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You are supposed to put (x+h)^(2/3) not (0+h)^(2/3) when finding the derivative using the definitions (f(x+h)f(x))/h.
Did you mean 'vertical' ?
Ok. And Example #2 does exactly that. Question #2. Yes, I did a mistake in the final step of calculating the limits. Sorry about that. It should be: But still, I have vs . Where did 2/3 disappeared? Question #3. Actually, I do not see why did you say that domain of the function is just positive numbers? I believe the domain is full R. After all, we can do : Not all graphing calculators are smart enough for such function. I am playing with QtiPlot and for "x^(2/3)" it draw the similar image as bob bundy showed. But if I rewrite formula into "(x^2)^(1/3)", I have a more correct graph. Which still has a strange horizontal part near the zero, which does not disappear when I zoom in.
Still, the f'(0) exists at that point.
hi Stefy,
Hi Bob
hi White_Owl
I don't know if substituting the value of directly into the limit is the right thingto do, but either way, your calculating of the limit is not correct in two ways. First, you reduced it incorrectly to h^(1/3) instead to h^(1/3). Second you claimed that the left and right limits of what you got aren't the same because they have the same absolute value, but different signs, whereas both limits have the same signed value 0 (which doesn't really have a sign, except when used in limits when denoting left and right limits).
The problem states: if g(x)=x^{2/3}, show that g'(0) does not exist. Now lets find left and right limits separately: From this we can see that left and right limits at point 0 are equal by an absolute value, but have opposite signs; therefore, the does not exists. But according to the rules of differentiation or in this case: My problem is: how correlate with ? They are both supposed to be equal to the slope of tangent line, aren't they? What am I missing? 