The Fundamental Theorem of Calculus

10 Key excerpts on "The Fundamental Theorem of Calculus"

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  Theoretical Introduction to Mathematical Theorems, A

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  The first part of the theorem, sometimes called the first fundamental theorem of calculus , shows that an indefinite integration can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions. The second part, sometimes called the second fundamental theorem of calculus , allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals. The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675). Isaac Barrow (1630–1677) proved the first completely general version of the theorem, while Barrow's student Isaac Newton (1643– 1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities. Physical intuition Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other quantity) adds up to the net change in the quantity. In the case of a particle traveling in a straight line, its position, x , is given by x ( t ) where t is time and x ( t ) means that x is a function of t . The derivative of this function is equal to the infinitesimal change in quantity, d x , per infinitesimal change in time, d t (of course, the derivative itself is dependent on time). This change in displacement per change in time is the velocity v of the particle. In Leibniz's notation: Rearranging this equation, it follows that: By the logic above, a change in x (or Δ x ) is the sum of the infinitesimal changes d x . It is also equal to the sum of the infinitesimal products of the derivative and time.

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  The Art and Craft of Problem Solving

  • Paul Zeitz(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  9 Calculus In this chapter, we take it for granted that you are familiar with the basic calculus ideas such as limits, continuity, differentiation, integration, and power series. On the other hand, we assume that you may have heard of, but not mastered: ∙ Formal “ − ” proofs ∙ Taylor series with “remainder” ∙ The mean value theorem In contrast to, say, Chapter 7, this chapter is not a systematic, self-contained treatment. Instead, we concentrate on just a few important ideas that enhance your understanding of how calculus works. Our goal is twofold: to uncover the practical meaning of some of the things that you have already studied, by developing useful reformulations of old ideas; and to enhance your intuitive understanding of calculus, by showing you some useful albeit non-rigorous “moving curtains.” The meaning of this last phrase is best understood with an example. 9.1 The Fundamental Theorem of Calculus To understand what a moving curtain is, we shall explore, in some detail, the most important idea of elementary calculus. This example also introduces a number of ideas that we will keep returning to throughout the chapter. Example 9.1.1 What is The Fundamental Theorem of Calculus (FTC), what does it mean, and why is it true? Partial Solution: You have undoubtedly learned about the FTC. One formulation of it says that if  is a continuous function, 1 then ∫    () =  () −  (), (9.1.1) where  is any antiderivative of  ; i.e.,  ′ () =  (). This is a remarkable statement. The left-hand side of equation (9.1.1) can be interpreted as the area bounded by the graph of  =  (), the -axis, and the vertical lines  =  and  = . The right-hand side has a completely different meaning, since 1 In this chapter, we will assume that the domain and range of all functions are subsets of the real numbers.

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  Crossroads in the History of Mathematics and Mathematics Education

  • Bharath Sriraman(Author)
  • 2007(Publication Date)
  • Information Age Publishing

    (Publisher)

  In this book a whole chapter has the title The fundamental theorem of the integral calculus. Later in the same book there is one chapter with the title The fundamental theorem of the integral calculus for the Lebesgue integral and another chapter called The fundamental theorem of the integral calculus for the Denjoy integral. In the textbook An introduction to the summation of differences of a function by Benjamin Feland Groat, printed in 1902, the expression the fundamental theorem of the integral calculus is used, as well as more shortly fundamental theorem: To find the limits of sums of the form φ ( x ) Δx ∑ , it was necessary to have an identity of the form: φ ( x ) Δx = ψ ( x ) − ψ ( x + Δx ) + F ( x , Δx ) Δx 2 . The fundamental theorem of the integral calculus puts into mathematical language a rule for finding the limit of any sum, of the kind considered, provided an identity of the right form can be found; and the rules an formulae of the integral calculus afford a method for the discovery of the essential form of the identity when it exists. 33. Fundamental theorem. ʹ2 f ( x ) dx = f ( b) − f ( a) a b ∫ . Or, more explicitly, Lt. Δx= 0 ʹ2 ψ ( x ) Δx a b ∑ = ψ ( b) − ψ ( a) , where ʹ2 ψ ( x ) is any function of x and ψ ( x ) any function whose differential coefficient with regard to x is ʹ2 ψ ( x ) (pp. 40-41). In addition to this quote, the fundamental theorem is also expressed without employing mathematical formulae. This indicates the effort of the author to address an audience that is not fluent to read specialized technical language, that is, an attempt of a didactic transposition. In his textbook, A course of pure mathematics from 1908, In a paragraph called areas of plane curves (derivatives and integral) the FTC is proved but not named.

