Proof Using Mean Value Theorem

In summary, The Mean Value Theorem can be used to prove that the derivative of the indefinite integral is f(x). The definition of derivative is F'(x) = (F(x+Δx) - F(x)) / Δx, and the definition of the indefinite integral is F(x) = ∫ f(x) dx. Therefore, plugging in the definitions, the derivative of the indefinite integral is equal to f(x), which is the anti-derivative of the function.
  • #1
courtrigrad
1,236
2
Hello all

Using the Mean Value Theorem, prove that the derivative of the indefinite integral [itex] \int f(x) \ dx [/itex] is [itex] f(x) [/itex]

So do I just use the fact that [itex] \int^b_a f(x) \ dx = f(\xi)(b-a) [/itex]?

Thanks
 
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  • #2
is this right?
 
  • #3
You CAN'T use the mean value theorem to prove the Fundamental Theorem of Calculus.
 
  • #4
not prove but maybe show
 
  • #5
hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer
 
  • #6
vincentchan said:
hey... do you forget our rule here... don't give out the answer... delete the link and give him some hints lead to the answer

Yes,you're right sorry about that.
 
  • #7
I believe your question should to prove that the derivative of this function:

[tex]F(x)=\int_a^{x} f(t)dt[/tex]

is f(x). Am I right? The above is a definite integral. The derivative of the indefinite integral is just f(x) by definition. Indefinite integral means anti-derivative.

What is F'(x) from first principles i.e: using the definition of derivative?
 
  • #8
[tex] F'(x) = \frac{F(x+\Delta x) - F(x)}{\Delta x} [/tex]

forgot to put limit
 
Last edited:
  • #9
courtrigrad said:
[tex] F'(x) = \frac{f(x+\Delta x) - f(x)}{\Delta x} [/tex]

forgot to put limit

Careful... we're looking for F'(x) not f'(x).
 
  • #10
is this right?
 
  • #11
courtrigrad said:
is this right?

Yes. So now use the definition of F(x) in post #7, to plug in the approriate F(x) and F(x+deltax) into your derivative equation...
 

What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints of the interval.

How is the Mean Value Theorem used in proofs?

The Mean Value Theorem is often used in proofs to show that a function has a critical point or to prove the existence of a solution to a problem. It is also frequently used in conjunction with other theorems, such as the Intermediate Value Theorem, to prove more complex results.

Can the Mean Value Theorem be applied to all functions?

No, the Mean Value Theorem can only be applied to functions that satisfy the conditions of the theorem, namely that they are continuous on a closed interval and differentiable on the open interval. If a function does not meet these criteria, the theorem cannot be used.

What is the difference between the Mean Value Theorem and the Rolle's Theorem?

The Mean Value Theorem is a more general version of Rolle's Theorem. While Rolle's Theorem states that there exists at least one point in the interval where the derivative of the function is equal to zero, the Mean Value Theorem states that there exists at least one point in the interval where the derivative of the function is equal to the average rate of change over the entire interval. In other words, Rolle's Theorem is a special case of the Mean Value Theorem.

What are the practical applications of the Mean Value Theorem?

The Mean Value Theorem has many practical applications, particularly in physics and engineering. It is used to calculate rates of change, to optimize functions, and to prove the existence of solutions to problems. It is also used in economics to analyze supply and demand curves and to calculate marginal utility.

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