Difference between first and second fundamental theorem of calculus In first fundamental theorem of calculus,it states if A(x)=int^x_a f(t) dt then A′(x)=f(x).But in second they say int^b_a f(t)dt=F(b)−F(a),But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?

tonan6e 2022-10-02 Answered
Difference between first and second fundamental theorem of calculus
In first fundamental theorem of calculus,it states if A ( x ) = a x f ( t ) d t then A ( x ) = f ( x ).But in second they say a b f ( t ) d t = F ( b ) F ( a ),But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Bestvinajw
Answered 2022-10-03 Author has 15 answers
They have different assumptions.
In the first part you mentioned, f is assumed to be continuous. In the second part, f can be assumed only Riemann integrable on the closed interval [a,b]. When f is continuous, the second part indeed follows from the first part.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-08-11
fundamental theorem of calculus and second derivative
Using the fundamental theorem of calculus, find the second derivative of
( x ) x x e t d t
I've looked up the theorem on wikipedia but I can't really see what I'm meant to do.
asked 2022-09-23
  1. 0π/4sin4tdt{"language":"en","toolbar":"                      "}
  2. -11xndx
asked 2022-09-29
The notation of the second fundamental theorem of Calculus
I am self studying calculus, and just finished the lesson on the second fundamental theorem of calculus.
the way the theorem is described is:
d d x ( a x f ( t ) d t ) = f ( x )
and it was told that the meaning is that the derivative of an integral of a function is the function itself.
I don't get how you can get that from this. the expression that I would think suggests this is:
d d x ( f ( x ) d t ) = f ( x )
so the derivative of an indefinite integral (as oppose to integrating over a range) of a function is the function itself.
another interpretation of the FToC2 I read here, is that it means that the derivative of the functions that gives the area under the curve of a different function is the different function. this is also something I don't understand how the FToC2 suggests of?
to me, it seems like what this means:
d d x ( a x f ( t ) d t ) = f ( x )
is how a very small change in x affects that area under f(t) between a (a constant) and x. how do I get from that to the right interpretation?
asked 2022-09-03
Second Fundamental Theorem of Calculus...
Let f have a continuous second derivative. Prove that
f ( x ) = f ( a ) + ( x a ) f ( a ) + a x ( x t ) f ( t ) d t .
Here is my attempt at the problem.
Since f has a continuous second derivative, then the first derivative is also continuous. Therefore, by the first fundamental theorem of calculus, we have that
f ( x ) = f ( a ) + a x f ( t ) d t .
Expanding out the right-hand side of the above using integration by parts, we see that
f ( x ) = f ( a ) + f ( t ) t a x t f ( t ) d t .
This is where I am confused.
asked 2022-11-02
Second Fundamental theorem of calculus
Theorem 3.20 (Second Foundamental Theorem of Calculus)
Let f be a continuous function on [a,b] and F any function on [a,b], differentiable on (a,b), continuous on [a,b] such that F ( x ) = f ( x ) for all x ( a , b ).
Then
a b f ( x ) d x = F ( b ) F ( a )
I need to use the second Fundamental theorem of calculus to work out:
0 π 8 tan ( 2 x ) d x
Firstly it is clear that tan(2x) is continuous on [ 0 , π 8 ]
Now F ( x ) = 1 2 ln | cos ( 2 x ) | = 1 2 ln cos ( 2 x )
where F ( x ) = f ( x )
To show that F(x) is differentiable x ( 0 , 1 ) is it enough to say that as f(x) is continuous on (0,1) the derivative exists?
asked 2022-08-19

Give Arrhenius and Bronsted definition of acid and base. Why are Bronsted's definition more useful in describing acid base properties