Evaluate where F(x) is differentiable

In summary, the conversation discusses a question about obtaining formulas and sketching graphs for f(x) and F(x). The missing definition of f(x) for x between 1 and 2 is mentioned and the possibility of differentiability at the joints is raised. The conversation also mentions using the definition of the derivative and suggests looking at one-sided limits. Finally, the relationship between f(x) and F(x) is questioned.
  • #1
sprite1608
2
0
Hi there, I cannot seem to figure this question out.

Homework Statement


Let f: [0,3] -> R be defined as follows

x if 0≤x<1,​
f(X)= 1≤x<2
x if 2≤x≤3​

obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where differentiable.I missed a day of class and am now totally lost. I've read through the sections in Intro to real analysis by Bartle that cover this section and I get no where. Any help on where to go would be greatly appreciated!
 
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  • #2
sprite1608 said:
Hi there, I cannot seem to figure this question out.

Homework Statement


Let f: [0,3] -> R be defined as follows

x if 0≤x<1,​
f(X)= 1≤x<2
You are missing the definition of f(x) for x between 1 and 2.

x if 2≤x≤3​

obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where differentiable.


I missed a day of class and am now totally lost. I've read through the sections in Intro to real analysis by Bartle that cover this section and I get no where. Any help on where to go would be greatly appreciated!
Hopefully, you know that f(x)= x is differentiable for all x. I suspect that the missing formula for x between 1 and 2 also defines a differentiable function. If that is the case, the only problem is whether the function is differentiable at the "joints", x= 1 and x= 2. Apply the definition of the derivative,
[tex]\lim_{h\to 0}\frac{f(a+h)- f(a)}{h}[/tex]
with a= 1 and then with a= 2. Look at the one sided limits.
 
  • #3
HallsofIvy said:
You are missing the definition of f(x) for x between 1 and 2.

shoot. I missed that when I checked it over. it is f(x) = 1 if 1≤x<2

With that information, would it really make much of a difference to what you said previously?
 
  • #4
sprite1608 said:
Hi there, I cannot seem to figure this question out.

Homework Statement


Let f: [0,3] -> R be defined as follows

x if 0≤x<1,​
f(X)= 1 if 1≤x<2
x if 2≤x≤3​

obtain formulas for F(x) = for 0≤x≤3 and sketch the graphs of f(x) and F(x). Where is F(x) differentiable? Evaluate F(x) where differentiable.

I missed a day of class and am now totally lost. I've read through the sections in Intro to real analysis by Bartle that cover this section and I get no where. Any help on where to go would be greatly appreciated!

You have f(x) and F(x). What relationship is being assumed between those two functions?
 

1. What does it mean for a function to be differentiable at a point?

When a function is differentiable at a point, it means that the derivative of the function exists at that point. This means that the function is smooth and has a well-defined slope at that point.

2. How do you determine if a function is differentiable?

A function is differentiable if its derivative exists at every point within its domain. This can be determined by taking the limit of the difference quotient as the change in x approaches 0. If the limit exists, the function is differentiable.

3. Can a function be differentiable at one point but not at another?

Yes, a function can be differentiable at one point but not at another. This can happen if the derivative exists at one point but is undefined or discontinuous at another point within the function's domain.

4. What are some common examples of functions that are not differentiable?

Some common examples of functions that are not differentiable include absolute value functions, step functions, and functions with sharp corners or breaks. These functions do not have a well-defined slope at certain points, making them non-differentiable.

5. Why is it important to know where a function is differentiable?

Knowing where a function is differentiable is important because it helps us understand the behavior of the function and how it changes over its domain. It also allows us to find the maximum and minimum values of the function, which can be useful in real-world applications.

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