CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.................... Section. Plane Curves and Parametric Equations.......... 8 Section. Parametric Equations and Calculus............ 9 Section. Polar Coordinates and Polar Graphs............ 7 Section. Area and Arc Length in Polar Coordinates........ Section. Polar Equations of Conics and Kepler s Laws....... Review Eercises............................. Problem Solving..............................
CHAPTER Conics, Parametric Equations, and Polar Coordinates Section. Conics and Calculus.. Verte:, p > Opens to the right Matches graph (h). 8. Verte:, p > Opens upward Matches graph (a). Verte:, p < Opens downward Matches graph (e).. Center:, Ellipse Matches. 9. 9 9 Center:, Circle radius. Ellipse Matches (g) Matches (f) 7. 8. 9 Hperbola Hperbola Center:, Center:, Vertical transverse ais Matches (c) Horizontal transverse ais Matches (d) 9.. 8 Verte:, Focus: 8, Directri: (, ) 8 Verte:, Focus:, Directri: 8 (, ) 8 8 8.. 8 Verte:, Focus:., Directri:.7 8 (, ) Verte:, Focus:, Directri: 8 8 8 (, )
Section. Conics and Calculus.. 8 9 8 9 Verte:, Focus:, Directri: (, ) Verte:, Focus:, Directri: 8 (, ) 8 8.. 8 8 Verte:, Focus:, Directri: (, ) Verte:, Focus:, Directri: 8 (, ) 7. 8., Verte: Focus:, Directri: ± Verte:, Focus:, Directri: 9 8 8 9.. 8 9 8 9 Verte: Focus:,, Directri: Verte:, Focus:, Directri: 8
Chapter Conics, Parametric Equations, and Polar Coordinates.. 8 8. h p k 9. Verte:, 8.. 7. Since the ais of the parabola is vertical, the form of the equation is a b c. Now, substituting the values of the given coordinates into this equation, we obtain c, 9a b c, a b c. Solving this sstem, we have Therefore, a, b, c. or 9. 8. From Eample : p 8 or p Verte:, 8 8 8 9.. 7 7 a, b, c (, ) a, b, c Center:, Foci: ±, Vertices: ±, Center:, Foci: ±, Vertices: ±, e e 7. 9 a, b 9, c Center:, Foci:, 9,, Vertices:,,, e 8 (, ) 8. b c a,, Center:, Foci: ±, Vertices:,,, (, ) e
Section. Conics and Calculus. 9 9 9 a 9, b, c Center:, Foci:, ± Vertices:,,, (, ) e 9. 79. 79 a, 8, b, c a b 9 Center:, Foci: ±, Vertices: ±, e c a 8 Center: Foci: 7 7 Vertices: Solve for : a, b, c, ±, ±, 7 (, ) 7 7 ± (Graph each of these separatel.)
Chapter Conics, Parametric Equations, and Polar Coordinates. b c a,, Center: Foci: Vertices: 9 8 7. 9 9, ± Solve for :,,,, 9 9 8 8 7 9 ± 8 7 Center: Foci: Vertices: Solve for :. 9 9 a, b, c, ±, (Graph each of these separatel.),, 7, 7 7 ± 8 (Graph each of these separatel.) 8..8.. 78 78 7 a, b, c Center: Foci: Vertices: Solve for :,, ±, ±...8... 7.. ± 7. 7 7 (Graph each of these separatel.)
Section. Conics and Calculus 9. Center:, Focus:, Verte:, Horizontal major ais a, c b 9. Vertices: Eccentricit: Horizontal major ais Center:, a, c b,,,. Vertices:,,, 9 Minor ais length: Vertical major ais Center:, Foci:, ± Major ais length: Vertical major ais Center:, a, b c, a 7 b 9 9. Center:,. Center:, Horizontal major ais Vertical major ais Points on ellipse:,,, Points on ellipse:,,, Since the major ais is horizontal, a b. Substituting the values of the coordinates of the given points into this equation, we have 9 a b, and. a From the sketch, we can see that h, k, a, b. (, ) (, ) (, ) The solution to this sstem is a, b 7. Therefore, 7,. 7.. 9 a, b, c a, b, c a b Center:, Vertices:, ± Foci:, ± Asmptotes: ± Center:, Vertices: ±, Foci: ±, Asmptotes: ± 8 8 8 8
Chapter Conics, Parametric Equations, and Polar Coordinates 7. 8. a, b, c a, b, c a b Center:, Vertices:,,, Foci: ±, Center:, Vertices:,,, Foci:,,, Asmptotes: ± Asmptotes: ± 7 8 9. 9 8. 9 7 9 9 8 9 9 7 9 a, b, c a, b, c a b Center:, Vertices:,,, Foci: ±, Asmptotes: ± Center:, Vertices:,,, Foci:,,, Asmptotes: ±. 9 9 8 8 Degenerate hperbola is two lines intersecting at,. 9 8 9 ±. 9 9 78 8 a b,, Center: Vertices: Foci: 9 c,,,,, ± 9 Asmptotes: ±
Section. Conics and Calculus 7. 9. 9 9 9 8 8 9 9 8 8 a, b, c 9 Center:, Vertices:, ± Foci:, ± Solve for : 9 9 8 9 9 7 7 a b,, Center: Vertices: Foci: Solve for : c, ±, ±, 8 ± 9 9 (Graph each curve separatel.) 9 8 ± 9 8 (Graph each curve separatel.). 7. 9 7 8 9 9 a, b, c Center:, Vertices:,,, Foci: ±, Solve for : 7 7 a, b, c Center:, Vertices:,,, Foci:,,, Solve for : 9 7 8 9 ± ± (Graph each curve separatel.) (Graph each curve separatel.) 7. Vertices: ±, Asmptotes: ± Horizontal transverse ais Center:, 8. Vertices:, ± Asmptotes: ± Vertical transverse ais a a, ± b a ±b ± b Slopes of asmptotes: ± a ± b Therefore,. 9 Thus, b. Therefore,. 9
8 Chapter Conics, Parametric Equations, and Polar Coordinates 9. Vertices:, ± Point on graph:, Vertical transverse ais Center:, a Therefore, the equation is of the form. 9 b Substituting the coordinates of the point,, we have 9 b or b 9.. Vertices:, ± Foci:, ± Vertical transverse ais Center:, a, c, b c a Therefore,. 9 Therefore, the equation is. 9 9. Center:, Verte:, Focus:, Vertical transverse ais a, c, b c a Therefore,.. Center:, Verte:, Focus:, Horizontal transverse ais a, c, b c a Therefore,. 9. Vertices:,,,. Focus:, Asmptotes:, Asmptotes: ± Horizontal transverse ais Center:, a Horizontal transverse ais Center:, since asmptotes intersect at the origin. c Slopes of asmptotes: ± b a ± Slopes of asmptotes: ±b and b a ± a Thus, b. Therefore,. 9 c a b Solving these equations, we have a and b. Therefore, the equation is.. (a) 9, 9, From part (a) we know that the slopes of the normal lines 9 must be 9. At : ±, At, : or At, : ± 9 9 9 or ± 9 At At, :, : 9 or 9 9 or 9
Section. Conics and Calculus 9. (a),,, 7. 9 9 At : ±, ± ± Ellipse At, : or At, : or From part (a) we know that the slopes of the normal lines must be. At, : or At, : or 8. 9. Hperbola Parabola 7. 9 7. 9 Parabola Circle (Ellipse) 7. 7. 9 9 9 9 9 9 9 9 Parabola Circle (Ellipse) 7. 7. ) 9 Hperbola Ellipse 7. 9 Ellipse 9 9 77. (a) A parabola is the set of all points, that are equidistant from a fied line (directri) and a fied point (focus) not on the line. h p k or k p h (c) See Theorem..
