Fundamental theorem Definition and 171 Threads

In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.

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  1. L

    Definite Integral w/ Second Fundamental Theorem of Calculus

    Homework Statement Find F'(x) if F(x)=\int_{0}^{x^3}(\sin (t^2))dt The Attempt at a Solution Here's what I did: F(x)= -\cos (t^2)\biggr]^{x^3}_{0} and I get: F(x)= -\cos (x^6) +1 F'(x)= sin (x^6)(6x^5) However, the book's answer is F'(x)= 3x^2 \sin(x^6) How did...
  2. E

    Fundamental theorem of algebra

    Homework Statement There are two versions of the fundamental theorem of algebra, one that says a polynomial of degree n has n roots and the other that says a polynomial can be factored into linear and irreducible quadratic factors. Is there a quick way to see how they are the same...
  3. F

    Fundamental theorem of calculus

    Homework Statement Find the derivative of cost/t (dt) evaluated from 3 to the sqrt of x The Attempt at a Solution using the fundamental theorem of calculus, I have [cos(sqrt x)]/(sqrtx) I know I have to use the chain rule but I don't know how to go about it from here. Any tips?
  4. I

    First fundamental theorem of calculus

    how do i interpret this geometrically F^\prime(x) = \lim_{h\to 0} \frac{F(x+h) - F(x)}{h} = f(x) that the change in area with respect to change in "width" is the "height"? i don't think that's right.
  5. K

    Fundamental theorem of calculus

    I have a big test tomorrow and as I was reviewing, I encountered the following confusions. I hope that someone can help me out. I really appreciate for your help!:smile: 1) http://www.geocities.com/asdfasdf23135/cal0007.JPG The answer is NO. But when I differentiate both sides and...
  6. T

    Factoring operator and fundamental theorem of algebra

    I haven't taken any abstract algebra course so I do not know if this is the right section to post this question in. Anyway, I am learning differential equation right now and my prof. recently showed factorization of differential operator. For example, let D be the differential operator, he...
  7. U

    How Do I Determine the Quantity on a Graph Using Fundamental Theorem?

    http://img521.imageshack.us/img521/4549/bbav9.png The problem states that it wants the upper and lower estimate of total distance. Therefore, I used rectangles to solve for it. However, let's say I'm working on upper limits. For my initial rectangles, I use the right endpoints, but then it...
  8. T

    Fundamental Theorem of Calculus

    http://img527.imageshack.us/img527/8089/fr2rl4.gif I know part a is the fundamental theorem of calculus, but I am not quite sure how to manipulate the integral to find part i or part ii. Part b is again the fundamental theorem of calculus, but I am having a hard time solving for the...
  9. B

    Why doesn't the derivative of an integral give the value at the lower limit?

    Can someone explain a concept to me? The derivative of an integral ( whose lower limit is a real constant and whose upper limit is the variable x), is the intergrand evaluated at x as per the FTofC. I always thought about this as the limit of the integral as x approached the lower limit becuase...
  10. B

    Part 2 of the Fundamental Theorem of Calculus

    Hello, I have a problem that I am getting stuck simplifying further. The problem asks me to find the integral if it exists using Part 2 of the FTC. I know that the second part of the FTC says:\int_{a}^{b} f(x)dx = F(b) - F(a) Where F is the anti-derivative of f. Here is the problem...
  11. K

    Directed Integration and the Fundamental Theorem of Calculus

    I'm still new to much of this stuff, so I do not claim to be an expert. But I thought I'd still comment. I'm trying to better understand the geometric calculus of David Hestenes: http://modelingnts.la.asu.edu/ A fairly common form for the Fundamental Theorem of Calculus is: \int_S...
  12. J

    Fundamental theorem of calculus

    (that's a 3 on the last integral) http://img131.imageshack.us/img131/2549/jesus1cj.png I need to find which of those are true, now I thought I and III were true for sure. But when I do II with an example f(x) = x^2 I get x^2 - 9, so it's not true right? (I and III are not choices given for the...
  13. D

    The Fundamental Theorem of Calculus

    Id like to know if the following argument is valid. Take an arbitrary function f(x). f(x)dx can be thought of an infinitesimal area of a certain form (I emphasise this because I use it later in the argument) determined by the form of the function f(x). Let's denote its integral by Y. \int{...
  14. M

    Applying the fundamental theorem of calc to calc III, confused

    Hello everyone, i have the following problem I'm confused about! Can anyone guide me to what I'm suppose to do? I tried the following but it was wrong, he then told me I can just apply the theorem to |r'(u)| insteed of the stuff under the square root which would be more difficult. This is...
  15. C

    An extension of the Fundamental Theorem of Calculus

    In a book of Introduction to Probability I found this statement: " Let be F(x) = \int_{-\infty}^{x} f(t)dt. Then, by the Fundamental Theorem of Calculus, F'(x) = f(x)." With the minus infinity on the lower limit, it is this a valid aplication of the FTC? Thanks.
  16. mattmns

    Fundamental Theorem of Line Integrals

    Quick question, what is the approach to this problem? Keep in mind I am supposed to use the Fundamental Theorem of Line Integrals. \int_{C} 2ydx + 2xdy Where C is the line segment from (0,0) to (4,4). Unless I am missing something I need to make that into the form of \vec{F} \cdot...
  17. C

    Fundamental Theorem of Calculus

    Could you check whether I am doing these questions right: 1. \int_{0}^{4} (2+x) dx . So I use the Fundamental Theorem of Calculus F(b)-F(a) and receive: \frac{(x+2)^{2}}{2} = F(4) - F(0) = 16 2. \int_{-1}^{1} (4t^{3} - 2t) dt = t^{4} - t^{2} = F(b)-F(a) = 0 3. \int_{0}^{3}...
  18. P

    Fundamental Theorem of Calculus

    Question :\int_0^{49pi^2} (sin(sqrt(x))/(sqrt(x)) dx should i just solve it as a regular integral like usally and then do F(b) - F(a)? if so, why is it called Fundamental Theorem of Calculus if it's just like a regular integral?
  19. H

    From indefinite integral to fundamental theorem of calculus, how does it all link?

    Hi there, Can someone explain to me what the following are and how each one is used as a tool for the next one: 1)Indefinite integral 2)Riemann Sum 3)Definite Integral 4)Fundamental Theorem of Calculus(The part which says that the derivative of the integral of f(t)dt from a to x is...
  20. tandoorichicken

    Calculating Derivative of h(x) using Fundamental Theorem

    Hello everyone, its been a while. It's been almost 4 months since I did anything calculus related so I forgot all of my skills. :bugeye: The problem is: Use the Fundamental Theorem of Calculus to find the derivative of the function h(x) = \int_{2}^{\frac{1}{x}} \arctan{t} \,dt
  21. S

    2nd fundamental theorem of calculus

    Can some on pleases explain this too me. I have an AP book, and i am to do a few problems out of it for class, and but can't find it in there ANY WHERE. Any help would be superb! -Jacob
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