|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''symmetric derivative''' is an [[Operator (mathematics)|operation]] related to the ordinary [[derivative]].
| | The name of the author is Numbers but it's not the most [http://www.thebody.com/h/list-of-curable-stds.html masucline] name out there. The thing she adores most is body building and now she is attempting std testing at home to make money at home std testing with it. North Dakota is her beginning [http://Www.youtube.com/watch?v=TLA6FAsXV9Q location] but she will have to transfer 1 working day or an additional. Managing individuals has [http://www.gaysphere.net/blog/167593 over the counter std test] been [http://tomport.ru/node/19667 over the counter std test] his working day occupation for a while.<br><br>Feel free to visit my website: [http://www.beasts-of-america.com/beasts/groups/curing-your-candida-how-to-do-it-easily/ www.beasts-of-america.com] |
| | |
| It is defined as:
| |
| | |
| :<math>\lim_{h \to 0}\frac{f(x+h) - f(x-h)}{2h}.</math>
| |
| | |
| A function is '''symmetrically differentiable''' at a point ''x'' if its symmetric derivative exists at that point. It can be shown that if a function is [[differentiable function|differentiable]] at a point, it is also symmetrically differentiable, but the converse is not true. The best known example is the [[absolute value]] function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. It can also be shown that the symmetric derivative at a point is the mean of the one-sided derivatives at that point, if they both exist.
| |
| | |
| ==Examples==
| |
| 1. The [[modulus function]],<math>f(x)= \left\vert x \right\vert</math> <br />
| |
| For [[absolute value function]], or the [[modulus function]], we have, at <math>x=0</math>,
| |
| :<math>\begin{matrix} | |
| \\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{\left\vert h \right\vert - \left\vert -h \right\vert}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{h-(-(-h))}{2h} \\
| |
| \\ f_s(0)= 0 \\
| |
| \end{matrix}</math>
| |
| | |
| only, where remember that <math> h>0 </math> and <math> h\longrightarrow 0</math>, and hence <math>\left\vert -h \right\vert</math> is equal to <math>-(-h)</math> only! So, we observe that the symmetric derivative of the modulus function exists at <math>x=0</math>,and is equal to zero, even if its ordinary derivative won't exist at that point (due to a "sharp" turn in the curve at <math>x=0</math>).
| |
| [[File:Modulusfunction.png|thumb|center|Graph of the [[Modulus Function]] y=|x|. Note the sharp turn at x=0, leading to non differentiability of the curve at x=0. The function hence possesses no ordinary derivative at x=0. Symmetric Derivative, however exists for the function at x=0.]]
| |
| | |
| 2. The function <math> f(x)=1/x^2</math> <br />
| |
| For the function <math> f(x)=1/x^2</math>, we have, at <math>x=0</math>,
| |
| :<math>\begin{matrix}
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{f(0+h) - f(0-h)}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{f(h) - f(-h)}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{1/h^2 - 1/(-h)^2}{2h} \\
| |
| \\ f_s(0)= \lim_{h \to 0}\frac{1/h^2-1/h^2}{2h} \\
| |
| \\ f_s(0)= 0 \\
| |
| \end{matrix}</math>
| |
| | |
| only, where again, <math> h>0 </math> and <math> h\longrightarrow 0</math>. See that again, for this function, its symmetric derivative exists at <math>x=0</math>, its ordinary derivative does not occur at <math>x=0</math>, due to discontinuity in the curve at <math>x=0</math> (i.e. essential discontinuity).
| |
| [[File:Graphinversesqrt.png|thumb|center|Graph of y=1/x². Note the discontinuity at x=0. The function hence possesses no ordinary derivative at x=0. Symmetric Derivative, however exists for the function at x=0.]]
| |
| | |
| 3. The [[Dirichlet function]], defined as:
| |
| | |
| <math>f(x) =
| |
| \begin{cases}
| |
| 1, & \text{if }x\text{ is rational} \\
| |
| 0, & \text{if }x\text{ is irrational} | |
| \end{cases}</math>
| |
| | |
| may be analysed to realize that it has symmetric derivatives <math> \forall x \in \mathbb{Q}</math> but not <math>\forall x \in \mathbb{R}-\mathbb{Q}</math>, i.e. symmetric derivative exists for rational numbers bur not for irrational numbers.
| |
| | |
| == See also ==
| |
| * [[Symmetrically continuous function]]
| |
| | |
| == References ==
| |
| * {{cite book
| |
| | first= Brian S.
| |
| | last= Thomson
| |
| | year= 1994
| |
| | title= Symmetric Properties of Real Functions
| |
| | publisher= Marcel Dekker
| |
| | isbn= 0-8247-9230-0
| |
| }}
| |
| | |
| ==External links==
| |
| *[http://demonstrations.wolfram.com/ApproximatingTheDerivativeByTheSymmetricDifferenceQuotient/ Approximating the Derivative by the Symmetric Difference Quotient (Wolfram Demonstrations Project)]
| |
| *[http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function]
| |
| *[http://math.feld.cvut.cz/mt/txtb/4/txe3ba4s.htm Dirichlet function and its modifications]
| |
| *[http://www.wolframalpha.com/input/?i=dirichlet+function&a=ClashPrefs_*MathWorld.DirichletFunction- Dirichlet function-Wolfram Alpha]
| |
| | |
| [[Category:Differential calculus]]
| |
The name of the author is Numbers but it's not the most masucline name out there. The thing she adores most is body building and now she is attempting std testing at home to make money at home std testing with it. North Dakota is her beginning location but she will have to transfer 1 working day or an additional. Managing individuals has over the counter std test been over the counter std test his working day occupation for a while.
Feel free to visit my website: www.beasts-of-america.com