7 Chapter Conics, Parametric Equations, and Polar Coordinates 78. (a) An ellipse is the set of all points,, the sum of whose distance from two distinct fied points (foci) is constant. 79. (a) A hperbola is the set of all points, for which the absolute value of the difference between the distances from two distance fied points (foci) is constant. h a k b or h b k a h a k b or k a h b (c) k ± b h a or k ± a h b 8. e c < e < a, c a b, For e, the ellipse is nearl circular. For e, the ellipse is elongated. 8. Assume that the verte is at the origin. The pipe is located 9 p p 9 p meters from the verte. (, ) (, ) Focus 8. Assume that the verte is at the origin. (a) p 8 p p The deflection is cm when ± 8 ±. meters. ( ) 8,, 8, ( ) ( ) 8. a a The equation of the tangent line is a a or a a. Let. Then: a a a Therefore, a a, is the -intercept. 8 8 = a (, a) (, )
Section. Conics and Calculus 7 8. (a) Without loss of generalit, place the coordinate sstem so that the equation of the parabola is p and, hence, p. Therefore, for distinct tangent lines, the slopes are unequal and the lines intersect. d At,, the slope is :. At,, the slope is : 9. Solving for, 9 9. d Point of intersection:, 8. (a) Consider the parabola p. Let m be the slope of the one tangent line at, and therefore, m is the slope of the second at,. Differentiating, p or and we have: p, m p or pm m p or p. m Substituting these values of into the equation p, we have the coordinates of the points of tangenc pm, pm and pm, pm and the equations of the tangent lines are pm m pm and The point of intersection of these lines is pm, p m ( ) p p, m m p p m m m. = p and is on the directri, p. 8 Verte: At,, At d. Tangent line at, : Tangent line at Since, d Point of intersection: d, :. m m,,, d.. the lines are perpendicular. = p ( pm, pm ) ( ) pm ( ), p m Directri: and the point of intersection lies on this line.,
7 Chapter Conics, Parametric Equations, and Polar Coordinates 8. The focus of 8 is,. The distance from a point on the parabola,, 8, and the focus,,, is d Since d is minimized when f f 8. f implies that 8. 8. d is minimized, it is sufficient to minimize the function. This is a minimum b the First Derivative Test. Hence, the closest point to the focus is the verte,,. (, ) (, 8 ) = 8 87. d At the point of tangenc on the mountain, m,. Also, Choosing the positive value for, we have. Thus, m m ±. The closest the receiver can be to the hill is.. ± ± m. (, ) (, ) (, )
Section. Conics and Calculus 7 88. (a) A.7t 9.t (c) da.t 9., dt t 8 8 Women are spending more time watching TV each ear. 89. Parabola Verte:, p p p 8 8 Circle Center:, k Radius: 8 k k k 8 k (Center is on the negative -ais.) ± Since the -value is positive when, we have. A arcsin 8 8 arcsin d. square feet 9. s ln ln ln.9 ln
7 Chapter Conics, Parametric Equations, and Polar Coordinates 9. (a) Assume that a. a a 8 8 f 8, S f 9 9 d 9 9 9 ln 9 9,7 9 ln,7 9 ln 9 9 8 9 ln 9 ln 9 9 ln 9 9 9 d 9 ln 8. m (Formula ) (, ) (, ) 9. 9. p, p,,,, As p increases, the graph becomes wider. S r d r r d r 8 p = p = p = p = p = 8 9. A h p ph p h 9. (a) At the vertices we notice that the string is horizontal and has a length of a. The thumbtacks are located at the foci and the length of string is the constant sum of the distances from the foci. Focus Verte Focus Verte 8 ph 9. c a b,, 97. The tacks should be placed. feet from the center. The string should be a feet long. 7 8 9 7 9 8 7 7
Section. Conics and Calculus 7 98. e c a 99. e c a.7 c 9,98, c,98,8. A P a a A P Least distance: a c 7,99,7. km Greatest distance: a c,9,8. km c a P A P e c a P A P A P A P A P A P A c P ( a, ). e A P A P, 9, 9,88.97,9. e A P.9.9.97 A P.9.9. a a b a a a a c a a a e As e, e and we have a a a b. or the circle a. 8 At 8, : The equation of the tangent line is 8. It will cross the -ais when and 8...... ± 9. ft ft V Area of bottomlength Area of toplength V... 9.. arcsin.. 9 (Recall: Area of ellipse is ab.). 9 ft 9.. arcsin. ft
7 Chapter Conics, Parametric Equations, and Polar Coordinates. 9 9. 8 9 when. At, or. Endpoints of major ais: At, or. Endpoints of minor ais: Note: Equation of ellipse is 8 9 is undefined when.,,,,,, 9 9 8 9 8 8 when. At, or. Endpoints of major ais: At, or. Endpoints of minor ais: Note: Equation of ellipse is 8 8 undefined when.,,,,,, 8 8 9 7. (a) A d Disk: arcsin V d 8 or, A ab S d (c) Shell: arcsin 9 9.8 S 8 ln V d d ln d.9 8
Section. Conics and Calculus 77 8. (a) A d Disk: (c) Shell: S 9 d V 87 7 7 arcsin 7 V 9 arcsin 9 d 9 d 9 8 8 7 7 d d 7 87 arcsin 87 8.9 9 S 9 99 99 9 99 9 8 7 7 78 7 8 ln 7 8 7 8 7 8 ln7 8 ln 9 8. 97 9. From Eample, For C a e sin d, 9 we have a 7, b, c 9, e c a 7. C 7 9 sin d 8.8 7.9
78 Chapter Conics, Parametric Equations, and Polar Coordinates. (a) Slope of line through c, and, : a b a b Slope of line through c, and, : m c m c b a (c) At P, b tan m m m m a b b c a b a c a b b c c a c b a c c c a b c arctan b tan m m m m a b b c a b b c b a c b a a c b a b a c c c a c arctan b, m. a c arctan b c c c b a c b a c b a c b a a b c a c b a b c a c b Since the tangent line to an ellipse at a point P makes equal angles with the lines through P and the foci.. Area circle r Area ellipse ab a a a Hence, the length of the major ais is a.. (a) e c Hence, a a b ea a a b. h a h a k b k. a e e 7 9 (c) As e approaches, the ellipse approaches a circle.. The transverse ais is horizontal since, and, are the foci (see definition of hperbola). Center:, c, a, b c a 7 Therefore, the equation is 9 7.
