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Why does taylor series have ample amount of importance in calculus?

I like to know some insights behind taylor series.

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    $\begingroup$ You should distinguish between Taylor polynomials (local polynomial approximations to a finite order of a function which might only have a finite number of derivatives) and Taylor series (the limit of the sequence of Taylor polynomials of a function which has all its derivatives). These are really sufficiently different subjects that they should have separate questions. Which one do you want this to be about? $\endgroup$ – Ian Aug 22 '15 at 20:56
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Taylor series use derivatives everywhere (that is one reason why they involve calculus). It is interesting to know that values of a function $f(x)$ can be approximated using only derivative information at $x=0$. For a wide collection of functions, the approximation has increasing accuracy as more derivative information is used:

$$f(x) \approx f(0) + xf’(0) + \frac{x^2}{2}f’’(0) + … + \frac{x^n}{n!} f^{(n)}(0)$$

Example (recall that $e^0=1$): \begin{align} &e^{0.1} \approx e^0 + (0.1)e^0 = 1.1\\ &e^{0.1} \approx e^0 + (0.1)e^0 + \frac{(0.1)^2}{2}e^0 = 1.105\\ &e^{0.1} \approx e^0 + (0.1)e^0 + \frac{(0.1)^2}{2}e^0 + \frac{(0.1)^3}{3!}e^0 = 1.105166666666667\\ &e^{0.1} \approx 1 + .1 + \frac{(.1)^2}{2} + \frac{(.1)^3}{3!} + \frac{(.1)^4}{4!} = 1.105170833333333 \end{align}

The approximations are converging to the true answer $e^{0.1} = 1.105170918075648…$.

In the limit (in cases when the Taylor expansion applies) we get that $f(x)$ can be exactly written as an infinite sum of polynomial functions. This is useful for understanding $f(x)$ in simpler ways. For example, differentiating or integrating $f(x)$ can be done simply by differentiating or integrating each term, which involves differentiating/integrating polynomial functions, which is easy.


Let $f(t)$ be the location of a car at time $t$. If you know where the car is at time $t=0$, and you know its velocity $v(0) = f'(0)$, then you can accurately predict where it will be at some small time later: $$ f(t) \approx f(0) + f'(0)t = f(0) + v(0)t $$ If you know both the velocity and the acceleration $a(0)=f''(0)$, then you can more accurately approximate: $$ f(t) \approx f(0) + f'(0)t + \frac{f''(0)}{2}t^2 = f(0)+v(0)t + \frac{a(0)t^2}{2} $$ You can see how this would be useful, for example, in a robotics control system that needs to guess where the robot will be in one second given its current location, velocity, and acceleration.

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Polynomials are about the simplest type of functions there are. They are closed under addition, multiplication, integration, and differentiation.

Taylor series allow a function to be approximated in terms of polynomials of a specified degree. This gives a good indication of how the function behaves locally.

Note that the Taylor series of degree 1 is just the tangent.

The more terms of the Taylor series, the closer the series fits the function at the point where the series is computed.

This is a start. There is lots more.

Also look up Chebychev approximation.

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