Lesson 32: The Fundamental Theorem Of Calculus

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Lesson 32: The Fundamental Theorem Of Calculus - Presentation Transcript

  1. Section 5.4 The Fundamental Theorem of Calculus Math 1a December 12, 2007 Announcements my next office hours: Today 1–3 (SC 323) MT II is graded. Come to OH to talk about it Final seview sessions: Wed 1/9 and Thu 1/10 in Hall D, Sun 1/13 in Hall C, all 7–8:30pm Final tentatively scheduled for January 17, 9:15am
  2. Outline The Area Function FTC1 Statement Proof Biographies Differentiation of functions defined by integrals “Contrived” examples Erf Other applications FTC2 Facts about g from f A problem
  3. An area function x Let f (t) = t 2 and define g (x) = t 3 dt. Can we evaluate the 0 integral in g (x)? x 0
  4. An area function x Let f (t) = t 2 and define g (x) = t 3 dt. Can we evaluate the 0 integral in g (x)? Dividing the interval [0, x] into n pieces x ix gives ∆x = and xi = 0 + i∆x = . n n So x x 3 x (2x)3 x (nx)3 · 3+ · + ··· + · Rn = n3 n3 nn n n 4 x = 4 13 + 2 3 + 3 3 + · · · + n 3 n x4 2 = 4 1 n(n + 1) n2 x 0 x 4 n2 (n + 1)2 x4 → = 4n4 4 as n → ∞.
  5. An area function, continued So x4 g (x) = . 4
  6. An area function, continued So x4 g (x) = . 4 This means that g (x) = x 3 .
  7. The area function Let f be a function which is integrable (i.e., continuous or with finitely many jump discontinuities) on [a, b]. Define t g (x) = f (t) dt. a When is g increasing?
  8. The area function Let f be a function which is integrable (i.e., continuous or with finitely many jump discontinuities) on [a, b]. Define t g (x) = f (t) dt. a When is g increasing? When is g decreasing?
  9. The area function Let f be a function which is integrable (i.e., continuous or with finitely many jump discontinuities) on [a, b]. Define t g (x) = f (t) dt. a When is g increasing? When is g decreasing? Over a small interval, what’s the average rate of change of g ?
  10. Outline The Area Function FTC1 Statement Proof Biographies Differentiation of functions defined by integrals “Contrived” examples Erf Other applications FTC2 Facts about g from f A problem
  11. Theorem (The First Fundamental Theorem of Calculus) Let f be an integrable function on [a, b] and define x g (x) = f (t) dt. a If f is continuous at x in (a, b), then g is differentiable at x and g (x) = f (x).
  12. Proof. Let h > 0 be given so that x + h < b. We have g (x + h) − g (x) = h
  13. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x
  14. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x Let Mh be the maximum value of f on [x, x + h], and mh the minimum value of f on [x, x + h]. From §5.2 we have x+h f (t) dt x
  15. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x Let Mh be the maximum value of f on [x, x + h], and mh the minimum value of f on [x, x + h]. From §5.2 we have x+h f (t) dt ≤ Mh · h x
  16. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x Let Mh be the maximum value of f on [x, x + h], and mh the minimum value of f on [x, x + h]. From §5.2 we have x+h mh · h ≤ f (t) dt ≤ Mh · h x
  17. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x Let Mh be the maximum value of f on [x, x + h], and mh the minimum value of f on [x, x + h]. From §5.2 we have x+h mh · h ≤ f (t) dt ≤ Mh · h x So g (x + h) − g (x) mh ≤ ≤ Mh . h
  18. Proof. Let h > 0 be given so that x + h < b. We have x+h g (x + h) − g (x) 1 = f (t) dt. h h x Let Mh be the maximum value of f on [x, x + h], and mh the minimum value of f on [x, x + h]. From §5.2 we have x+h mh · h ≤ f (t) dt ≤ Mh · h x So g (x + h) − g (x) mh ≤ ≤ Mh . h As h → 0, both mh and Mh tend to f (x). Zappa-dappa.