Section. Conics and Calculus 79. The transverse ais is vertical since, and, are the foci. Center:, c a, b c a, Therefore, the equation is.. a a c b 7 8 9 7 9 8 7 7. Center:, Horizontal transverse ais Foci: ±c, Vertices: ±a, The difference of the distances from an point on the hperbola is constant. At a verte, this constant difference is a c c a a. Now, for an point, on the hperbola, the difference of the distances between, and the two foci must also be a. c c a c a c c a a c c c a a c c a a c c a c a a c c (, ) c a a a c a a c a ( c, ) ( a, ) ( a, ) (, c ) Since a b c, we have a b. 7. Time for sound of bullet hitting target to reach c Time for sound of rifle to reach, : Since the times are the same, we have:, : c v m c v s v s c v m c v s c v s ( c, ) rifle (, ) (, c ) target c c c v m v m v c c s v s v s c v m v s c v s v m v m v s v s v m c c v s v m c v m v s v m
8 Chapter Conics, Parametric Equations, and Polar Coordinates 8. c, a.8,, a 9, b 9,8 9,8 When 7, we have 9 7,8. miles. 9. The point, lies on the line between, and,. Thus,. The point also lies on the hperbola. Using substitution, we have: 9 7 7 8 7 Choosing the positive value for we have: 9 9 7 9 7 8 ± 8 77 7.8 and. 8 ± 9 9 ± 9 7. a b a or b b a b a a a b b b a b a a b b a a b. Let c a b. a b b a. a b b a a a b a b ± aa b ac ± a b a b b a c a a b b b a b There are four points of intersection: a b a b b CONTINUED b a b a a b b a b ± b a b a b e b a c b h b c a b ac a b, ± b a b, ac a b, ± b a b
Section. Conics and Calculus 8. CONTINUED At ac b the slopes of the tangent lines are: a b, a b, b ac e b a c a a b a b and b ac h b c a c. a b a b Since the slopes are negative reciprocals, the tangent lines are perpendicular. Similarl, the curves are perpendicular at the other three points of intersection.. A C D E F (Assume A and C ; see below) A D D A A C E E D E C C F A C R D A C (a) If A C, we have A D A C E C F C E D A E C R A which is the standard equation of a circle. (c) If AC <, we have D A R A which is the equation of an ellipse. A C E R C R AC If C, we have A D A F E D A. If A, we have C E C F D E C. These are the equations of parabolas. (d) If AC <, we have D A R A C E R C which is the equation of a hperbola. ±. False. See the definition of a parabola.. True. True. False. 7. True 8. True ields two intersecting lines: ± 9. Let be the equation of the ellipse with a > b >. Let ±c, be the foci, a b c a b. Let u, v be a point on the tangent line at P,, as indicated in the figure. b a a b b a b a Slope at P, b (u, v) a d P(, ) c F F c a CONTINUED b a + b =
8 Chapter Conics, Parametric Equations, and Polar Coordinates 9. CONTINUED Now, Since there is a right angle at u, v, We have two equations: Multipling the first b v and the second b u, and adding, Similarl, v u a b vb a u. v u b a a a v b b u a b a v b u a v b u a b a u b v. a b a v b u a v a u a b v u v b v d b v u d a. v d b. From the figure, u d cos and v d sin. Thus, cos d and cos sin d a d b b d a d a b a sin d b. Let r PF and r PF, d a b b a r r a. r r r r r r a c c a c a b a Finall, d r r b a b a b a b b b a a a b a b b a b a a a b b a b a b a b a b a b a b, a constant!
Section. Plane Curves and Parametric Equations 8. Consider circle and hperbola 9. v, 9 v Let u, u and be points on the circle and hperbola, respectivel. We need to minimize the distance between these points: Distance f u, v u v u 9 v. 9 = ( ) 9 v, v ( u, u ) + = The tangent lines at, and, are both perpendicular to, and hence parallel. The minimum value is 8. Section. Plane Curves and Parametric Equations. t, t (a) t (c) (d) t,. cos sin (a) (c) (d) cos sin, (e) The graph would be oriented in the opposite direction.. t. t t t 8
8 Chapter Conics, Parametric Equations, and Polar Coordinates. t. t 7. t t t t, For t <, the orientation is right to left. t implies t For t >, the orientation is left to right. 8. t t, t t Subtracting the second equation from the first, we have t or t. t Since the discriminant is B AC, the graph is a rotated parabola. 9. t, t. t, t. t t, t, t t. t t implies t t. t t 8 8
Section. Plane Curves and Parametric Equations 8. t. e t, >. e t, > t e t, > e t, > 7. sec 8. tan 9. cos, sin cos <, < sec sec tan Squaring both equations and adding, we have 9.,. cos. sin. cos sin cos sin cos sin ellipse sin cos sin cos sin ±
8 Chapter Conics, Parametric Equations, and Polar Coordinates. cos sin. cos sin cos sin cos sin 7 8. cos. sec 7. sec sin tan tan cos sin sec tan sec 9 tan 9 9 9 8. cos 9. t. ln t sin ln t t cos sin ln ln t e. e r e.
