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# Lesson 28: The Fundamental Theorem of Calculus

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• 1. Section 5.4 The Fundamental Theorem of Calculus V63.0121.027, Calculus I December 8, 2009 Announcements Final Exam: Friday 12/18, 2:00-3:50pm, Tisch UC50 . . . . . .
• 2. Redemption policies Current distribution of grade: 40% &#xFB01;nal, 25% midterm, 15% quizzes, 10% written HW, 10% WebAssign Remember we drop the lowest quiz, lowest written HW, and 5 lowest WebAssign-ments [new!] If your &#xFB01;nal exam score beats your midterm score, we will re-weight it by 50% and make the midterm 15% . . . . . .
• 3. Outline Recall: The Evaluation Theorem a/k/a 2FTC The First Fundamental Theorem of Calculus The Area Function Statement and proof of 1FTC Biographies Differentiation of functions de&#xFB01;ned by integrals &#x201C;Contrived&#x201D; examples Erf Other applications . . . . . .
• 4. The de&#xFB01;nite integral as a limit De&#xFB01;nition If f is a function de&#xFB01;ned on [a, b], the de&#xFB01;nite integral of f from a to b is the number &#x222B; b &#x2211;n f(x) dx = lim f(ci ) &#x2206;x a &#x2206;x&#x2192;0 i =1 . . . . . .
• 5. Theorem (The Second Fundamental Theorem of Calculus) Suppose f is integrable on [a, b] and f = F&#x2032; for another function F, then &#x222B; b f(x) dx = F(b) &#x2212; F(a). a . . . . . .
• 6. The Integral as Total Change Another way to state this theorem is: &#x222B; b F&#x2032; (x) dx = F(b) &#x2212; F(a), a or the integral of a derivative along an interval is the total change between the sides of that interval. This has many rami&#xFB01;cations: . . . . . .
• 7. The Integral as Total Change Another way to state this theorem is: &#x222B; b F&#x2032; (x) dx = F(b) &#x2212; F(a), a or the integral of a derivative along an interval is the total change between the sides of that interval. This has many rami&#xFB01;cations: Theorem If v(t) represents the velocity of a particle moving rectilinearly, then &#x222B; t1 v(t) dt = s(t1 ) &#x2212; s(t0 ). t0 . . . . . .
• 8. The Integral as Total Change Another way to state this theorem is: &#x222B; b F&#x2032; (x) dx = F(b) &#x2212; F(a), a or the integral of a derivative along an interval is the total change between the sides of that interval. This has many rami&#xFB01;cations: Theorem If MC(x) represents the marginal cost of making x units of a product, then &#x222B; x C(x) = C(0) + MC(q) dq. 0 . . . . . .
• 9. The Integral as Total Change Another way to state this theorem is: &#x222B; b F&#x2032; (x) dx = F(b) &#x2212; F(a), a or the integral of a derivative along an interval is the total change between the sides of that interval. This has many rami&#xFB01;cations: Theorem If &#x3C1;(x) represents the density of a thin rod at a distance of x from its end, then the mass of the rod up to x is &#x222B; x m(x) = &#x3C1;(s) ds. 0 . . . . . .
• 10. My &#xFB01;rst table of integrals &#x222B; &#x222B; &#x222B; [f(x) + g(x)] dx = f(x) dx + g(x) dx &#x222B; &#x222B; &#x222B; x n +1 xn dx = + C (n &#x338;= &#x2212;1) cf(x) dx = c f(x) dx n+1 &#x222B; &#x222B; 1 ex dx = ex + C dx = ln |x| + C x &#x222B; &#x222B; ax sin x dx = &#x2212; cos x + C ax dx = +C ln a &#x222B; &#x222B; cos x dx = sin x + C csc2 x dx = &#x2212; cot x + C &#x222B; &#x222B; sec2 x dx = tan x + C csc x cot x dx = &#x2212; csc x + C &#x222B; &#x222B; 1 sec x tan x dx = sec x + C &#x221A; dx = arcsin x + C 1 &#x2212; x2 &#x222B; 1 dx = arctan x + C 1 + x2 . . . . . .
