Derivative of a symetrical function

In summary, if f(x) is a symmetric function and C^{\infty}, then the terms of odd powers in the Taylor expansion will vanish. This can be proven by looking at f(-x) and using the fact that Taylor series are equal in an interval if and only if the coefficients are equal. Additionally, any odd derivative of an even function is an odd function, and for any odd function, f(0)=0. Rotational symmetry in odd functions means that rotating the graph of the function 180 degrees about the origin will result in the same graph.
  • #1
quasar987
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If f(x) is symetrical, then

[tex]f^{(n)}(x) =0, \ \ \ n=1,3,5,...[/tex]

What would be a proof of that?
 
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  • #2
ummm, no that doesn't make sense, does it. How about

If f(x) is symetrical and [itex]C^{\infty}[/itex], then the terms of odd powers in the Taylor expansion vanish.

What would be a proof of that?
 
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  • #3
Suppose that the coefficients of the odd terms are not all zero. Look at [itex]f(-x)[/itex] and use the fact that (convergent) Taylor series are equal in an interval iff the coefficients are equal.
 
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  • #4
Great, thank you Data.
 
  • #5
So you meant "even" instead of "symmetric"...The function is "even",its graph is "symmetric" wrt the vertically chosen Oy axis...:wink:

Daniel.
 
  • #6
One can also show that any odd derivative (1, 3, 5, etc.) of an even function is an odd function. Of course, for any odd function, f, f(0)= 0.
 
  • #7
And odd functions are also symmetric, rotationally,
 
  • #8
Yeah, what does rotational symmetry, or symmetry about the origin mean again? Thinking of graphs of odd functions I can remember (e.g. y = x3), the only thing I can think of is that it means rotating the graph of the function 180o about the origin, gives an image the same as the original.
 

Related to Derivative of a symetrical function

1. What is a symmetrical function?

A symmetrical function is a mathematical function that remains unchanged when its input is replaced with its negative counterpart. This means that if you reflect the graph of a symmetrical function across the y-axis, it will look exactly the same.

2. How is the derivative of a symmetrical function calculated?

The derivative of a symmetrical function is calculated by using the power rule, which states that the derivative of a power function is equal to the exponent multiplied by the coefficient, and then subtracting 1 from the exponent. In the case of a symmetrical function, the coefficient will be 0 since the function is unchanged when its input is negative.

3. Are all symmetrical functions differentiable?

No, not all symmetrical functions are differentiable. A function must be continuous in order to be differentiable, and there are some symmetrical functions that are not continuous at certain points.

4. Can a symmetrical function have a negative derivative?

Yes, a symmetrical function can have a negative derivative. This occurs when the function is decreasing on one side of the y-axis and increasing on the other side. The derivative will be negative on the decreasing side and positive on the increasing side.

5. What is the significance of the derivative of a symmetrical function?

The derivative of a symmetrical function is significant because it can help determine the behavior of the function at any given point. It can also be used to find the slope of the tangent line at a specific point on the function, which is useful in many applications such as optimization and curve fitting.

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