Section. Plane Curves and Parametric Equations 87. e t. e t e t e t e t e t >, > > >. B eliminating the parameters in (a) (d), we get. The differ from each other in orientation and in restricted domains. These curves are all smooth ecept for. (a) t, t cos cos d when d d, ±, ±,.... (c) e t e t (d) e t e t > > > >
88 Chapter Conics, Parametric Equations, and Polar Coordinates. B eliminating the parameters in (a) (d), we get. The differ from each other in orientation and in restricted domains. These curves are all smooth. (a) cos, sin t t t t, (c) t t (d) e t e t < > <.. The curves are identical on < The are both smooth. Represent. The orientations are reversed. The graphs are the same. The are both smooth. 7. (a) The orientation of the second curve is reversed. (c) The orientation will be reversed. (d) Man answers possible. For eample, t, t, and t, t. 8. The set of points, corresponding to the rectangular equation of a set of parametric equations does not show the orientation of the curve nor an restriction on the domain of the original parametric equations. 9. t t t m. cos h r sin k r cos sin h k r h r cos k r sin h r k r
Section. Plane Curves and Parametric Equations 89. h a cos. h a sec k b sin k b tan h a cos h a sec k b sin k b tan h a k b h a k b. From Eercise 9 we have t t. Solution not unique. From Eercise 9 we have t t. Solution not unique. From Eercise we have cos sin. Solution not unique. From Eercise we have cos sin. Solution not unique 7. From Eercise we have a, c b cos sin. Center:, Solution not unique 8. From Eercise we have a, c b cos sin. Center:, Solution not unique 9. From Eercise we have. From Eercise we have. a, c b a, c b Eample sec tan t, t tan. sec. t, t Center:, Solution not unique Center:, Solution not unique The transverse ais is vertical, therefore, and are interchanged.. Eample t, t. Eample t, t t, t. Eample t, t t, t t, t tan t, tan t. sin 7.. sin cos cos sin cos Not smooth at n Not smooth at n 7
9 Chapter Conics, Parametric Equations, and Polar Coordinates 8. sin 9. cos. sin cos sin cos 9 9 9 Not smooth at, ±, and, ±, or n. Smooth everwhere. cot sin. t t t t Smooth everwhere Smooth everwhere. See definition on page 79.. Each point, in the plane is determined b the plane curve f t, gt. For each t, plot,. As t increases, the curve is traced out in a specific direction called the orientation of the curve.. A plane curve C, represented b f t, gt, is smooth if and are continuous and not simultaneousl. See page 7. f g. (a) Matches (iv) because, is on the graph. Matches (v) because, is on the graph. (c) Matches (ii) because and. (d) Matches (iii) because, is on the graph. (e) Matches (vi) because undefined at. (f) Matches (i) because for all. 7. When the circle has rolled radians, we know that the center is at a, a. sin sin8 C b BD b cos cos8 AP or b Therefore, a b sin and a b cos. or BD b sin AP b cos P b A C θ a B D 8. Let the circle of radius be centered at C. A is the point of tangenc on the line OC. OA, AC, OC is the point on the curve being traced out as the angle changes AB. P, AP, AB and AP. Form the right triangle CDP. The angle OCE and DCP OE E sin sin. EC CD sin cos sin sin cos cos θ A C α D B E P = (, ) Hence, cos cos, sin sin.
Section. Plane Curves and Parametric Equations 9 9. False t t The graph of the parametric equations is onl a portion of the line. 7. False. Let t and t. Then and is not a function of. 7. (a) (c) mihr 8 h v sin t t It is not a home run when, <. v cos t cos t sin t t ftsec (d) We need to find the angle (and time t) such that From the first equation t cos. Substituting into the second equation, We now solve the quadratic for tan : tan.8 cos t sin t t. sin cos cos 7 tan sec tan tan. tan tan 7 9. Yes, it s a home run when, >. 7. (a) v cos t h v sin t t t. h tan sec v h, v cos tan Hence, 8 cost h v sin v cos h tan sec v and,. sec v v v. v 8. 8 sint t. v cos (c) 8 (d) Maimum height: Range:.88 at
9 Chapter Conics, Parametric Equations, and Polar Coordinates Section. Parametric Equations and Calculus. dt d ddt t t. dt d ddt t t. d d dd cos sin sin cos Note: and d d. e e d dd e e. t, t. t, t dt d ddt when t. d t t d d Line d d t t concave upwards 7. t, t t 8. t t, t t when t. d d t when t. d d concave upwards d t d t t 7 when t. concave downward 9. cos, sin. cos, sin d cos cot when sin. d cos cot sin d is undefined when. d d csc csc sin when. d d csc d sin sin d is undefined when. concave downward. sec, d sec sec tan sec csc tan d d d d d d when cot when concave downward tan csc cot sec tan... t, t t d t t when t. t d t t tt t d t when t. t concave downward
Section. Parametric Equations and Calculus 9. cos, sin. sin, cos d sin cos cos sin d sin cos when. tan when d d sec cos sin cos sin sec csc when.. cos cos sin d d cos cos when cos concave downward. concave upward. cot, sin. cos, sin d sin cos sin csc cos d cos sin cot At,, Tangent line:, and d 8. 8 8 8 At,, and is undefined. d Tangent line: At,,,, and. d At,, Tangent line:, and. d Tangent line: At,, 7, and d. At,, Tangent line:, and d 8. 8 8 Tangent line: 7. t, t, t 8. t,, t t (a) (a) At t,,,, (d) d, dt, dt. d and (c). At,, d. (d) d, dt At t,,,, dt, and d. (c) At,, d.. (, )
9 Chapter Conics, Parametric Equations, and Polar Coordinates 9. t t, (a) t t, t. cos, (a) sin, 8 At t,,,, (d) d dt, (, ), dt. d 8 and (c). At,, d At (c) (d), d dt, At d.,,,,, dt, and d... sin t, sin t crosses itself at the origin,,,. At this point, t or t. cos t d cos t At t : and Tangent Line d. At t, and Tangent Line d.. cos t, t sin t crosses itself at a point on the -ais:,. The corresponding t-values are t ±. dt cos t, At t : d. Tangent line: At t : d. Tangent line: d dt sin t, d cos t sin t. t t, t t crosses itself at the point,,. At this point, t or t. d t t At t, and. Tangent Line d At t, and or. Tangent Line dt 9
Section. Parametric Equations and Calculus 9. t t, t crosses itself at,. The corresponding t-values are t ±. At At d t t t, d. Tangent line: t, d. Tangent line:. cos Horizontal tangents: Vertical tangents: Note: d sin, when Points:, n,, n where n is an integer. Points shown:,, when corresponds to the cusp at,,. sin cos d d sin d sin cos cos tan at ± ±,, ± ±,, ±,... ±,... Points: n n, Points shown:,,,,, n,,,., cos 7. t, t Horizontal tangents: ±,... sin when d, ±, Horizontal tangents: Point:, t when t dt Points: n,, n, where n is an integer Points shown: Vertical tangents:,,,,, d ; none d Vertical tangents: d ; dt none (, ) 8. t, t t Horizontal tangents: Point:, 9 Vertical tangents: t when t dt d ; dt none