• 11. Outline Recall: The Evaluation Theorem a/k/a 2FTC The First Fundamental Theorem of Calculus The Area Function Statement and proof of 1FTC Biographies Differentiation of functions de&#xFB01;ned by integrals &#x201C;Contrived&#x201D; examples Erf Other applications . . . . . .
• 12. An area function &#x222B; x 3 Let f(t) = t and de&#xFB01;ne g(x) = f(t) dt. Can we evaluate the 0 integral in g(x)? . 0 . x . . . . . . .
• 13. An area function &#x222B; x 3 Let f(t) = t and de&#xFB01;ne g(x) = f(t) dt. Can we evaluate the 0 integral in g(x)? Dividing the interval [0, x] into n pieces x ix gives &#x2206;t = and ti = 0 + i&#x2206;t = . So n n x x3 x (2x)3 x (nx)3 Rn = &#xB7; 3 + &#xB7; 3 + &#xB7;&#xB7;&#xB7; + &#xB7; 3 n n n n n n x4 ( 3 ) = 4 1 + 2 3 + 3 3 + &#xB7; &#xB7; &#xB7; + n3 n x4 [ 1 ]2 = 4 2 n(n + 1) . n 0 . x . x4 n2 (n + 1)2 x4 = &#x2192; 4n4 4 as n &#x2192; &#x221E;. . . . . . .
• 14. An area function, continued So x4 g(x) = . 4 . . . . . .
• 15. An area function, continued So x4 g(x) = . 4 This means that g &#x2032; (x ) = x 3 . . . . . . .
• 16. The area function Let f be a function which is integrable (i.e., continuous or with &#xFB01;nitely many jump discontinuities) on [a, b]. De&#xFB01;ne &#x222B; x g(x) = f(t) dt. a The variable is x; t is a &#x201C;dummy&#x201D; variable that&#x2019;s integrated over. Picture changing x and taking more of less of the region under the curve. Question: What does f tell you about g? . . . . . .
• 17. Envisioning the area function Example Suppose f(t) is the function graphed below v . . . . . . . t .0 t .1 c . t .2 t t .3 . . &#x222B; x Let g(x) = f(t) dt. What can you say about g? t0 . . . . . .
• 18. features of g from f Interval sign monotonicity monotonicity concavity of f of g of f of g [ t0 , t 1 ] + &#x2197; &#x2197; &#x2323; [t1 , c] + &#x2197; &#x2198; &#x2322; [c, t2 ] &#x2212; &#x2198; &#x2198; &#x2322; [ t2 , t 3 ] &#x2212; &#x2198; &#x2197; &#x2323; [t3 , &#x221E;) &#x2212; &#x2198; &#x2192; none . . . . . .
• 19. features of g from f Interval sign monotonicity monotonicity concavity of f of g of f of g [ t0 , t 1 ] + &#x2197; &#x2197; &#x2323; [t1 , c] + &#x2197; &#x2198; &#x2322; [c, t2 ] &#x2212; &#x2198; &#x2198; &#x2322; [ t2 , t 3 ] &#x2212; &#x2198; &#x2197; &#x2323; [t3 , &#x221E;) &#x2212; &#x2198; &#x2192; none We see that g is behaving a lot like an antiderivative of f. . . . . . .
• 20. Theorem (The First Fundamental Theorem of Calculus) Let f be an integrable function on [a, b] and de&#xFB01;ne &#x222B; x g(x) = f(t) dt. a If f is continuous at x in (a, b), then g is differentiable at x and g&#x2032; (x) = f(x). . . . . . .
• 21. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have g(x + h) &#x2212; g(x) = h . . . . . .
• 22. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; g(x) 1 = f(t) dt. h h x . . . . . .
• 23. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; 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 &#xA7;5.2 we have &#x222B; x +h f(t) dt x . . . . . .
• 24. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; 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 &#xA7;5.2 we have &#x222B; x +h f(t) dt &#x2264; Mh &#xB7; h x . . . . . .
• 25. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; 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 &#xA7;5.2 we have &#x222B; x +h mh &#xB7; h &#x2264; f(t) dt &#x2264; Mh &#xB7; h x . . . . . .
