Proving F'(x)= f(x) using the definition of integral?

In summary, the conversation discusses using the definition of integrals to prove a mathematical statement involving a differentiable function and integral. The use of the "partition" preposition and the "integral mean value theorem" are suggested as potential methods. The conversation also mentions moving the discussion to a different forum and provides a proof using the limit as h goes to 0.
  • #1
irresistible
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Hey guys,:smile:
Can you help me prove this?:confused:

Suppose that f:[a.b] -> R is integrable and that F:[a,b]->R is a differentiable function such thet F'(x)= f(x) for all x[tex]\in[/tex] [a,b].
Prove from the definition of the integral that;

F(b)-F(a) =[tex]\int[/tex] f(x) dx ( integral going from a to b)

I can prove this using the Fundamental theorem of calculus;however, this question specifically asks that we use the definition of integral to prove this:

I'm thinking that I have to use the "partition" prepositions to prove this.
Any ideas?
Thank you in advance guys!:wink:
 
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  • #2
Why is this in the Topology & Geometry forum?
 
  • #3
sorry, I'm new here
I'm going to post it over there and delete this one if possible
 
  • #4
I'll move this to Calculus
 
  • #5
Now: choose any given value for [itex]x_0[/itex]. [itex]F(x_0)= F(a)+ \int_a^{x_0} f(t)dt[/itex]. For any h> 0, [itex]F(x_0+ h)= \int_a^{x_0+h}f(t)dt[/itex] and [itex] F(x_0+h)- F(x_0)= \int_{x_0}^{x_0+ h} f(t)dt[/itex].

By the "integral mean value theorem" (that's where you use the definition of "integral"), there exist an [itex]\overline{x}[/itex], between [itex]x_0[/itex] and [itex]x_0+ h[/itex] such that [itex]\int_{x_0}^{x_0+ h} f(t)dt= f(\overline{x})((x_0+h)- x_0)= f(\overline{x}h[/itex]. Then [itex]F(x_0+h)- F(x_0)= f(\overline{x})h[/itex] and
[tex]\frac{F(x_0+h)- F(x_0)}{h}= f(\overline{x})[/tex]
Taking the limit as h goes to 0, since [itex]\overline{x}[/itex] must always be between [itex]x_0[/itex] and [itex]x_0+ h[/itex], [itex]f(\overline{x})[/itex] goes to f(x). That is, [itex]dF/dx[/itex], at [itex]x= x_0[/itex] is [itex]f(x_0)[/itex].
 
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FAQ: Proving F'(x)= f(x) using the definition of integral?

1. What is the definition of integral?

The definition of an integral is a mathematical concept that represents the area under a curve in a given interval. It is represented by the symbol ∫ and is used to calculate the total accumulated value of a function over a specific interval.

2. How is F'(x) related to the definition of integral?

F'(x) is the derivative of a function, which is the rate of change of the function at a specific point. When we use the definition of integral to prove that F'(x) = f(x), we are essentially showing that the derivative and the integral are inverse operations of each other.

3. What is the process for proving F'(x) = f(x) using the definition of integral?

The process involves breaking down the integral into smaller and smaller intervals, then using the limit definition of a derivative to calculate the derivative of each interval. After that, we add up all the derivatives to get the total derivative, which should equal the original function, f(x).

4. Why is it important to prove that F'(x) = f(x) using the definition of integral?

Proving this relationship is important because it helps us understand the fundamental concept of integration and its connection to differentiation. It also provides a rigorous way to solve problems involving integrals and derivatives.

5. Are there any limitations to using the definition of integral to prove F'(x) = f(x)?

Yes, there are some limitations. The function must be continuous and differentiable in the given interval for the proof to be valid. Additionally, the process can be time-consuming and complex for more complicated functions, so alternative methods may be used in those cases.

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