9 Chapter Conics, Parametric Equations, and Polar Coordinates 9. t, t t. t t, t t Horizontal tangents: when t ±. dt t Horizontal tangents: when t ±. dt t Points:,,, Points:,,, Vertical tangents: d ; dt none Vertical tangents: d t when t dt. (, ) Point: 7, 8 (, ). cos, sin. cos, sin Horizontal tangents: Points: Vertical tangents: Points: (, ) (, ) (, ) cos when d,,, d sin when d,,,,.,. Horizontal tangents: Points: Vertical tangents: Points:,,, cos when d,,,, d sin when d,,, 7.,,,,. (, ). cos, sin. cos, sin Horizontal tangents: Points:,,, cos when d,. Horizontal tangents: cos when,. Since at and, eclude them. dd d Vertical tangents: Points:,,, d sin when d,. Vertical tangents: d 8 cos sin when d,. (, ) 8 Point:, (, ) (, ) (, ). sec, tan Horizontal tangents: Vertical tangents: Points:,,, sec ; none d d sec tan when,. d (, ) (, )
Section. Parametric Equations and Calculus 97. cos,7. cos t, t t Horizontal tangents: sin when,. d t t Since at these values, eclude them. tt t d d Vertical tangents: cos sin when d t t dd Point:, d d,. Eclude,. t 8t t t Concave upward for t > Concave downward for t < 8. t, t t 9. t t d t t d d t t Concave upward for t > Concave downward for t < t ln t, t ln t, t > t t d t t d d t t t t t t t t t d Because t >, d > Concave upward for t >. t, ln t, t >. sin t, cos t, < t < t d t t d sin t cos t tan t d d t d t t t d sec cos t cos t d Because t >, d < Concave upward on < t < Concave downward on < t < Concave downward for t >. cos t, sin t, < t <. t t, t, t cos t d sin t cot t d d csc t sin t sin t Concave upward on < t < Concave downward on < t < d t, t dt dt b s d a dt dt dt t t dt 8t t 9t dt t t dt
98 Chapter Conics, Parametric Equations, and Polar Coordinates. ln t, t, t. d dt t, dt b s d a dt dt dt dt t e t, t, t d dt et, dt b s d a dt dt dt e t dt. t sin t, t cos t, t 7. d cos t, sin t dt dt b s d a dt dt dt cos t sin t dt cos t sin t dt t, t, t d t, dt, dt s t dt d dt dt t t tt ln t t ln.9 8. t, t, t d t, dt s d dt dt t, t t dt t 7.9 dt t t t t dt 9. e t cos t, e t sin t, t d dt et cos t sin t dt et sin t cos t, s d dt dt dt e t dt e t dt e t e.. arcsin t, ln t t, d dt t, ln t t dt t t t t s d dt dt dt t dt ln ln.9 t dt. t, s u du ln u u u u ln7 7.9 u t, t, t 9 dt du t dt d dt t, dt t dt t
Section. Parametric Equations and Calculus 99. t, S t dt t t dt t t t, t t dt t 779 t d, dt dt t t. d a cos, a sin, a cos sin, a sin cos d s 9a cos sin 9a sin cos d a sin cos cos sin d a d sin d a cos a. d a cos, a sin, a sin, S a sin a cos d a a d a d a cos d. a sin, d a cos, a sin d s a cos a sin a cos d sin a cos a cos, d d d a cos 8a. cos d S sin sin, d sin cos sin d cos, d d cos 7. 9 cos t, (a) 9 sin t t Range: 9. ft, t d (c) 9 sin t dt 9 cos, dt for t. s 9 cos 9 sin t dt.8 ft
Chapter Conics, Parametric Equations, and Polar Coordinates 8. 9 sin t t t, 9 sin 9 cos t 9 cos 9 sin 9 sin cos 9 sin 9 cos B the First Derivative Test, 8 maimizes the range. feet. d To maimize the arc length, we have dt 9 cos, dt 9 sin t. 9sin s 9 cos 9 sin t dt Using a graphing utilit, we see that s is a maimum of approimatel.7 feet at.98.. sin cos ln sin sin 9. t t t, t (a) (c) s t t t t dt t 8 t 8 t t t t t dt.7 dt t 8t t t t t t t Points:,, when t or t.,.799,. t t 8 t t t t dt. cot, sin, tan (a) d 8 sin cos (c) Arc length over t :.8 d csc d for d, ± Horizontal tangent at,, ± (Function is not defined at
Section. Parametric Equations and Calculus. (a) t sin t cos t t t sint cost t The average speed of the particle on the second path is twice the average speed of a particle on the first path. (c) t t sin cos t The time required for the particle to traverse the same path is t.. (a) First particle: Second particle: cos t, sin t, t sin t, cos t, t There are points of intersection. (c) Suppose at time t that cos t sin t tan t and and sin t cos t tan t Yes, the particles are at the same place at the same time for tan t. t.,.78. The intersection points are.,. and.,. (d) The curves intersect twice, but not at the same time.. d t, dt. t, dt S t dt.9 7t dt 87 d dt t, t t, dt S t t dt t t dt 9.. d cos,. cos sin cos, sin S cos cos sin sin d. d d sin, d cos, d S. d cos sin cos cos sin d cos cos sin d 7. t, (a) t, S d, dt dt t dt t dt t S t dt t t dt
Chapter Conics, Parametric Equations, and Polar Coordinates 8. t, (a) t, d, dt S t dt t t dt 8 S t dt t 9. d cos, sin, sin, cos S cos sin cos d d d cos d sin 7. t, t, t, -ais d dt t, dt S t t dt 9 7.8 9 7. d a cos, a sin, a cos sin, a sin cos S d d a sin 9a cos sin 9a sin cos d a sin a cos d sin a 7. d a cos, b sin, a sin, b cos (a) S b sin a sin b cos d ab sin a b ab b a b a b arcsin a b a b ab e arcsine e a b a S a cos a sin b cos d a cos b c sin d a a e e cos e cos arcsine cos d c a : eccentricit a cos d ab e c c c sin b c sin b ln c sin b c sin a c c b c b ln c b c b ln b a ab a b ln a a b b a b e ln e d ab e e e arcsine c cos b c sin d e sin e cos d e dt 7. d ddt See Theorem.7. 7. (a)
Section. Parametric Equations and Calculus 7. One possible answer is the graph given b t, t. 7. One possible answer is the graph given b t, t. 77. s b d a dt dt dt See Theorem.8. 78. (a) b S gt a d dt dt dt b S f t d dt dt dt a 79. Let be a continuous function of on a b. Suppose that f t, gt, and ft a, f t b. Then using integration b substitution, d ft dt and b d t gtft dt. a t 8. r cos, S r sin r sin r cos d r sin d r sin r cos θ r cos 8. sin sin tan d sin cos d A sin tan sin cos d 8 sin cos 8 sin d 8 sin cos 8 θ < 8. d cot, sin, csc A d sin csc d 8 d 8 8. ab8. is area of ellipse (d). 8a is area of asteroid. 8.is a area of cardioid (f). 8 8. a is area of deltoid (c). 87. ab is area of hourglass (a). 88. ab is area of teardrop (e).
Chapter Conics, Parametric Equations, and Polar Coordinates 89. t, A t, d A A d,, 8 < t < d t t dt tt t t dt t t t dt 8t tt dt t dt t t t 8t t dt t tt t t 8 9. t, A Let u t, then du t dt and t u. tt dt t dt t t 8 t t dt dt t t, 8, 8 t, t t dt d dt t, t u du u u arcsin u 8 t t 8 9. d cos,9. sin sin d V sin sin d sin d cos sin d cos cos (Sphere) d cos, sin, sin d V sin sin d 8 sin d 8 cos cos 9. a sin, a cos (a) At d a sin, a cos d, Tangent line: d d a sin a cos sin cos d d cos cos sin sin cos a cos cos a cos acos CONTINUED a, a a, d a.