• 26. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; 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 &#xA7;5.2 we have &#x222B; x +h mh &#xB7; h &#x2264; f(t) dt &#x2264; Mh &#xB7; h x So g(x + h) &#x2212; g(x) mh &#x2264; &#x2264; Mh . h . . . . . .
• 27. Proof. Let h &gt; 0 be given so that x + h &lt; b. We have &#x222B; x+h g(x + h) &#x2212; 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 &#xA7;5.2 we have &#x222B; x +h mh &#xB7; h &#x2264; f(t) dt &#x2264; Mh &#xB7; h x So g(x + h) &#x2212; g(x) mh &#x2264; &#x2264; Mh . h As h &#x2192; 0, both mh and Mh tend to f(x). . . . . . .
• 28. Meet the Mathematician: James Gregory Scottish, 1638-1675 Astronomer and Geometer Conceived transcendental numbers and found evidence that &#x3C0; was transcendental Proved a geometric version of 1FTC as a lemma but didn&#x2019;t take it further . . . . . .
• 29. Meet the Mathematician: Isaac Barrow English, 1630-1677 Professor of Greek, theology, and mathematics at Cambridge Had a famous student . . . . . .
• 30. Meet the Mathematician: Isaac Newton English, 1643&#x2013;1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687 . . . . . .
• 31. Meet the Mathematician: Gottfried Leibniz German, 1646&#x2013;1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute . . . . . .
• 32. Differentiation and Integration as reverse processes Putting together 1FTC and 2FTC, we get a beautiful relationship between the two fundamental concepts in calculus. &#x222B; x d f(t) dt = f(x) dx a . . . . . .
• 33. Differentiation and Integration as reverse processes Putting together 1FTC and 2FTC, we get a beautiful relationship between the two fundamental concepts in calculus. &#x222B; x d f(t) dt = f(x) dx a &#x222B; b F&#x2032; (x) dx = F(b) &#x2212; F(a). a . . . . . .
• 34. Outline Recall: The Evaluation Theorem a/k/a 2FTC The First Fundamental Theorem of Calculus The Area Function Statement and proof of 1FTC Biographies Differentiation of functions de&#xFB01;ned by integrals &#x201C;Contrived&#x201D; examples Erf Other applications . . . . . .
• 35. Differentiation of area functions Example &#x222B; 3x Let h(x) = t3 dt. What is h&#x2032; (x)? 0 . . . . . .
• 36. Differentiation of area functions Example &#x222B; 3x Let h(x) = t3 dt. What is h&#x2032; (x)? 0 Solution (Using 2FTC) 3x t4 1 h(x) = = (3x)4 = 1 4 &#xB7; 81x4 , so h&#x2032; (x) = 81x3 . 4 0 4 . . . . . .
• 37. Differentiation of area functions Example &#x222B; 3x Let h(x) = t3 dt. What is h&#x2032; (x)? 0 Solution (Using 2FTC) 3x t4 1 h(x) = = (3x)4 = 1 4 &#xB7; 81x4 , so h&#x2032; (x) = 81x3 . 4 0 4 Solution (Using 1FTC) &#x222B; u We can think of h as the composition g k, where g(u) = &#x25E6; t3 dt 0 and k(x) = 3x. . . . . . .
• 38. Differentiation of area functions Example &#x222B; 3x Let h(x) = t3 dt. What is h&#x2032; (x)? 0 Solution (Using 2FTC) 3x t4 1 h(x) = = (3x)4 = 1 4 &#xB7; 81x4 , so h&#x2032; (x) = 81x3 . 4 0 4 Solution (Using 1FTC) &#x222B; u We can think of h as the composition g k, where g(u) = &#x25E6; t3 dt 0 and k(x) = 3x. Then h&#x2032; (x) = g&#x2032; (k(x))k&#x2032; (x) = (k(x))3 &#xB7; 3 = (3x)3 &#xB7; 3 = 81x3 . . . . . . .
• 39. Differentiation of area functions, in general by 1FTC &#x222B; k(x) d f(t) dt = f(k(x))k&#x2032; (x) dx a by reversing the order of integration: &#x222B; b &#x222B; h(x) d d f(t) dt = &#x2212; f(t) dt = &#x2212;f(h(x))h&#x2032; (x) dx h (x ) dx b by combining the two above: &#x222B; (&#x222B; &#x222B; ) k(x) k (x ) 0 d d f(t) dt = f(t) dt + f(t) dt dx h (x ) dx 0 h(x) = f(k(x))k&#x2032; (x) &#x2212; f(h(x))h&#x2032; (x) . . . . . .
