Bounded Derivative of f(x) = xCos(x) for 0<= x<=5

In summary, a bounded derivative is a type of derivative with a finite limit as the interval of the function approaches a certain value. To find the bounded derivative of a function, one must take the derivative of the function using calculus rules and then determine if it is bounded using the definition. For the function f(x) = xCos(x) for 0<= x<=5, the bounded derivative is -sin(x) + xCos(x), which approaches a finite limit within the interval. The main difference between a bounded derivative and a regular derivative is that a bounded derivative has a finite limit at any point within the interval, while a regular derivative may have an infinite limit at certain points within the interval. The bounded derivative is important in mathematical analysis
  • #1
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Hi all, my first post so go easy on me! Doing some revision and have been tripped up on a really simple question!

Where f(x) = xCos(x)

Show the bound f '(x)<=6 is valid for 0<= x<=5

I suspect this is an easy solution using the intermediate value theorem!

Thanks in advance,

slippers
 
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  • #2
f'(x) = cos(x) + x*sin(x) <= 1 + 5*1 = 6
 
  • #3
Ahhh, something just moved in my brain...

Thanks very much! :D
 

1. What is the definition of a bounded derivative?

A bounded derivative is a type of derivative that has a finite limit as the interval of the function approaches a certain value. In other words, the derivative is not infinitely large or small at any point within the interval.

2. How do you find the bounded derivative of a function?

To find the bounded derivative of a function, you must first take the derivative of the function using the rules of calculus. Then, you can use the definition of a bounded derivative to determine if the derivative is bounded or not.

3. What is the bounded derivative of f(x) = xCos(x) for 0<= x<=5?

The bounded derivative of f(x) = xCos(x) for 0<= x<=5 is -sin(x) + xCos(x). This derivative is bounded because it approaches a finite limit as x approaches any value within the interval of 0 to 5.

4. How does the bounded derivative differ from the regular derivative?

The main difference between a bounded derivative and a regular derivative is that a bounded derivative has a finite limit at any point within the interval, while a regular derivative may have an infinite limit at certain points within the interval.

5. Why is the bounded derivative important in mathematical analysis?

The bounded derivative is important in mathematical analysis because it allows us to determine the behavior of a function at any given point within an interval. It also helps us to determine the continuity and differentiability of a function, which are essential concepts in calculus and real analysis.

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