Proving the Derivative Function for a Differentiable Function on an Interval

In summary, the conversation discusses a differentiable function f defined for all x>=0 with certain conditions. Part a asks to find the integral of f from 0 to 3, which is equal to 6. Part b asks to prove a given equation involving f and its derivative, while part c uses this equation to find f.
  • #1
tdwp
4
0

Homework Statement


Let f be a differentiable function defined for all x>=0 such that f(0)=5 and f(3)=-1. Suppose that for any number b>0, the average value of f(x) on the interval 0<=x<=b is (f(0)+f(b))/x

a. Find the integral of f(x) from 0 to 3.
b. Prove that f'(x)=(f(x)-5)/x for all x<0.
c. Using part b), find f(x)

Homework Equations


(b-a)(f((ave)x))= the integral of f(x) from a to b

The Attempt at a Solution


Part a is easy, I got 6 as my answer. I'm completely at a loss on how to do part b/c. If anyone would at least point me in the right direction, I would greatly appreciate it.
 
Last edited:
Physics news on Phys.org
  • #2
Suppose that for any number b>0, the average value of f(x) on the interval 0<=x<=b is (f(0)+f(b))/x
Either you copied out the question wrong or the question asked doesn't make sense.
 
  • #3
Argh, yes I did copy it wrong. It should be (f(0)+f(b))/2.
 
  • #4
Use your "relevant equations" together with the given formula for the average of f over [0,b] to get an equation. Differentiate both sides of the equation.
 

FAQ: Proving the Derivative Function for a Differentiable Function on an Interval

What is the Mean Value Theorem of Integrals in Calculus?

The Mean Value Theorem of Integrals is a fundamental theorem in calculus that states that if a function is continuous on a closed interval, then there exists a point within that interval where the average value of the function is equal to the function's value at that point.

How is the Mean Value Theorem of Integrals used in real-world applications?

The Mean Value Theorem of Integrals is used in various fields such as physics, economics, and engineering to find the average value of a function over a given interval. It is also used in optimization problems to find the maximum or minimum value of a function.

What is the difference between the Mean Value Theorem of Integrals and the Mean Value Theorem of Derivatives?

The Mean Value Theorem of Derivatives applies to a function's derivative, while the Mean Value Theorem of Integrals applies to the function itself. The Mean Value Theorem of Derivatives states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point within the interval where the slope of the tangent line is equal to the average rate of change of the function.

How is the Mean Value Theorem of Integrals related to the Fundamental Theorem of Calculus?

The Mean Value Theorem of Integrals is a key component of the Fundamental Theorem of Calculus. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral of the function on that interval can be evaluated by subtracting the antiderivative at the endpoints.

Is the Mean Value Theorem of Integrals applicable to all functions?

No, the Mean Value Theorem of Integrals is only applicable to continuous functions on a closed interval. If a function is not continuous on the interval, then the theorem does not hold. Also, if the function is not integrable on the interval, then the theorem does not apply.

Similar threads

Non-Differentiable Function proof
Replies
8
Views
788
Proving piecewise function is k-differentiable
Replies
5
Views
1K
Linearly independent functions with identically zero Wronskian
Replies
1
Views
995
Proof given ##x < y < z## and a twice differentiable function
Replies
8
Views
1K
Show function series involving arctan is not differentiable at x=0
Replies
26
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
Replies
6
Views
1K
Derivative Problem: f’(1), f’(2), f’(3), and f’(5)
Replies
11
Views
774
Help me prove integral answer over infinitesimal interval
Replies
9
Views
1K
A few questions to make sure I understand Fourier series
Replies
3
Views
1K
A question regarding continuous function on a closed interval
Replies
9
Views
962

Hot Threads

Recent Insights

Back
Top