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


by irresistible
Tags: integrable, integral, prove
irresistible
irresistible is offline
#1
Feb7-09, 12:25 PM
P: 15
Hey guys,
Can you help me prove this?

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!
Phys.Org News Partner Science news on Phys.org
Going nuts? Turkey looks to pistachios to heat new eco-city
Space-tested fluid flow concept advances infectious disease diagnoses
SpaceX launches supplies to space station (Update)
shoehorn
shoehorn is offline
#2
Feb7-09, 12:40 PM
P: 448
Why is this in the Topology & Geometry forum?
irresistible
irresistible is offline
#3
Feb7-09, 12:46 PM
P: 15
sorry, i'm new here
I'm gonna post it over there and delete this one if possible

HallsofIvy
HallsofIvy is offline
#4
Feb7-09, 02:05 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,881

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


I'll move this to Calculus
HallsofIvy
HallsofIvy is offline
#5
Feb7-09, 02:18 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,881
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].


Register to reply

Related Discussions
proving limits of recursive sequences using definition Calculus & Beyond Homework 5
definition of integral Calculus 1
computer science proving big-O definition Calculus & Beyond Homework 6
Proving the definition of abs. value Introductory Physics Homework 2
help with proving limits using Epsilon-Delta definition Calculus 4