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  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  C h a p te r 17 The Fundamental Theorems of Calculus Here it is: the big kahuna. I’m talking about the Fundamental Theorems of Calculus, which not only provide the key for finding definite integrals without using messy Riemann sums, but also show how differentiation and integration are connected to each other. Without further ado, here’s the roadmap for the chapter: we’ll investigate • functions which are based on integrals of other functions; • the First Fundamental Theorem, and the basic idea of antiderivatives; • the Second Fundamental Theorem; and • indefinite integrals and their properties. After all this theoretical stuff, we’ll look at a lot of different examples in the following categories: • problems based on the First Fundamental Theorem; • finding indefinite integrals; and • finding definite integrals and areas using the Second Fundamental The-orem. 17.1 Functions Based on Integrals of Other Functions In the previous chapter, we used Riemann sums to show that Z 1 0 x 2 dx = 1 3 and Z 2 0 x 2 dx = 8 3 . (Actually, we only did the second one; I left the first one to you!) Unfortu-nately, the method of Riemann sums was really nasty. It would be nice to have an easier method to find the above integrals. Why stop there, though? Let’s try to find Z any number 0 x 2 dx. 356 • The Fundamental Theorems of Calculus So we want to allow the right-hand limit of integration to be variable . Ev-eryone’s favorite variable is x , but you can’t write down Z x 0 x 2 dx unless you want to be really confusing. After all, x is the dummy variable, so it can’t be a real variable too. So let’s start over, this time using t as the dummy variable. First, we have Z 1 0 t 2 dt = 1 3 and Z 2 0 t 2 dt = 8 3 . Remember, the letter we use for the dummy variable is irrelevant—we’ve just renamed the x -axis to be the t -axis. The actual area doesn’t change. Now we want to consider the quantity Z x 0 t 2 dt.

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  Real Analysis and Applications

  Including Fourier Series and the Calculus of Variations

  • Frank Morgan(Author)
  • 2005(Publication Date)
  • American Mathematical Society

    (Publisher)

  Part III Calculus Part III: Calculus. It is satisfying to see how naturally the theory of calculus follows from the basic concepts of real analysis, especially when you remember that after Newton and Leibniz invented the calculus in the late 1600s, it took mathematicians two hundred years to get the theory straight. This part focuses on R 1 rather than R n , although Chapters 16 and 17 hold for real-valued functions on R n as well. Chapter 13 The Derivative and the Mean Value Theorem Differentiation and integration are the two distinguishing processes of calcu-lus. The first major theorem about the derivative, the Mean Value Theorem, follows easily from the compactness of the interval [ a, b ], via Proposition 13.2 and Rolle’s Theorem 13.3. We begin by recalling the definition of the deriv-ative of a function on an interval in R . 13.1. Definition. Let f be a real-valued function on an open interval ( a, b ). Define the derivative f = df /dx by f ( x ) = lim t → x f ( t ) − f ( x ) t − x = lim ∆ x → 0 ∆ f ∆ x . If this limit exists, we say that f is differentiable at x . One important interpretation of the derivative is the slope of the graph. More generally, the derivative gives a rate of change. We assume the easy and familiar properties of the derivative and now state and prove the important theorems. We mention that if f is differen-tiable at x then f is continuous at x . If f has a continuous derivative, we say that f is continuously differentiable or C 1 . If f has k continuous derivatives, we say that f is C k . If f has derivatives of all orders, we say that f is C ∞ or smooth. (Sometimes smooth is used for C 1 . It is a somewhat vague term that means one does not want to worry about differentiability and wants to assume whatever is needed.) We say that f is piecewise C k or smooth if the domain is the union of finitely many closed intervals on which f is C k or smooth. (For this 61

