What is Fundamental theorem: Definition and 171 Discussions

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. C

    The Fundamental Theorem for Line Integrals

    Homework Statement Determine whether or not f(x,y) is a conservative vector field. f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) > If F is a conservative fector field find F = gradient of f Homework Equations N/A The Attempt at a Solution Fx = -3e^(-3x)(-3)cos(-3y) Fy =...
  2. nicolauslamsiu

    Fundamental theorem of calculus

    Homework Statement Using Fundamental Theorem of Calculus to find the derivative 2. Homework Equations upper limit=x^2, lower limit=4x ∫ { 1 / [1+ (sin t)^2] }dt The Attempt at a Solution two independent variables are involved, how should i find the derivative? [/B]
  3. PcumP_Ravenclaw

    Proof of fundamental theorem of arithmetic

    Dear all, Please help me understand the proof by induction for only one way of expressing the product of primes up to the order of the factors. Please see the two attachments from the textbook "alan F beardon, algebra and geometry" A is a set of all natural numbers excluding 1 and 0?? r and s...
  4. T

    So-called Fundamental Theorem of Algebra

    This came up in one of my readings: "Neither the so-called fundamental theorem [of algebra] itself nor its classical proof by the theory of functions of a complex variable is as highly esteemed as it was a generation ago, and the theorem seems to be on its way out of algebra to make room for...
  5. J

    Fundamental theorem of calculus for double integral

    The popular fundamental theorem of calculus states that \int_{x_0}^{x_1} \frac{df}{dx}(x)dx = f(x_1)-f(x_0) But I never see this theorem for a dobule integral... The FTC for a univariate function, y'=f'(x), computes the area between f'(x) and the x-axis, delimited by (x0, x1), but given a...
  6. P

    Fundamental theorem of Calculus

    Suppose a is a constant. If G(x) = \int_a^x \Big [ f(t) \int_t^x g(u) du \Big ] dt, what is G\,'(x)? My attempt, G\,'(x) = f(x) \int_x^x g(u) du = 0, and I am sure this is wrong.
  7. evinda

    MHB First fundamental theorem of Calculus

    Hello! :) I am looking at the theorem: "$f:[a,b] \to \mathbb{R}$ integrable We suppose the function $F:[a,b] \to \mathbb{R}$ with $F(x)=\int_a^x f$.If $x_0$ a point where $f$ is continuous $\Rightarrow F$ is integrable at $ x_0$ and $F'(x_0)=f(x_0)$". There is a remark that the theorem stands...
  8. L

    Fundamental theorem of calculus

    Homework Statement Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx##...
  9. M

    What is the geometric interpretation of the fundamental theorem of calculus?

    hey pf! i'm trying to get a geometric understanding of the fundamental theorem: \int_a{}^{b}f'(x)dx=f(b)-f(a) basically, isn't the above just saying that if we add up a lot of slopes on a line at every point we will get the difference of the y values? thanks! feel free to add more or correct me
  10. J

    Fundamental theorem of calculus for surface integrals?

    Hellow! A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense?
  11. MarkFL

    MHB Solve Derivative of Integral with F(x) - SnowPatrol Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  12. Q

    Fundamental Theorem of Calculus

    Homework Statement The derivative of an integral with a constant as its lower bound and a function as its upper bound is the function at its upper bound multiplied by the derivative of the upper bound. The Attempt at a Solution How come the constant term has no bearing? I understand...
  13. T

    Integration using the fundamental theorem of calculus

    Hello PF. Homework Statement Find a function g such that \int_0^{x^2} \ tg(t) \, \mathrm{d}t = x^2+x Homework Equations From the fundamental theorem of calculus: f(x) = \frac{d}{dx}\int_a^x \ f(t) \, \mathrm{d}t The Attempt at a Solution After taking the derivative of...
  14. J

