ddiional Eercises for Caper 5 bou Lines, Slopes, and Tangen Lines 39. Find an equaion for e line roug e wo poins (, 7) and (5, ). 4. Wa is e slope-inercep form of e equaion of e line given by 3 + 5y + =? 4. Find an equaion of e line wi slope 3 roug ( 6, 7). 4. Skec e line wi slope and y-inercep. 43. Two lines L and L, neier of em verical nor orizonal, are given. Le eir slopes be m and m respecively. Sow a if e lines are perpendicular, en m = m. Conversely, sow a if m = m, en e lines are perpendicular. One way of doing is is oulined below. Move e lines wiou canging eir slopes so a ey bo go roug e origin O. Le e poins P = (, y ) and P = (, y ) and e angles α, α, β, and β be as sown y P = (, y ) α β' O α' β P = (, y ) L L in e figure. i. Suppose a P OP = 9 and sow a m = m. [Hin: If P OP = 9, en e wo riangles depiced are similar. Wy?] ii. Sow a if α and α are wo acue angles wi an α = an α, en α = α. [Hin: Use wo rig riangles wi e same base o illusrae is.] iii. ssume a m = m and sow a P OP = 9. [Hin: Noice a an α = an α and use (ii).] 44. Skec e grap of e equaion y = + 7. Le P = (, y) be any poin on is grap wi
y, and use Leibniz s meod o compue e slope of e angen o e curve a a poin. Find e slope of e angen a e poin (, 3) and deermine an equaion of is angen line. ns: ( 3, 9 4 ) 45. Consider e circle + y = r. Le (, y ) be any poin on e circle wi y and use Leibniz s angen meod o sow a e slope of e angen line o e circle a (, y ) is equal o y. i. Consider e radius of e circle from e origin o e poin (, y ) and sow a i is perpendicular o is angen. ii. Suppose a r =. Consider a line y = + b wi slope and y-inercep b. For 3 3 wic consans b is e line angen o e circle? ns: b = ± 3 ( ) 46. Use e Leibniz angen meod for y 3 = 3 + 7 o sow a e slope of e angen a a poin P = (, y) on e grap of is equaion is equal o m P =. (3 +7) 3 Use Facs (no limis) o Compue e Required Derivaives and niderivaives 47. Compue e derivaives of e funcions f() = 3, g() = 4 + 5, () = 6 3 7 3. 48. Use e conclusion of Eercises 45 and 46 o sow a e derivaive of f() = is f () = and a e derivaive of f() = (3 + 7) 3 is f () =. (3 +7) 3 49. Consider e funcion f() = 5 3 and le P = (, y) be some poin on is grap. Deermine an equaion for e angen line o e grap a P. [Suggesion: Cange e noaion for e coordinaes of P.] 5. Consider e parabola y = and e line y = 3 4. Sow a ey do no inersec. Move e line oward e parabola wiou canging is slope. wa poin will i firs ouc e parabola? 5. Find aniderivaives for e funcions f() = 3 3 +, g() = 3 + 8 3, and () = 4 + 3 + 7. 5. Use e resuls of Eercise 49 o find aniderivaives for g() = and g() = (3 +7) 3 53. Le P = (, y) be a poin on e grap of some funcion f. Illusrae e meaning of e raio f(+ ) f( ) f(+ ) f( ). Wa do you ink a lim is equal o? Ten prove a your answer is correc..