Section. Parametric Equations and Calculus 9. CONTINUED (c) sin, cos d sin cos Points of horizontal tangenc:, an, a (d) Concave downward on all open -intervals:...,,,,,,,... (e) a sin a cos d a cos d s sin d a sin d a a cos 8a 9. t, t t (a) (c) d t, dt d d tt t t at d,, 8 t. 8 dt t, t t t 8 t d t t t t (d) s t t dt t t 9 t dt t dt t dt (e) S t t t dt 8 9. t u r cos r sin rcos sin v w r sin r cos rsin cos r w θ r t v u θ θ (, )
Chapter Conics, Parametric Equations, and Polar Coordinates 9. Let s focus on the region above the -ais. From Eercise 9, the equation of the involute from, to, is cos sin. At,, the string is full etended and has length. So, the area of region A is. We now need to find the area of region B. d sin sin. d Hence, the far right point on the involute is,. The area of the region B C D is given b d d d where sin Thus, we can calculate sin sin cos cos and d cos cos d cos cos Since the area of C D is, we have Total area covered cos d... is cusp. (, A (, ) (, ) (, ) ( B DC (, ( 97. (a) t t, t t, The graph for < t < is the circle, ecept the point,. Verif: t t t t t t t t t t (c) As t increases from t to, the speed increases, and as t increases from to, the speed decreases. 98. (a) ln < t sech, t tanh t, t CONTINUED Same as the graph in (a), but has the advantage of showing the position of the object and an given time t.
Section. Polar Coordinates and Polar Graphs 7 98. CONTINUED (c) d sech t secht tant sinh t Tangent line: t tanh t sinh t sech t (, ) t sinh t 8 (, ) -intercept:, t Distance between, t and, : d sech t tanh t d for an t. d 99. False. d dt gt d ft ft ftgt gtft ft. False. Both ddt and dt are zero when t. B eliminating the parameter, we have which does not have a horizontal tangent at the origin. Section. Polar Coordinates and Polar Graphs. 7,.,. cos sin,, cos 7 sin 7,,, sin cos,, ( ), (, ) ( ),., 7 cos 7.,..,.7 cos.. sin..99 cos.7. sin.7 sin 7,,,.,.99,., (,.) (., ) (, )
8 Chapter Conics, Parametric Equations, and Polar Coordinates 7. r,, 8. r,, 9.,.,. (.,.),.7, (.7, ) r,.,.,.8,.9 (.8,.9). r, 8.,...,,,.9, 7.99 8 8 (.9, 7.99) r ± tan,,, (, ),,,, r ± tan undefined,,,,, (, ).,, r ±9 ±.,, r ± ± tan.,.,,.,,. (, ) tan.,.,,.78 (, ).,,,, r tan r,,, (, ) r 9 tan r,,,, ) )
Section. Polar Coordinates and Polar Graphs 9 7.,, 8.,, r,.,.88 r,,.78 9.,,.,, r,.8,.9 r,,.7. (a),,. r,,. (,.) (,.). (a) Moving horizontall, the -coordinate changes. Moving verticall, the -coordinate changes. Both r and values change. (c) In polar mode, horizontal (or vertical) changes result in changes in both r and.. r sin circle. r cos. r cos. r sec Matches (c) Rose curve Cardioid Line Matches Matches (a) Matches (d) 7. a r a 8. a r ar cos rr a cos a r a cos a a 9. r sin. r cos r csc r sec 8. r cos r sin r cos sin r cos sin
Chapter Conics, Parametric Equations, and Polar Coordinates.. 9 r cos r sin r sin 9r cos r sec csc 8 csc r 9 cos sin r 9 csc cos 7. 9. r r 9r cos r sin r 9 r r 9cos 9 r 9 cos. r 7. r sin 8. r r r sin r cos r r cos 9. r tan r tan tan arctan
Section. Polar Coordinates and Polar Graphs. tan tan. r sec r cos. r csc r sin. r cos. r sin. r sin < < < 8. r cos < 7. r cos Traced out once on < < 8. r sin Traced out once on 9.. r sin r cos. r sin r sin r sin < < <
Chapter Conics, Parametric Equations, and Polar Coordinates. r. Graph as r, r. It is traced out once on,.... r h cos k sin r rh cos k sin r hr cos kr sin h k h k h h k k h k h k h k Radius: Center: h, k h k. (a) The rectangular coordinates of r are r cos, r sin,. The rectangular coordinates of r, are r cos, r sin. d If r cos r cos r sin r sin r cos r r cos cos r cos r sin r r sin sin r sin r cos sin r cos sin r r cos cos sin sin r r r r cos d r r r r cos, the points lie on the same line passing through the origin. In this case, d r r r r cos r r r r. (c) If 9, then cos and d r r, the Pthagorean Theorem! (d) Man answers are possible. For eample, consider the two points r,, and r,,. d cos Using r,, and r,,, You alwas obtain the same distance. d cos.
Section. Polar Coordinates and Polar Graphs.,,, d cos cos.. 7,,, d cos 7 9 cos 9 7. 7.,., 7,. 8.,.,, d 7 7 cos.. d cos. 8 cos.7. 9 cos.. 9. d cos sin cos sin cos cos sin sin At, At,, At r sin cos sin cos sin cos sin cos sin,,,. d d.. d. r sin d cos sin cos sin cos cos sin sin At At At,, 7,,,, d. d is undefined.. d. (a), r cos. (a), r cos 8 8 r,,,, Tangent line: (c) At, d.. (c) At r,,,, Tangent line:, d does not eist (vertical tangent).. (a), r sin (c) At,.7. d r,,, Tangent line:, 9 9 9
Chapter Conics, Parametric Equations, and Polar Coordinates. (a), r. 8 8 r,, Tangent line: (c) At, d.,, sin cos cos sin d r sin cos sin cos or sin Horizontal tangents:,,,,, d sin sin cos cos d sin sin sin sin sin sin sin sin or sin Vertical tangents:, 7,,,,,, 7,. r a sin 7. r csc a sin cos a cos sin d csc cos csc cot sin d a sin cos cos,,,, d a sin a cos a sin d sin ±,,,, 7 Horizontal:,,, Horizontal: Vertical: a,,, a,, a, 8. a sin cos a sin cos a cos sin d r a sin cos, asin cos sin cos a sin cos cos sin tan, Horizontal: a,, a,,,, 9. r sin cos Horizontal tangents: r,,,.,.78,.,.
