Find derivative of floor function using limit definition of derivative?

In summary, the conversation discusses finding the derivative of a function using the limit definition. The attempt at a solution involves using the limit as h approaches zero of a specific equation. The question of whether a non-continuous function is differentiable is raised and it is determined that for a function to be differentiable, it must also be continuous. The function in question is discontinuous at every integer value, so the solution is to restrict the domain to intervals that do not include integers. It is also mentioned that the function is not differentiable at integer values.
  • #1
wills921
1
0

Homework Statement



I have been asked to find the derivative of f(x) = 0.39 + 0.24*floor(x-1) using the limit definition of a derivative. Is this possible?

Homework Equations





The Attempt at a Solution



The limit as h approaches zero of 0.24(floor(x+h-1)-floor(x-1))/h is as far as I have got. It seems the two floor functions will cancel out when substituting in 0 for h, but I'm stuck on how to get rid of the h in the denominator.
 
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  • #2
You can't substitute 0 in for h in the numerator if h is in the denominator.

Can you define a piecewise function for floor(a+b)?
 
  • #3
Is a non-continuous function differentiable?
 
  • #4
JonF said:
Is a non-continuous function differentiable?

If a function is differentiable (along an interval), it is continuous (along that interval). If it is continuous, it is not necessarily differentiable.
 
  • #5
Is this function continuous?
 
  • #6
JonF said:
Is this function continuous?

By definition, it has infinitely many step discontinuities.
 
  • #7
So we are left with two options:
We can say that function isn’t differentiable as an answer.

Or we can guess that the author of the problem meant for us to look at restrictions of the domain. So for what values is this function discontinuous?
 
  • #8
JonF said:
So we are left with two options:
We can say that function isn’t differentiable as an answer.

Or we can guess that the author of the problem meant for us to look at restrictions of the domain. So for what values is this function discontinuous?

It is discontinuous at every integer value...floor(1.9)=1, floor(2)=2, floor(2.1)=2, floor(2.9)=2, floor(3)=3, floor(3.1)=3...
 
  • #9
So let fa(x) be a restriction on the domain of f(x) to (a,a+1) i.e. the continuous intervals. Can you write fa(x) without a floor function?
 
  • #10
JonF is correct! Adding to what he said... if a function is differentiable on an interval is has to be continuous there.
Hence, since the original function is not continuous our recourse for getting a derievative is restricting the domain to not include integers.
 
  • #11
Let [itex]x_0[/itex] be a non-integer. Then there exist [itex]\delta> 0[/itex] such that the interval [itex][x_0- \delta, x_0+ \delta][/itex] contains only non-integers. If x and y are both in that interval, what is f(x)- f(y)?

And, of course, as other said the function is not differentiable at integer values.
 

What is the limit definition of derivative?

The limit definition of derivative is the mathematical process of finding the instantaneous rate of change of a function at a specific point. It is expressed as the limit of the difference quotient as the change in the input variable approaches zero.

What is the floor function?

The floor function, denoted as ⌊x⌋, is a mathematical function that rounds a real number down to the nearest integer. For example, the floor of 3.14 is 3, and the floor of -2.5 is -3.

How do you find the derivative of a floor function using the limit definition?

To find the derivative of a floor function, we first use the limit definition of derivative to calculate the difference quotient. Then, we take the limit as the change in the input variable approaches zero. Finally, we use the properties of the floor function to simplify the expression and find the derivative.

Can the derivative of a floor function be negative?

Yes, the derivative of a floor function can be negative. This occurs when the floor function has a decreasing slope at a specific point. For example, the derivative of the floor function at x=2 would be -1, as the function is decreasing at that point.

Is the derivative of a floor function always a whole number?

No, the derivative of a floor function is not always a whole number. The derivative can be a fractional or decimal value, depending on the input variable and the rate of change of the floor function at that point.

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