    Maximally strong fundamental theorem

    This is still not clear to me. Here's the conjecture: Assume that f:[a,b]\to\mathbb{R} is such function that it is differentiable at all points of its domain, and that \int\limits_{[a,b]}|f'(x)|dm(x) < \infty holds, where the integral is the ordinary Lebesgue integral. Then also...
  15. S

    Use fundamental theorem of calculus to compute definite integral

    Homework Statement The problem and my (incorrect) work are typed and attached as TheProblemAndMyWorkTypedUp.jpg. Homework Equations Integral from a to b of f(t) = F(b) – F(a) The Attempt at a Solution As mentioned above, my work is attached as TheProblemAndMyWorkTypedUp.jpg. (The (2 –...
  16. S

    Fundamental Theorem of Calculus - Variables x and t

    Hello, I'm getting slightly confused by the following so was hoping someone may be able to clear my problem up. For integrals, if b is the upper limit and a is the lower limit, I will write ∫[b,a]. From the Fundamental Theorem of calculus part 1 we can show that: if F(x) = ∫[x,a]...
  17. P

    First Fundamental Theorem of Calculus

    Hi, I just learned about the First Fundamental theorem of calculus. From my understanding, it talks specifically about definite integrals. I was wondering if there is any sort of theorem that proves that the derivative of the indefinite integral of a function is equal to the function itself...
  18. B

    Fundamental theorem in 2 dimensions.

    Hello I have heard that Greens, Stokes and the Divergence theorem is the equivalent of the fundamental theorem in multiple dimensions. But is there some way to show the result under: if F(x,y) = \int_{-\infty}^x\int_{-\infty}^yf(x^{*},y^{*})dx^{*}dy^{*} this implies that...
  19. 1

    Proving/deriving first fundamental theorem of calc

    I know this is going to be atrociously bad, but I like to try to prove things. \frac{f(x+\Delta x) - f(x)}{ \Delta x} = \frac{\Delta y}{\Delta x} => f(x+\Delta x) - f(x) = \frac{\Delta y}{\Delta x} \Delta x => f(x+\Delta x) = \frac{\Delta y}{\Delta x} \Delta x + f(x) Now...
  20. A

    MHB Fundamental Theorem of Calculus Questions

    Last one for the night! These are the questions: This is my work: I think question 5 is correct (I hope), but I'm not entirely sure about question 6. Any help would be appreciated!
  21. Fernando Revilla

    MHB Patrick's question at Yahoo Answers (First fundamental theorem of Calculus)

    Here is the question: Here is a link to the question: Integration by parts? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  22. Darth Frodo

    Proof of the Fundamental Theorem of Calculus.

    Hi all I'm currently working my way through proving the FToC by proving something that is a foundation for it. So I need to prove that; L(f,P_{1}) ≥ L(f,P) where P\subsetP_{1} i.e where P_{1} is a refinement of P. So, Let P_{1} = P \cup {c} where c \in [x_{k-1},x_{k}] Let L' = inf{x|x...
  23. G

    Fundamental Theorem of Line Integration

    Homework Statement Suppose that F is the inverse square force field below, where c is a constant. F(r) = c*r/(|r|)^3 r = x i + y j + z k (a) Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the...
  24. R

    Fundamental Theorem Of Calculus problems help

    Fundamental Theorem Of Calculus problems help! Homework Statement A)))) Find the derivative of g(x)=∫[8x to 4x] (u+7)/(u-4) dx B))) Use part I of the Fundamental Theorem of Calculus to find the derivative of h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt C))) F(x) = ∫[ 1 to √3]...
  25. L

    Use part 1 of the Fundamental Theorem of Calculus to find the derivative.

    1. h(x) = ∫-3 to sin(x) of (cos(t^3) + t)dt 2. Okay, I know that you are supposed to replace t with the upper limit, and then I think you multiply that term by the derivative of the upper limit. So I thought it would be: cos(sinx)^3 * cos(x) + sinxcosx But what even is cos(sinx)...
  26. B

    The fundamental theorem of calculus(I think;) )

    Homework Statement Been doing some old exams lately and found out that something I have problems with is questions of the type ( example): Differente the function: ∫ (x^2 ),(1), ln(t^2) dt Sorry for the bad writing. (x^2 ),(1), is the intgral from 1 to X^2 It should be fairly...
  27. O

    Second fundamental theorem of calculus.

