Difinition of a derivative

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In summary: This is a piecewise function. f(x) has one rule when x is not zero (!=0), and another rule when x=0.wait, you just said that x!=0, how can "f(x) = 0 when x = 0" when you just said x can't equal zero?No, he didn't say that. He said that as long as x is NOT 0, you define f(x) to be x2 sin(1/x) (which obviously can't be correct for x=0) but that if x IS 0, f(0)= 0.
  • #1
UrbanXrisis
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let f be a function such that [tex]lim_{ h->x} \frac{f(2+h)-f(2)}{h} = 5[/tex]

Which of the following must be true?

I. f is continuous at x=2.
II. F is differentiable at x=2.
III. The derivative of f is continuous at x=2.

I know (I) is true because it can be differentiated
I know that (II) is ture to because the derivative was found so it can be differentiable.
I don't know if III is true because the it doesn't tell me the limit of f'(x)

are these correct?
 
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  • #2
That isn't the limit definition for a derivative unless x = 0, or maybe you typed it in wrong.
 
  • #3
UrbanXrisis said:
let f be a function such that [tex]lim_{ h->0} \frac{f(2+h)-f(2)}{h} = 5[/tex]

Which of the following must be true?

I. f is continuous at x=2.
II. F is differentiable at x=2.
III. The derivative of f is continuous at x=2.

I know (I) is true because it can be differentiated.I know that (II) is ture to because the derivative was found so it can be differentiable.
I don't know if (III) is true because the it doesn't tell me the limit of f'(x)

are these correct?

It's okay...At least,for me...I edited your typo and advise u to use the code

\lim_{...} for the limit and \rightarrow for the "->"...

Daniel.
 
  • #4
UrbanXrisis said:
let f be a function such that [tex]lim_{ h->x} \frac{f(2+h)-f(2)}{h} = 5[/tex]

Which of the following must be true?

I. f is continuous at x=2.
II. F is differentiable at x=2.
III. The derivative of f is continuous at x=2.
I think this must be:

[tex]\lim_{h\rightarrow{0}} \frac{f(2+h)-f(2)}{h} = 5[/tex] (ie. 0, not x)

I know (I) is true because it can be differentiated
I agree, with one caveat: We don't know the domain of h. If h can be positive or negative, f would be differentiable at 2 and, therefore, continous.

I know that (II) is ture to because the derivative was found so it can be differentiable.
True, subject to the above.
I don't know if III is true because the it doesn't tell me the limit of f'(x)
I think that if a function f(x) is differentiable at x, its derivative must be continuous at x.

AM
 
  • #5
Andrew Mason said:
I think that if a function f(x) is differentiable at x, its derivative must be continuous at x.
AM

Not true. Counterexample: Let f(x) = x2sin(1/x) for x != 0, and f(x) = 0 when x = 0. Then f is differentiable at zero, but the derivative is not continuous at 0.
 
  • #6
hypermorphism said:
Not true. Counterexample: Let f(x) = x2sin(1/x) for x != 0, and f(x) = 0 when x = 0. Then f is differentiable at zero, but the derivative is not continuous at 0.

wait, you just said that x!=0, how can "f(x) = 0 when x = 0" when you just said x can't equal zero?
 
  • #7
UrbanXrisis said:
wait, you just said that x!=0, how can "f(x) = 0 when x = 0" when you just said x can't equal zero?

This is a piecewise function. f(x) has one rule when x is not zero (!=0), and another rule when x=0.
 
  • #8
UrbanXrisis said:
wait, you just said that x!=0, how can "f(x) = 0 when x = 0" when you just said x can't equal zero?

No, he didn't say that. He said that as long as x is NOT 0, you define
f(x) to be x2 sin(1/x) (which obviously can't be correct for x=0) but that if x IS 0, f(0)= 0.
 

What is the definition of a derivative?

The derivative of a function at a given point is the slope of the tangent line to the graph of the function at that point.

How is the derivative of a function calculated?

The derivative of a function can be calculated using the formula: f'(x) = lim(h->0) (f(x+h)-f(x))/h, where h represents the change in x.

What does the derivative represent?

The derivative represents the rate of change of a function at a specific point. It can also be interpreted as the instantaneous rate of change or the slope of the function at that point.

Why is the derivative important?

The derivative is important because it allows us to analyze the behavior of a function and understand how it changes at a particular point. It is also used to solve optimization problems and to find the maximum and minimum values of a function.

Can the derivative be negative?

Yes, the derivative can be negative. This means that the function is decreasing at that point. A positive derivative indicates that the function is increasing at that point, while a derivative of zero means that the function is neither increasing nor decreasing at that point.

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