Problem from Tokyo 54. Waer is poured a a consan rae ino eac of e ree drinking glasses and ree vases sown below. Eac vessel is empy a e sar. Te grap under e glass labeled (a) represens e eig of e waer in e glass as a funcion of ime. Tink carefully wa is going on in e oer five cases and draw e ime/eig graps for eac. [Tis problem was adaped from a 6 adverisemen in a Tokyo subway for a uorial service for middle scool sudens.] a b c d e f 3
More reas and Differenials 55. Consider e funcion g() = 9 wi. Selec e poins.3.5.8 on e -ais beween and and compue e sum of e areas y d = g() d of all e recangles a is se of poins deermines. Do so wi ree decimal place accuracy. Tis sum is an approimaion of e area under e grap of g() = 9 over. Repea is compuaion (again wi ree decimal accuracy) wi e poins..3.5.7.8 o ge anoer approimaion of e area under e grap. Wic of ese wo approimaions would you epec o be beer? Use rcimedes s Teorem o ge e eac answer. ns: 8.779, 8.75, 6 3 8.667. 56. Consider e funcion f() = 6 wi. Selec e poins.6.4.5.4 on e -ais beween - and. Compue e sum of e areas y d = f() d of all e recangles a is seup deermines. Do so wi ree decimal place accuracy. Tis sum is an approimaion of e area under e grap of f() = 6 over. Use rcimedes s Teorem o compue is area precisely. ns: 58.499, 76 3 58.667. 57. Consider e funcion y = f() = 4. Is grap is e upper alf of e circle of radius wi cener e origin. Selec e poins <. <.4 <.5 <.6 <.8 < <. <.3 <.5 <.6 <.8 <.9 < on e -ais beween and, and compue e sum of e areas y d = f() d of all e recangles a is seup deermines (again wi ree decimal place accuracy). Observe a is sum is an esimae of e area under e upper alf of e circle and over e segmen from o on e -ais. Wa is is area equal o precisely? ns: 3.68, π 3.46. 58. Consider e funcion y = f() = 9. Is grap is e upper alf of e circle of radius 3 wi cener e origin. Selec e poins < < < < < < on e -ais beween and, and compue e sum of e areas y d = f() d of all e recangles a is seup deermines (again wi ree decimal place accuracy). Observe a is sum is an esimae of e definie inegral 9 d. Use e formula for e area of a circular secor o find e precise value of is inegral. 4
Compuing reas and Inegrals by using e Fundamenal Teorem 59. Skec e grap of e funcion f() = +. Compue e area under e grap and over e -ais from = o = 9. ns: 36 6. Consider e funcion f() = 3 + for 4. Skec is grap and compue e area under e grap and over e -ais. ns: 68 6. Consider e parabolic secion obained by cuing e parabola y = 3 + + wi e -ais. Epress e area of e parabolic secion as a definie inegral. Compue is area by applying e Fundamenal Teorem of Calculus and en again by using rcimedes s Teorem. ns: 3 7 6. Consider e parabolic secion obained by cuing e parabola y = + 7 6 wi e line y =. Epress e area of e parabolic secion as a definie inegral. To compue e area consider e applicaion of bo e Fundamenal Teorem of Calculus and rcimedes s Teorem. Coose e simpler of e wo meods. ns: 7 6 7 63. Eplain wy b a (f() + g()) d is equal o b a f() d + b a g() d firs by appealing o e definiion, and en again by using o e Fundamenal Teorem of Calculus. Using Derivaives and Inegrals 64. Consider a polynomial of e from f() = a + b + c wi a > and suppose a i as wo (real) roos. Locae e wo roos on e -ais and compue e midpoin beween em. How is is poin relaed o e minimum value of e funcion f()? Wa propery of e grap of f() confirms is connecion. ns: ( b a, ), f () = for = b a. 65. Use derivaives o compue e disance beween e line y = + 5 and e poin ( 4, 3). [Hin: Le (, y) be any poin on e line. Epress e disance beween ( 4, 3) and (, y) as a funcion of. Ten deermine e smalles value of e square of is disance.] ns: 8 5 66. Consider e parabola y = and cu i wi e line from (, ) o (4, 8) o obain e parabolic secion sown below. Sow a e equaion of e line of e cu is y = 3 +. 5
