Find f(0), f'(0) Calculating Homework Answers

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In summary, if f(x) is a continuous and differentiable function and f(1/n)=0 for all nεN, then the correct answer is b) f(0)=f'(0)=0 as the limit of the derivative at 0 is 0.
  • #1
utkarshakash
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Homework Statement


If f(x) is a continuous and differentiable function and f(1/n)=0, for all nεN, then
a)f(x)=0 for all x ε (0,1]
b) f(0)=f'(0)=0
c) f'(0)=f"(0)
d)f(0)=0 and f'(0) need not be zero.


Homework Equations



The Attempt at a Solution


I would say d but the correct answer is b. Why should I believe that f'(0)=0 if f(0)=0?
 
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  • #2
If I instead told you that f(1/n) = 1/n, then f(0) = 0 still but the actual values of the sequence probably suggest the function looks a bit different near zero (can you guess what f'(0) will be?). You need to use the actual values of the sequence to show that the derivative is zero.
 
  • #3
Clearly f(0) = 0 because the limit as n->infinity is zero ...

Also it is zero at an unlimited number of points near zero; for every finite interval specified, no matter how small, there are an infinite number of zeros in that interval.

But it was given that the function was differentiable as well as continuous; so form the derivative as
f'(0) = lim n-> inf [f(0+1/n) - f(0)]/[1/n] ... but the numerator is zero, and so is f'(0).

Hence (b) is correct.
 
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  • #4
Another way of looking at it: 1/n is a sequence of points converging to 0. Since f is continuous, f(0)= lim f(1/n)= 0.

As UltrafastPED said, [f(0+ 1/n)- f(0)]/(1/n)= 0. Either the limit, as n goes to 0, does not exist or it is 0. Since we are told that the function is differentiable, the limit exists, so is 0, so f'(0)= 0.
 

1. How do I find f(0)?

To find f(0), you need to substitute 0 into the function wherever there is an x. This will give you the value of the function at x=0.

2. What does f'(0) mean?

f'(0) represents the derivative of the function at x=0. It is also known as the slope of the function at that point.

3. How do I calculate f'(0)?

To calculate f'(0), you need to take the derivative of the function and then substitute 0 for x in the resulting equation. This will give you the value of the derivative at x=0.

4. Can I use a calculator to find f(0) and f'(0)?

Yes, you can use a calculator to find f(0) and f'(0) if the function is given in a format that the calculator can understand, such as a polynomial or trigonometric function.

5. Why is finding f(0) and f'(0) important?

Finding f(0) and f'(0) allows us to evaluate the function at a specific point and understand the behavior of the function at that point, such as whether it is increasing or decreasing. It is also important for finding the equation of a tangent line and for solving optimization problems in calculus.

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