    Let f(x) be a non-stochastic mapping f: \mathbb{R} \to \mathbb{R}. The second fundamental theorem of calculus states that: \frac{d}{dx} \int_a^x f(s)ds = f(x). *QUESTION 1* Is the following true? \frac{d}{dx} \int_x^a f(s)ds = f(x). *QUESTION 2* Related to this, how can I...
  28. P

    Fundamental Theorem of Calculus

    Homework Statement Homework Equations The Attempt at a Solution I know this is not right. Could someone help me out here? Thanks
  29. V

    Second fundamental theorem of calculus viewed as a transform?

    You see this picture of the second fundamental theorem of calculus and you are taught in high school / early college calculus that the t is a dummy variable. However, couldn't you view this as some sort of transform? You convert a function f(t) into a function of f(x). Is this a valid way to...
  30. D

    Fundamental theorem for line integrals

    Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is...
  31. E

    Fundamental theorem of calculus

    \frac{d}{dx} \int_a^b f(x) dx=f(b) This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.
  32. R

    Fundamental Theorem of Calculus

    Homework Statement All this information is in the attached file. Homework Equations All this information is in the attached file. The Attempt at a Solution What I tried to do was take the anti-derivative of the first equation and plug in the number 5. I'm not sure if that was...
  33. C

    Another quick question, Fundamental Theorem of Calculus

    I am guessing the fundamental theorem of calculus, isn't not valid, if the integrand f depends on x. Right? For example if he had: \int^{x}_{0} f(u) ( x-u) du. If one would make F(x) = \int^{x}_{0} g(u) du, with g(u) = f(u) ( x-u). Then F`(x) = g(x) = f(x) (x-x) = 0. But this is not...
  34. A

    Questions regarding the Fundamental Theorem of Calculus

    If you define a function g(x) = \int_a^x \! f(t) \, \mathrm{d} t then from what I currently understand, g(x) gives the value of the area under the curve y=f(t) When you differentiate both sides, g'(x) gives the rate of change of the area underneath y=f(t), however, I don't understand...
  35. M

    Fundamental Theorem of Calculus Part II

    1. Find the derivative of: ∫cos3(t) where a = 1/x and b = ∏/3 This was a part of a question on my first calc exam and I just wanted to know if I did it correctly. We can solve this using the Fundamental Theorem of Calculus, Part II The solution would be to simply plug in the values for a and...
  36. V

    The Fundamental Theorem of Algebra

    I just wanted to say first of all that I am not looking for any specific answers, just hoping someone could shed a light on the subjects at hand. Is the quadratic formula a specific example of some general root finding algorithm that solves for the n (or n-1?) roots of a nth degree...
  37. R

    Proof of the fundamental theorem of calculus

    Homework Statement This is supposed to be a proof of the fundamental theorem of calculus. I'm not really sure what that proves, but to me at least it does not prove that the area under a curve is the antiderivative of the function and then inserting the upper x value and...
  38. A

    Something weird with the fundamental theorem of calculus

    Suppose I know my function G is infinitely differentiable on the closed interval [a,b] and that all derivatives of G (including G itself) vanish at b. For any z in [a,b], I have by the FTC that \int_z^b G'(w) dw = G(b) - G(z). Or, switching limits, \int_b^z G'(w) dw = G(z) - G(b). One...
  39. I

    Fundamental Theorem of Calculus

    Homework Statement F(x) = ∫ cos (1+t^2)^-1) from 0 to 2x - x^2 Determine whether F has maximum or minimum value Homework Equations The Attempt at a Solution I tried finding F'(x) = Dx (∫ cos (1+t^2)^-1) from 0 to 2x - x^2) = (2-2x)cos[(1+(2x-x^2))^-1] What do I do...
  40. Q