  19. Meet the Mathematician: Isaac Barrow English, 1630-1677 Professor of Greek, theology, and mathematics at Cambridge Had a famous student
  20. Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687
  21. Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute
  22. Outline The Area Function FTC1 Statement Proof Biographies Differentiation of functions defined by integrals “Contrived” examples Erf Other applications FTC2 Facts about g from f A problem
  23. Differentiation of area functions Example x t 3 dt. We know g (x) = x 3 . What if instead we Let g (x) = 0 had 3x t 3 dt. h(x) = 0 What is h (x)?
  24. Differentiation of area functions Example x t 3 dt. We know g (x) = x 3 . What if instead we Let g (x) = 0 had 3x t 3 dt. h(x) = 0 What is h (x)? Solution We can think of h as the composition g ◦ k, where u t 3 dt and k(x) = 3x. Then g (u) = 0 h (x) = g (k(x))k (x) = 3(k(x))3 = 3(3x)3 = 81x 3 .
  25. Example sin2 x (17t 2 + 4t − 4) dt. What is h (x)? Let h(x) = 0
  26. Example sin2 x (17t 2 + 4t − 4) dt. What is h (x)? Let h(x) = 0 Solution We have sin2 x d (17t 2 + 4t − 4) dt dx 0 d = 17(sin2 x)2 + 4(sin2 x) − 4 · sin2 x dx = 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
  27. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0
  28. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0 It turns out erf is the shape of the bell curve.
  29. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0 It turns out erf is the shape of the bell curve. We can’t find erf(x), explicitly, but we do know its derivative. erf (x) =
  30. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0 It turns out erf is the shape of the bell curve. We can’t find erf(x), explicitly, but we do know its derivative. 2 2 erf (x) = √ e −x . π
  31. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0 It turns out erf is the shape of the bell curve. We can’t find erf(x), explicitly, but we do know its derivative. 2 2 erf (x) = √ e −x . π Example d erf(x 2 ). Find dx
  32. Erf Here’s a function with a funny name but an important role: x 2 2 e −t dt. erf(x) = √ π 0 It turns out erf is the shape of the bell curve. We can’t find erf(x), explicitly, but we do know its derivative. 2 2 erf (x) = √ e −x . π Example d erf(x 2 ). Find dx Solution By the chain rule we have d d 2 4 22 4 erf(x 2 ) = erf (x 2 ) x 2 = √ e −(x ) 2x = √ xe −x . dx dx π 2π
  33. Other functions defined by integrals The future value of an asset: ∞ π(τ )e −r τ dτ FV (t) = t where π(τ ) is the profitability at time τ and r is the discount rate. The consumer surplus of a good: p∗ CS(p ∗ ) = f (p) dp 0 where f (p) is the demand function and p ∗ is the equilibrium price (depends on supply)
  34. Outline The Area Function FTC1 Statement Proof Biographies Differentiation of functions defined by integrals “Contrived” examples Erf Other applications FTC2 Facts about g from f A problem
  35. Theorem (The Second Fundamental Theorem of Calculus, Weak Form) If f is continuous on [a, b] and f = F for another function F , then b f (t) dt = F (b) − F (a). a
  36. Theorem (The Second Fundamental Theorem of Calculus, Weak Form) If f is continuous on [a, b] and f = F for another function F , then b f (t) dt = F (b) − F (a). a Proof. Let g be the area function. Since f is continuous on [a, b], g is differentiable on (a, b), and g = f = F on (a, b). Hence g (x) = F (x) + C for all x in [a, b] (remember this requires the Mean Value Theorem!). Since g (a) = 0, we have C = −F (a). Therefore g (b) = F (b) − F (a).
  37. Outline The Area Function FTC1 Statement Proof Biographies Differentiation of functions defined by integrals “Contrived” examples Erf Other applications FTC2 Facts about g from f A problem
  38. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 3 • (3,3) 2 • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  39. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 What is the particle’s velocity 3 • (3,3) at time t = 5? 2 • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  40. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 What is the particle’s velocity 3 • (3,3) at time t = 5? 2 • • (2,2) (5,2) Solution 1 • (1,1) Recall that by the FTC we have 1 2 3 4 5 6 7 8 9 s (t) = f (t). So s (5) = f (5) = 2.
  41. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 Is the acceleration of the par- 3 • (3,3) ticle at time t = 5 positive or 2 negative? • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  42. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 Is the acceleration of the par- 3 • (3,3) ticle at time t = 5 positive or 2 negative? • • (2,2) (5,2) 1 • (1,1) Solution We have s (5) = f (5), which 1 2 3 4 5 6 7 8 9 looks negative from the graph.