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  Theories of Integration

  The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane

  • Douglas S Kurtz, Charles W Swartz;;;(Authors)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  Chapter 4 Fundamental Theorem of Calculus and the Henstock-Kurzweil integral In Chapter 2, we gave a brief discussion of The Fundamental Theorem of Calculus for the Riemann integral. In the first part of this chapter, we consider Part I of The Fundamental Theorem of Calculus for the Lebesgue integral and show that the Lebesgue integral suffers from the same defect with respect to Part I of The Fundamental Theorem of Calculus as does the Riemann integral. We then use this result to motivate the discussion of the Henstock-Kurzweil integral for which Part I of The Fundamental Theorem of Calculus holds in full generality. Recall that Part I of The Fundamental Theorem of Calculus involves the integration of the derivative of a function f and the formula In Example 2.31, we gave an example of a derivative which is unbounded and is, therefore, not Riemann integrable, and we showed in Theorem 2.30 that if f' is Riemann integrable, then (4.1) holds. That is, in order for (4.1) to hold, the assumption that the derivative f' is Riemann integrable is required. It would be desirable to have an integration theory for which Part I of The Fundamental Theorem of Calculus holds in full generality. That is, we would like to have an integral which integrates all derivatives and satisfies (4.1). Unfortunately, the example below shows that the general form of Part I of The Fundamental Theorem of Calculus does not hold for the Lebesgue integral. Example 4.1 In Example 2.31, we considered the function f defined by f (0) = 0 and f (z) = z2cos$ for 0 < x 5 1. The function f is differentiable with derivative f' satisfying f' (0) = 0 and f' (x) = 22 cos $+ 2n sin 5 for 0 < x 5 1. We show that f' is not Lebesgue integrable. 133 134 Theories of Integration If 0 < a < b < 1, then f' is continuous on [a,b] and is, therefore, Riemann integrable with r 2 r b2 a2 f' = b2cos - -a cos-. Setting bk = I / & and a k = , / -, we see that J : : f' = 1/2k.

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  A Mathematics Course for Political and Social Research

  • Will H. Moore, David A. Siegel(Authors)
  • 2013(Publication Date)
  • Princeton University Press

    (Publisher)

  To get at this question, we offer the grand-sounding fundamental theorem of calculus: Z b a f (x)dx = F (b) - F (a). In other words, the definite integral of a function from a to b is equal to the antiderivative of that function evaluated at b minus the same evaluated at a. The theorem is fundamental because it bridges (or links) differential and integral calculus. We can make the connection even more clear with a little notation R b a f (x)dx = F (x)| b a , where the vertical line means “evaluate the antiderivative F (x) at b, and subtract the antiderivative evaluated at a.” With this theorem, to calculate the area under the curve—a value that will prove very important for statistical inference—all we need to know is the indef- inite integral of the function. We don’t even have to worry about the constant C: since it appears in both indefinite integrals on the RHS, it cancels when they are subtracted. 5 Given this, let’s return to the example of the figures above. There f (x) = x 2 . What’s the antiderivative of this? Well, we know that the derivative of x 3 is 3x 2 from the previous chapter, so an x 3 will be involved. If we further divide by 3 we’ll get x 2 without the 3 in front. Thus, F (x) = 1 3 x 3 . Consequently, by The Fundamental Theorem of Calculus, R 2 1 x 2 dx = F (2) - F (1) = 1 3 (2) 3 - 1 3 (1) 3 = 8 3 - 1 3 = 7 3 , and that is the area under the curve of x 2 between 1 and 2. Let’s try another example. Consider some function f (x) = 1 + 2x + x 2 . Suppose we want to know the area under the curve over the range from x = 0 to x = 3. This area is the shaded region in Figure 7.3. We first need to compute the indefinite integral. Recalling that the derivative is linear, we note that x differentiates to 1, x 2 differentiates to 2x, and, as we just saw, 1 3 x 3 differentiates to x 2 . Putting them together yields F (x) = x + x 2 + 1 3 x 3 . Evaluated at a = 0, this is 0. At b = 3, this is 21.

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  The Fundamentals of Mathematical Analysis

  • G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  186 6. THEOREMS OF DIFFERENTIAL CALCULUS can be found on the curve at which the tangent is parallel to the x-axis (Fig. 39). We emphasize that the continuity of the function f(x) in the closed interval [a, b] and the existence of the derivative in the whole open interval (a, b) are essential for the validity of the theorem. The function /(JC) = x — E(x) satisfies in the interval [0,1] all conditions of the theorem, except that it has a discon-tinuity at x = 1, and the derivative /'(*) = 1, everywhere in (0,1). The function defined by the relations f(x) = x for 0 < x < 1/2 and f(x) = 1 — x for 1/2 < < # < 1 also satisfies all conditions in the considered interval, except that at x — 2 a finite (two-sided) derivative does not exist; now in the left half of the interval /'(*) = + 1 while in the right one/'(*) = — 1. Similarly, condition (3) of the theorem is important: the function /(JC) = x in the interval [0,1] satisfies all conditions of the theorem except the third, and everywhere its derivative/'(*) = 1. The construction of the appropriate diagrams is left to the reader. 102. Theorem on finite increments. We now consider the direct consequences of Rolle's theorem. The first is the following theorem on finite increments announced by Lagrange. LAGRANGE'S THEOREM. Suppose that (1) f(x) is defined and is continuous in the closed interval [a, b], (2) there exists a finite deriv-ative fix) at least in the open interval (a, b). Then between a and b a point c can be found (a