(4, 8) (-, ) O i. Make use of e Fundamenal Teorem o compue e area of e parabolic secion. ii. Compue e coordinaes of e vere V of e parabolic secion. iii. Use calculus o compue e disance from V o e line of e cu. sraegy of Eercise 65.] [Hin: Use e iv. Compue e area of e parabolic secion again, is ime wi rcimedes s Teorem. ns: i. 5, ii. ( 3, 9 8 ), iii. 5 4 3 67. Draw e grap of e funcion f() =. Le Q be a poin on e grap and ake e poin P on e -ais so a e segmen QP is perpendicular o e ais. Le be e area of e region under e grap of f and over e segmen from e origin O o P. Le B be e area of e riangle deermined by e angen o e grap a Q, e segmen P Q, and e -ais. Sow a = B no maer were Q is aken. 3 68. Consider e funcion f() = 4 = ( ). i. Sow a f() = for =,, and, bu for no oer. ii. Sow a f() for and a f() < for all oer. iii. Sow a f () = 4( ). iv. Wa is e slope of e angen o e grap a =? =? Find ose a wic e grap as a orizonal angen. v. For wa values of does f acieve is larges value? Wa is e larges value of f? 69. Coninue o consider e funcion f() = 4 = ( ) and is derivaive f () = 4( ). i. Sar wi a large negaive on e -ais. Move owards =. Is f () increasing or decreasing in e process? Now sar a = and move o e rig. Is f () increasing or decreasing? Wa do your answers ell you abou e slopes of e angen lines of e grap of f? ii. Use all e informaion you ave abou f and f o skec e grap of f. iii. pply e Fundamenal Teorem o sow a e area under e grap of f() from 6
o is equal o e area under e grap of f() from o. Wa basic feaure of e grap of f is is fac relaed o? Definie Inegrals and Lengs of Circular rcs 7. Consider e circle + y = 4. Refer o e diagrams below, and sow a e -coordinae B o 3 6 o (a) B 45 o 45 o (b) B 6 o 3 (a) o of e poin B is in (a), in (b), and 3 in (c). I follows, as in e discussion of e circle of radius 5 a concludes Secion 5., a e slope of e angen o e circle a any poin (, y) wi y is. Use is o sow a e derivaive of f() = 4 y is f () = 4. Deduce from is and e diagrams a d = π 4 6, d = π 3 4 4, and 4 d = π 3. 7. Consider e funcion f() = r. Te figure below sows is grap, e poin = (, r), a poin B in e firs quadran, and e angle θ wi θ π a B deermines. Use e B θ r fac a f () = r, o verify e equaliy r cos θ r d = π θ. Definie Inegrals as reas, Volumes, and Lengs of Curves 7. Te definie inegral 5 + 4 d is bo i. e area under e grap of e funcion f() = over e inerval, and ii. e leng of e grap of e funcion f() = poin (, ). 7 from e poin (, ) o e
73. Te definie inegral 3 + d is i. e area under e grap of f() = from = o =, as well as ii. e volume obained by roaing a region under e grap of g() = revoluion abou e -ais, and also iii. e leng of a piece of e grap of () =. 74. Sow a e grap of f() = + 4 is e upper alf of a yperbola and skec i. i. Compue e volume of e solid obained by roaing e region below e grap and above e segmen 6 one complee revoluion around e -ais. ii. Use Leibniz s angen meod for e curve y = + 4 o sow a e derivaive of e funcion f is f () =. +4 iii. Epress as a definie inegral e leng of e yperbolic arc from e poin (, 4) o e poin (3, 5). iv. Compue e area under e grap of e funcion g() = 5. ns: i. 4π 3, iv. 4 7 +4 one and above e segmen 75. Epress as definie inegrals e volumes obained by roaing e graps of e funcions y = sin, π, and y = cos, π, one revoluion around e -ais. Use e relaionsip beween em, o compue eac of ese volumes. bou Hyperbolas 76. Deermine e and y inerceps of e yperbolas 5 y 3 = and y 3 5 = and eir asympoes. Skec eir graps. 77. Skec e asympoes of e yperbolas y = for b equal o,, and on e same b ais sysem. Draw in e graps of e ree yperbolas. 78. Skec e asympoes of e yperbolas y = for b equal o,, and 4 on e same a 4 ais sysem. Draw in e graps of e ree yperbolas. 79. Skec e asympoes of e yperbolas y = for b equal o,, and on e same b ais sysem. Draw in e graps of e ree yperbolas. 8. Sudy e soluion of Eercise 6 of Caper 4 and en sow a e focal poins of e yperbola y = are ( a a b + b, ) and ( a + b, ). 8
n sserion of Leibniz 8. Leibniz assers e following (see e boom of e rig column on page 5): nd i was no difficul for me o figure ou a e descripion of is curve could be reduced o e quadraure of e yperbola. Refer o Eercise 9. Te Fundamenal Teorem of Calculus ells us a e problem of finding an aniderivaive of y = a (e sign is no relevan in is regard) is closely relaed o e problem of finding e area under e grap of is funcion. So is is e quadraure a Leibniz appears o ave ad in mind. Tus Leibniz seems o ink a e grap of y = a, for < a, lies on a yperbola. Is is correc? [Hin: Recall from Secion 5. a any yperbola (in fac any conic secion) is given by an equaion of e form + By + Cy + D + Ey + F = for some consans, B, C, D, E, and F, no all of wic are zero. ssuming a Leibniz is correc, y = a saisfies suc an equaion. Tis implies, afer subsiuing and rearranging ings, a C(a ) + E a = 4 B a D 3 F for all wi < a. Now pus o zero and conclude a C mus be zero. fer cancelling, pus o zero again and conclude a E =. So B a = D F. Sow a e grap of y = B a as a verical angen a = a and conclude a B =. Bu is implies a + D + F = for < a. So all e consan, B, C, D, E, and F are zero.] 9