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Basis of Taylor's expansion of a function about a point

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Can anyone explain the basis of Taylor's expansion of a function about a point. A graphical illustration would be better.
 

taylor series graphical

Any function can be approximated by an infinite series. The greater the accuracy and the greater range of the independent variable required, the greater number of terms of the series are required.

If you restrict the independent variable, such as the region of one value, you can use fewer terms in the expansion for a given accuracy.
 

Re: Taylor series

Thanks!
Would you please cite a graphically representation how a taylor series approximates a function.
 

Re: Taylor series

Essentially, the tylor expansion of function at one point is to equate the derivatives of the function at that point up to certain degree to those of a polynomial. If the function and the polynomial are equal at the point up to zeroth degree, it means they only have the same value. Likewise, if they are equal up to the first degree, then they have the same tangent while if they are equal up to second degree, they have the same curvature, and so on. You can draw pictures based on these.
 

Re: Taylor series

This is the way I visualize functions: I draw them in Excel.
Here is a representation of the sin(x) function around zero, together with approximations given by the Taylor series, truncated after the first, second, third terms. Note how the values used to plot the chart approach the true value of the function, as the number of terms increases. The straight line approximation leads to errors rather quickly, but the others are not so bad. The chart is from -1 to +1 radian. As you can see, the last curve approximates the function pretty well, and not only around zero.
 

Re: Taylor series

For simplcity consider only a taylor expansion truncated after the first derivative.


You perform the taylor expansion about a given point Xo.

This means we are approximating the function (curve) at Xo with a straight line with a slope equal to the derivative at Xo.

Draw any curve , and draw a tangent line to some point on the curve. The tangent line will approximate the curve well for regions around the tangent point. If you want it to be more accurate for a wider range then you can include more terms of the taylor expansion.
 

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