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  Differential and Integral Calculus, Volume 1

  • Richard Courant(Author)
  • 2011(Publication Date)
  • Wiley-Interscience

    (Publisher)

  Ill, however, we know that: Every indefinite integral (x) of the function f(x) is a primitive ofi(x). Yet this result does not completely solve the problem of find-ing primitive functions. For we do not yet know if we have found all the solutions of the problem. The question about the S (K798) ii4 FUNDAMENTAL IDEAS [CHAP. group of all primitive functions is answered by the following theorem, sometimes referred to as the fundamental theorem of the differential and integral calculus: The difference of two primitives F 1 (x) and F 2 (x) of the same func-tion f(x) is always a constant: F&) - F 2 (x) = c. Thus from any one primitive function ~F(x) we can obtain all the others in the form F(x) + c by suitable choice of the constant c. Conversely, for every value of the constant c the expression F x (x) = F(x) + c represents a primi-tive function of f(x). It is clear that for any value of the constant c the function F(x) + c is a primitive, provided that F{x) itself is one. For (cf. p. 96) we have {F(x + h) + c} - {F(x) + c}_ F(x +h)- F(x) h k and since by hypothesis the right-hand side tends to/(x) as h -*■ 0, so does the left-hand side, and therefore ^{F(x) + c}=f(x) = F'(x). Thus to complete the proof of our theorem it only remains to show that the difference of two primitive functions F t (x) and F 2 (x) is always a constant. For this purpose we consider the difference F 1 (x)-F 2 (x) = G(x) and form the derivative G'(x) = lim f*'i(*+*)--g'i(») _ F 2 (x+h)-F 2 (x) *->o h h ) Both the expressions on the right-hand side, by hypothesis, have the same limit/(x) as A-> 0; thus for every value of x we have G'(x) = 0. But a function whose derivative is everywhere zero must have a graph whose tangent is everywhere parallel to the x-axis, i.e. must be a constant; and therefore we have G(x) = c, II] THE INDEFINITE INTEGRAL 115 as we stated above. We can prove this last fact without relying upon intuition, by using the mean value theorem.

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  Calculus

  One and Several Variables

  • Saturnino L. Salas, Garret J. Etgen, Einar Hille(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  A function G is called an antiderivative for f on [a , b] if G is continuous on [a , b] and G  (x ) = f (x ) for all x ∈ (a , b). Theorem 5.3.5 tells us that if f is continuous on [a , b], then F (x ) =  x a f (t ) dt is an antiderivative for f on [a , b]. This gives us a prescription for constructing anti- derivatives. It tells us that we can construct an antiderivative for f by integrating f . The theorem below, called the “fundamental theorem,” goes the other way. It gives us a prescription, not for finding antiderivatives, but for evaluating integrals. It tells us that we can evaluate the integral  b a f (t ) dt from any antiderivative of f by evaluating the antiderivative at b and at a . THEOREM 5.4.2 THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS Let f be continuous on [a , b]. If G is any antiderivative for f on [a , b], then  b a f (t ) dt = G(b) − G(a ). 5.4 THE FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS ■ 255 PROOF From Theorem 5.3.5 we know that the function F (x ) =  x a f (t ) dt is an antiderivative for f on [a , b]. If G is also an antiderivative for f on [a , b], then both F and G are continuous on [a , b] and satisfy F  (x ) = G  (x ) for all x in (a , b). From Theorem 4.2.4 we know that there exists a constant C such that F (x ) = G(x ) + C for all x in [a , b]. Since F (a ) = 0, G(a ) + C = 0 and thus C = −G(a ). It follows that F (x ) = G(x ) − G(a ) for all x in [a , b]. In particular,  b a f (t ) dt = F (b) = G(b) − G(a ). ❏ We now evaluate some integrals by applying the fundamental theorem. In each case we use the simplest antiderivative we can think of. Example 1 Evaluate  4 1 x 2 dx . SOLUTION As an antiderivative for f (x ) = x 2 , we can use the function G(x ) = 1 3 x 3 . (Verify this.) By the fundamental theorem,  4 1 x 2 dx = G(4) − G(1) = 1 3 (4) 3 − 1 3 (1) 3 = 64 3 − 1 3 = 21. NOTE: Any other antiderivative of f (x ) = x 2 has the form H (x ) = 1 3 x 3 + C for some constant C.

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