• 40. Example &#x222B; sin2 x Let h(x) = (17t2 + 4t &#x2212; 4) dt. What is h&#x2032; (x)? 0 . . . . . .
• 41. Example &#x222B; sin2 x Let h(x) = (17t2 + 4t &#x2212; 4) dt. What is h&#x2032; (x)? 0 Solution We have &#x222B; sin2 x d (17t2 + 4t &#x2212; 4) dt dx 0 ( ) d = 17(sin2 x)2 + 4(sin2 x) &#x2212; 4 &#xB7; sin2 x ( ) dx = 17 sin4 x + 4 sin2 x &#x2212; 4 &#xB7; 2 sin x cos x . . . . . .
• 42. Example &#x222B; ex Find the derivative of F(x) = sin4 t dt. x3 . . . . . .
• 43. Example &#x222B; ex Find the derivative of F(x) = sin4 t dt. x3 Solution &#x222B; ex d sin4 t dt = sin4 (ex ) &#xB7; ex &#x2212; sin4 (x3 ) &#xB7; 3x2 dx x3 . . . . . .
• 44. Example &#x222B; ex Find the derivative of F(x) = sin4 t dt. x3 Solution &#x222B; ex d sin4 t dt = sin4 (ex ) &#xB7; ex &#x2212; sin4 (x3 ) &#xB7; 3x2 dx x3 Notice here it&#x2019;s much easier than &#xFB01;nding an antiderivative for sin4 . . . . . . .
• 45. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 . . . . . .
• 46. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 It turns out erf is the shape of the bell curve. . . . . . .
• 47. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 It turns out erf is the shape of the bell curve. We can&#x2019;t &#xFB01;nd erf(x), explicitly, but we do know its derivative. erf&#x2032; (x) = . . . . . .
• 48. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 It turns out erf is the shape of the bell curve. We can&#x2019;t &#xFB01;nd erf(x), explicitly, but we do know its derivative. 2 2 erf&#x2032; (x) = &#x221A; e&#x2212;x . &#x3C0; . . . . . .
• 49. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 It turns out erf is the shape of the bell curve. We can&#x2019;t &#xFB01;nd erf(x), explicitly, but we do know its derivative. 2 2 erf&#x2032; (x) = &#x221A; e&#x2212;x . &#x3C0; Example d Find erf(x2 ). dx . . . . . .
• 50. Erf Here&#x2019;s a function with a funny name but an important role: &#x222B; x 2 2 erf(x) = &#x221A; e&#x2212;t dt. &#x3C0; 0 It turns out erf is the shape of the bell curve. We can&#x2019;t &#xFB01;nd erf(x), explicitly, but we do know its derivative. 2 2 erf&#x2032; (x) = &#x221A; e&#x2212;x . &#x3C0; Example d Find erf(x2 ). dx Solution By the chain rule we have d d 2 2 2 4 4 erf(x2 ) = erf&#x2032; (x2 ) x2 = &#x221A; e&#x2212;(x ) 2x = &#x221A; xe&#x2212;x . dx dx &#x3C0; &#x3C0; . . . . . .
• 51. Other functions de&#xFB01;ned by integrals The future value of an asset: &#x222B; &#x221E; FV(t) = &#x3C0;(&#x3C4; )e&#x2212;r&#x3C4; d&#x3C4; t where &#x3C0;(&#x3C4; ) is the pro&#xFB01;tability at time &#x3C4; and r is the discount rate. The consumer surplus of a good: &#x222B; q&#x2217; CS(q&#x2217; ) = (f(q) &#x2212; p&#x2217; ) dq 0 where f(q) is the demand function and p&#x2217; and q&#x2217; the equilibrium price and quantity. . . . . . .
• 52. Surplus by picture c . onsumer surplus p . rice (p) s . upply .&#x2217; . p . . quilibrium e . emand f(q) d . . .&#x2217; q q . uantity (q) . . . . . .