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varygoode
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[SOLVED] Is |x|^3 differentiable?
Is [tex] |x|^3 [/tex] differentiable?
[tex] Def: \ Let \ f \ be \ defined \ (and \ real-valued) \ on [a,b]. \ \ For \ any \ x \in [a,b], \ form \ the \ quotient [/tex]
[tex]\phi(t)=\frac{f(t)-f(x)}{t-x} \ \ \ \ (a<t<b, \ t\neqx), \\ [/tex]
[tex] and \ define \\ [/tex]
[tex] f^{'}(x)=\lim_{\substack{t\rightarrow x}} \phi(t) [/tex]
Well, using the definition, I calculated that the right-hand limit and left hand limit are different. But I'm not sure if that means anything or what I can conclude here. Nor am I sure how I should define the left and right limits here.
Any help would be great, thanks!
Homework Statement
Is [tex] |x|^3 [/tex] differentiable?
Homework Equations
[tex] Def: \ Let \ f \ be \ defined \ (and \ real-valued) \ on [a,b]. \ \ For \ any \ x \in [a,b], \ form \ the \ quotient [/tex]
[tex]\phi(t)=\frac{f(t)-f(x)}{t-x} \ \ \ \ (a<t<b, \ t\neqx), \\ [/tex]
[tex] and \ define \\ [/tex]
[tex] f^{'}(x)=\lim_{\substack{t\rightarrow x}} \phi(t) [/tex]
The Attempt at a Solution
Well, using the definition, I calculated that the right-hand limit and left hand limit are different. But I'm not sure if that means anything or what I can conclude here. Nor am I sure how I should define the left and right limits here.
Any help would be great, thanks!