Section. Polar Coordinates and Polar Graphs 7. r cos sec 7. r csc Horizontal tangents: r,., ±. Horizontal tangents: r, 7,,, 7. r cos 7. r sin r r sin 9 Horizontal tangents: r,.89,.77,.7,.9,.998,.,.,.7 Circle r Center:, Tangent at the pole: 7. r cos r r cos 7. r sin Cardioid Smmetric to -ais, 9 Circle: Center: r, Tangent at pole: 7. r cos Cardioid Smmetric to polar ais since r is a function of cos. 9 r
Chapter Conics, Parametric Equations, and Polar Coordinates 77. r cos Rose curve with three petals Smmetric to the polar ais Relative etrema:,,,, r, Tangents at the pole:,, 78. r sin 79. r sin Rose curve with five petals Rose curve with four petals Smmetric to Smmetric to the polar ais, and pole, Relative etrema occur when dr cos at d Tangents at the pole:,,,, 7, 9.,,, Relative etrema: ±, Tangents at the pole: give the same tangents.,,, ±, 8. r cos Rose curve with four petals Smmetric to the polar ais, Relative etrema:,,, Tangents at the pole: and 7, given the same tangents. and pole,,,,, 8. r 8. r Circle radius: Circle radius:
Section. Polar Coordinates and Polar Graphs 7 8. r cos 8. r sin Cardioid Cardioid 8. r cos Limaçon Smmetric to polar ais r 8. r sin Limaçon Smmetric to r 9 7 87. r csc r sin Horizontal line 88. r sin r cos Line r sin cos 89. r Spiral of Archimedes Smmetric to r 9. r Hperbolic spiral r Tangent at the pole:
8 Chapter Conics, Parametric Equations, and Polar Coordinates 9. r cos r cos, Lemniscate Smmetric to the polar ais, Relative etrema: ±, and pole, r ± ± Tangents at the pole:, 9. r sin Lemniscate Smmetric to the polar ais, Relative etrema: ±, Tangent at the pole: and pole, r ± ± ± 9. Since the graph has polar ais smmetr and the tangents at the pole are Furthermore, r as Also, r sec, cos r as r r r cos r cos r r r,... Thus, r ± as. = 9. Since the graphs has smmetr with respect to Furthermore, r as r as Also, Thus, r ± as. r csc, sin r r r sin sin r r r.. =.
Section. Polar Coordinates and Polar Graphs 9 9. r Hperbolic spiral r as lim r r sin sin r sin sin sin lim = cos 9. r cos sec Strophoid r as r as r cos sec cos sec r cos cos cos lim ± cos = 97. The rectangular coordinate sstem consists of all points of the form, where is the directed distance from the -ais to the point, and is the directed distance from the -ais to the point. Ever point has a unique representation. The polar coordinate sstem uses r, to designate the location of a point. r is the directed distance to the origin and measured clockwise. Points do not have a unique polar representation. is the angle the point makes with the positive -ais, 98. r cos, r sin r, tan 99. r a, circle b, line. Slope of tangent line to graph of r f at r, is d f cos fsin. f sin fcos If f and f, then is tangent at the pole.. r sin (a) (c)
Chapter Conics, Parametric Equations, and Polar Coordinates. r cos (a), r cos (c) 9 9 9, r cos r cos cos cos sin sin sin 9 The graph of r cos is rotated through the angle. The graph of r cos is rotated through the angle.. Let the curve r f be rotated b to form the curve r g. If r is a point on r f, then r,, is on r g. That is, g r f. (, r θ + φ), Letting, or we see that g g f f. φ (, r θ ) θ. (a) sin sin cos cos sin cos r f sin sin sin cos cos sin sin r f sin f sin f cos (c) sin sin cos cos sin cos r f sin f cos. r sin (a) r sin sin cos r sin cos cos CONTINUED
Section. Polar Coordinates and Polar Graphs. CONTINUED (c) r sin sin sin (d) r sin cos. r sin sin cos (a) r sin cos r sin cos sin cos (c) r sin cos (d) r sin cos sin cos 7. (a) r sin r sin Rotate the graph of r sin through the angle. 8. B Theorem 9., the slope of the tangent line through A and P is f cos f sin. f sin f cos This is equal to tan tan tan tan tan sin cos tan. cos sin tan Equating the epressions and cross-multipling, ou obtain f cos f sin cos sin tan sin cos tan f sin f cos f cos f cos sin tan f sin cos f sin tan f sin f sin cos tan f sin cos f cos tan f cos sin f tan cos sin tan f f r drd. Polar curve r= f( θ) θ Radial line ψ P = ( r, θ) A Tangent line
Chapter Conics, Parametric Equations, and Polar Coordinates 9. tan r At, drd tan is undefined. cos sin. tan r At drd tan, cos sin arctan..78 7. 8. r cos tan r At drd cos cot sin, tan cot. arctan 8.. tan r At drd, arctan tan sin cos. sin 8 cos.77.89 θ ψ. At r tan r cos cos dr sin sin cos cos cos d tan,,. dr d sin cos ψ θ. tan r undefined drd 9 9. True. True 7. True 8. True
Section. Area and Arc Length in Polar Coordinates Section. Area and Arc Length in Polar Coordinates. A f d sin d sin d. A f d cos d. A f d sin d. A f d cos d. (a) r 8 sin A 8 sin d sin d cos d sin A. (a) r cos A 9 9 9 sin cos d cos d cos d 9 A 9 7. A cos d sin 8. A sin d cos sin 8 9 8 d sin d
Chapter Conics, Parametric Equations, and Polar Coordinates 9. A cos d. sin 8 A cos d sin. A sin d cos sin. A sin d cos sin 8. A sin sin cos d. A sin d arcsin arcsin 8 sin sin d 8 sin arcsin cos d 8 cos 9 sin.7 arcsin 8 8. The area inside the outer loop is cos d sin sin From the result of Eercise, the area between the loops is A... Four times the area in Eercise, A. More specificall, we see that the area inside the outer loop is The area inside the inner loop is 7 sin d sin sin d 8. sin d. Thus, the area between the loops is 8.
Section. Area and Arc Length in Polar Coordinates 7. r cos 8. r sin r cos r sin Solving simultaneousl, Solving simultaneousl, cos cos sin sin cos sin,. Replacing r b r and b in the first equation and solving, cos cos, cos, Both curves pass through the pole,,, and,, respectivel. Points of intersection:,,,,,. Replacing r b r and b in the first equation and solving, sin sin, sin, Both curves pass through the pole,,, and,, respectivel..,. Points of intersection:,,,,, 9. r cos. r cos r sin r cos Solving simultaneousl, Solving simultaneousl, cos sin cos cos cos sin tan, 7. Replacing r b r and b in the first equation and solving, cos sin, sin cos, which has no solution. Both curves pass through the pole,,, and,, respectivel. Points of intersection:,,, 7,, cos,. Both curves pass through the pole, (, arccos ), and,, respectivel. Points of intersection:,,,,,. r sin r sin Solving simultaneousl, sin sin. r cos r cos Solving simultaneousl, cos cos sin,. Both curves pass through the pole,, arcsin, and,, respectivel. Points of intersection:,,,,, cos,. Both curves pass through the pole,,, and,, respectivel. Points of intersection:,,,,,
Chapter Conics, Parametric Equations, and Polar Coordinates. r. r r Solving simultaneousl, we have,. Points of intersection:,,, Line of slope passing through the pole and a circle of radius centered at the pole. Points of intersection:,,,. r sin r r sin is the equation of a rose curve with four petals and is smmetric to the polar ais, and the pole. Also, r is the equation of a circle of radius centered at the pole. Solving simultaneousl,, sin,,. Therefore, the points of intersection for one petal are, and,. B smmetr, the other points of intersection are, 7,,,,,, 7,, 9, and,.. r sin r csc Points of intersection: 7, arcsin 7, 7, arcsin 7,.,.9,.,. The graph of r sin is a limaçon smmetric to and the graph of r csc is the horizontal line. Therefore, there are two points of intersection. Solving simultaneousl,, sin csc sin sin sin ± 7 arcsin 7.9.