    Fundamental Theorem of Calc., Inc./Dec., and concavity

    Homework Statement I am having extreme trouble with the following problems: http://i.minus.com/iYs6ix6otGtLV.png Homework Equations For 26: If the first derivative is positive, then the function is increasing. If the first derivative is negative, then the function is...
  41. K

    Rieman Integral: The Fundamental Theorem of Calculus

    Homework Statement Let I := [a,b] and let f: I→ℝ be continuous on I. Also let J := [c,d] and let u: J→ℝ be differentiable on J and satisfy u(J) contained in I. Show that if G: J→ℝ is defined by G(x) :=∫u(x)af for x in J, then G'(x) = (f o u)(x)u'(x) for all x in J. 2. The attempt...
  42. 1

    How bad is this statement regarding the Fundamental Theorem for Line Integrals?

    State the Fundamental Theorem: Let F be a vector field. If there exists a function f such that F = grad f, then \int_{C} F \cdot dr = f(Q) - f(P) where P and Q are endpoints of curve C. _________________________________ I didn't receive any credit for this answer. Admittedly...
  43. A

    Fundamental Theorem for Line Integrals

    Vector field F(bar)= <6x+2y,2x+5y> fx(x,y)= 6x+2y fy(x,y)= 2x+5y f(x,y)= 3x^2+2xy+g(y) fy(x,y)=2x+g'(y) 2x+g'(y)= 2x+5y g'(y)= 5y g(y)= 5/2*y^2 f(x,y)=3x^2+2xy+(5/2)y^2 Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1 I'm stuck on finding the last part for the F(bar)...
  44. F

    Help understanding the First Fundamental Theorem of Calculus

    The first fundamental theorem of calculus begins by defining a function like this: http://i.imgur.com/aWXql.png (sorry was not sure how to write this legibly in this post so I just uploaded on imgur) I kind of have a hard time wrapping my mind aruond this. How do you chose a? I...
  45. M

    Proof of a corollary of fundamental theorem of algebra

    Homework Statement Assuming the validity of the fundamental theorem of algebra, prove the corollary that: Every polynomial of positive degree n has a factorization of the form: P(x)=a_{n}(x-r_{1})...(x-r_{n}) where r_{i} aren't necessarily distinct. Homework Equations Fundamental...
  46. B

    Proof for part 2 of fundamental theorem of calculus

    The proof my book gives for the 2nd part of the FTC is a little hard for me to understand, but I was wondering if this particular proof (which is not from my book) is valid. I did the proof myself, I'm just wondering if it's valid. \frac{d}{dx}\int^{x}_{0}f(t) \ dt = f(x) So suppose that the...
  47. T

    Number Theory fundamental theorem of arithemetic

    I have two full questions on some number theory questions I've been working on, I guess my best bet would be to post them separately. 1) Suppose that n is in N (natural numbers), p1,...,pn are distinct primes, and l1,...ln are nonnegative integers. Let m = p1l1p2l2...pnln. Let d be in N such...
  48. J

    Fundamental Theorem of Calculus

    1. Homework Statement Prove that if f(x) is a differentiable real-valued function, and f'(x) is continuous and integrable. Then: \int_a^bf'(x)\,dx = f(b) - f(a) 2. Hint Provided Use the Cauchy Criterion for integrals, the mean value theorem applied to subintervals of a partition, and the...
  49. A

    Why is the fundamental theorem of calculus astounding?

    Homework Statement I just learned this idea from my lecture in calculus. I think I understand it at a surface level but don't know much about why it is an astounding discovery as my lecturer suggested. Homework Equations So, it states that let (f) be a continuous on an interval [a,b]. Let...
  50. J

    Fundamental Theorem of Calculus

    Homework Statement I am working on some problems with the fundamental theorem. I need to refresh my memory though, because I have forgotten exactly how to do these problems. I actually am not even sure if I use part 1 or part 2 of the FTC for these problems, so if you could help me in the...
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