  43. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 What is the particle’s position 3 • (3,3) at time t = 3? 2 • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  44. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 What is the particle’s position 3 • (3,3) at time t = 3? 2 • • (2,2) (5,2) Solution 1 • (1,1) Since on [0, 3], f (x) = x, we have 1 2 3 4 5 6 7 8 9 3 9 s(3) = x dx = . 2 0
  45. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 At what time during the first 9 3 • (3,3) seconds does s have its largest 2 value? • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  46. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 At what time during the first 9 3 • (3,3) seconds does s have its largest 2 value? • • (2,2) (5,2) 1 • (1,1) Solution 1 2 3 4 5 6 7 8 9
  47. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 At what time during the first 9 3 • (3,3) seconds does s have its largest 2 value? • • (2,2) (5,2) 1 • (1,1) Solution The critical points of s are 1 2 3 4 5 6 7 8 9 the zeros of s = f .
  48. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 At what time during the first 9 3 • (3,3) seconds does s have its largest 2 value? • • (2,2) (5,2) 1 • (1,1) Solution By looking at the graph, we 1 2 3 4 5 6 7 8 9 see that f is positive from t = 0 to t = 6, then negative from t = 6 to t = 9.
  49. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 At what time during the first 9 3 • (3,3) seconds does s have its largest 2 value? • • (2,2) (5,2) 1 • (1,1) Solution Therefore s is increasing on 1 2 3 4 5 6 7 8 9 [0, 6], then decreasing on [6, 9]. So its largest value is at t = 6.
  50. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 Approximately when is the ac- 3 • (3,3) celeration zero? 2 • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  51. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 Approximately when is the ac- 3 • (3,3) celeration zero? 2 • • (2,2) (5,2) Solution 1 • (1,1) s = 0 when f = 0, which happens at t = 4 and t = 7.5 1 2 3 4 5 6 7 8 9 (approximately)
  52. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 When is the particle moving 3 • (3,3) toward the origin? Away from 2 the origin? • • (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  53. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 When is the particle moving 3 • (3,3) toward the origin? Away from 2 the origin? • • (2,2) (5,2) 1 • (1,1) Solution The particle is moving away 1 2 3 4 5 6 7 8 9 from the origin when s > 0 and s > 0.
  54. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 When is the particle moving 3 • (3,3) toward the origin? Away from 2 the origin? • • (2,2) (5,2) 1 • (1,1) Solution Since s(0) = 0 and s > 0 on 1 2 3 4 5 6 7 8 9 (0, 6), we know the particle is moving away from the origin then.
  55. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 When is the particle moving 3 • (3,3) toward the origin? Away from 2 the origin? • • (2,2) (5,2) 1 • (1,1) Solution After t = 6, s < 0, so the 1 2 3 4 5 6 7 8 9 particle is moving toward the origin.
  56. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 On which side (positive or neg- 3 • (3,3) ative) of the origin does the 2 • • particle lie at time t = 9? (2,2) (5,2) 1 • (1,1) 1 2 3 4 5 6 7 8 9
  57. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 On which side (positive or neg- 3 • (3,3) ative) of the origin does the 2 • • particle lie at time t = 9? (2,2) (5,2) 1 • (1,1) Solution We have s(9) = 1 2 3 4 5 6 7 8 9 6 9 f (x) dx + f (x) dx, 0 6 where the left integral is positive and the right integral is negative.
  58. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 On which side (positive or neg- 3 • (3,3) ative) of the origin does the 2 • • particle lie at time t = 9? (2,2) (5,2) 1 • (1,1) Solution In order to decide whether 1 2 3 4 5 6 7 8 9 s(9) is positive or negative, we need to decide if the first area is more positive than the second area is negative.
  59. Facts about g from f Let f be the function whose graph is given below. Suppose the the position at time t seconds of a particle moving t along a coordinate axis is s(t) = f (x) dx meters. Use the 0 graph to answer the following questions. 4 On which side (positive or neg- 3 • (3,3) ative) of the origin does the 2 • • particle lie at time t = 9? (2,2) (5,2) 1 • (1,1) Solution This appears to be the case, 1 2 3 4 5 6 7 8 9 so s(9) is positive.

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