value or height above the
x-axis and its slope or derivative there
in other words the best linear
approximation of a function at a point
is one whose value and derivative both
match those of the function at that
point now the value of e to the X at xal
0 is just 1 and the same goes for its
derivative so the ideal linear
approximation for E to X near xal 0 is
just the function 1 +
x from here we can improve the
approximation by using a higher degree
polinomial a linear approximation is