Section. Area and Arc Length in Polar Coordinates 7 7. r cos r sec The graph of r cos is a limaçon with an inner loop b > a and is smmetric to the polar ais. The graph of r sec is the vertical line. Therefore, there are four points of intersection. Solving simultaneousl, cos sec cos cos cos ± arccos arccos.7.8. Points of intersection:.8, ±.7,.8, ±.7 r = sec θ 8 r =+ cosθ 8. r cos r cos The graph of r cos is a cardioid with polar ais smmetr. The graph of r cos is a parabola with focus at the pole, verte,, and polar ais smmetr. Therefore, there are two points of intersection. Solving simultaneousl, cos cos cos cos ± arccos. Points of intersection:, arccos.,.998,, arccos.,.8 r = cos θ r = ( cos θ) 9. r cos r sin Points of intersection:,,.9,.,.,. The graphs reach the pole at different times ( r = cos θ values).. r sin r sin Points of intersection:,,, The graphs reach the pole at different times ( r = sin θ values). r = sin θ r = ( + sin θ)
8 Chapter Conics, Parametric Equations, and Polar Coordinates. From Eercise, the points of intersection for one petal are, and,. The area within one petal is sin d r = d sin d. 8 sin d sin. Total area A 8 (from Eercise ) 7 9 sin d sin d 9 8 d (b smmetr of the petal). A sin d r = + sin θ r = sin θ cos sin 9 9 7 7 r = sin θ 7. r sin and r cos intersect at and A 9. sin d cos 9 9 9. sin 9 9 9. A sin 8 r = sin θ sin d d 8 r = 8. A sin d cos sin d sin cos sin d 7. A a a a cos d sin a a sin a a
Section. Area and Arc Length in Polar Coordinates 9 8. Area Area of r a cos Area of sector twice area between r a cos and the lines,. A a a a a cos d a cos d 9. a 8 a 8 a a A 8 a cos d a a 8 cos cos d a sin sin a a sin a a a a a a a θ = a a = θ. r a cos, r a sin. (a) r a cos tan, A a sin d a sin a a 8 a a a cos r = a sin θ r = a cos θ a d (c) r ar cos a A cos cos d cos d cos d cos cos d a = a = sin 8 sin
Chapter Conics, Parametric Equations, and Polar Coordinates. B smmetr, A A and A A. A A a a A a a a a A a a a sin A 7 sin A A a sin a a cos a d a cos a sin d cos d a cos d a a a a sin d cos d a sin d cos d a a a sin a d a a sin d a sin a [Note: A A A 7 A a area of circle of radius a] sin a a d a a a a a r= asin θ θ = a A a A A A A θ = θ = a A = θ A 7 θ = r = r= acos θ a. r a cosn For n : r a cos A a a For n : r a cos A 8 a cos d a a a CONTINUED
Section. Area and Arc Length in Polar Coordinates. CONTINUED For n : r a cos A a cos d a For n : r a cos A 8 a cos d a a a In general, the area of the region enclosed b r a cosn for n,,,... is a if n is odd and is a if n is even.. r sec cos, < r cos cos < r cos r A sec cos d sec cos d sec cos d tan sin. r a. r a cos r s a d a (circumference of circle of radius a) a r a sin s a cos a sin d a d a a 7. r sin r cos s sin cos d sin d cos sin d sin 8
Chapter Conics, Parametric Equations, and Polar Coordinates 8. r 8 cos, r 8 sin s 8 cos 8 sin d cos cos sin d cos d cos cos cos d sin cos cos d 9. r,. r sec,. r,... Length. Length.7 eact. Length.7. r e,. r sin cos,. r sin cos, Length. Length.9 Length 7.78. r cos. r a cos r sin S cos sin cos sin d 7 sin sin cos d r a sin S a cos cos a cos a sin d a cos d a cos d sin a a
Section. Area and Arc Length in Polar Coordinates 7. r e a r ae a S e a cos e a ae a d a e a cos d a ea a a cos sin a a ea a 8. r a cos r a sin S a cos sin a cos a sin d a sin cos cos d a cos sin d a cos a 9. r cos r 8 sin S cos sin cos sin d cos sin cos sin d.87. r r S sin d.. Area f d r d Arc length f f d r dr d d. The curves might intersect for different values of : See page 7.. (a) is correct: s... (a) f sin f f d S f cos f f d S. Revolve r about the line r sec. f, f S cos d cos d sin r = r = sec cos
Chapter Conics, Parametric Equations, and Polar Coordinates. Revolve r a about the line r b sec where b > a >. f a a f S b a cos a d a b a sin a a b ab ab 7. r 8 cos, (a) A r d (Area circle r ) cos d cos....8... A.. 7..8.7..8 d sin (c), (d) For of area For of area For of area.7:. 8.:.7 7.7:.7 (e) No, it does not depend on the radius. 8. r sin, (a) A Note: radius of circle is r d 9 sin d 9 A cos d 9 sin....8... A.9.9..7.7.9.77 9 (c), (d) For of area 9 8 8.88: For of area 9.77: For of area 9.:.88..7 9. r a sin b cos r ar sin br cos a b b a represents a circle.
Section. Area and Arc Length in Polar Coordinates 7. r sin cos,converting Circle to rectangular form: A r r d r sin r cos sin cos d sin cos d sin Circle of radius Area and center, 7. (a) As a increases, the spiral opens more rapidl. If <, the spiral is reflected about the -ais. r a,, crosses the polar ais for n and integer. To see this (c) r for f, f s, r a r sin a sin n. n, The points are r, an, n, n,,,.... d ln ln. (d) A r dr d 7. r e A e d e d e e e d e d = = = e e e e 9. 7. The smaller circle has equation r a cos. The area of the shaded lune is: a A a a cos d sin a cos d r = θ = r = a cos θ This equals the area of the square, a a a a a Smaller circle: r cos.
Chapter Conics, Parametric Equations, and Polar Coordinates 7. t t t, t (a) 7t t 7t (c) t t 7t t Hence,. r cos r sin r cos r sin r cos sin cos sin A r d 7. False. f and g have the same graphs. 7. False. f and g sin have onl one point of intersection. 77. In parametric form, b s d a dt dt dt. Using instead of t, we have r cos f cos and r sin f sin. Thus, d f cos f sin and f sin f cos. d It follows that d d d f f. Therefore, s d f f d. Section. Polar Equations of Conics and Kepler s Laws. r (a) (c) e e cos e, r e., r e., r, cos parabola,. cos cos,. cos cos ellipse hperbola e =. 8 e =. e =.. r (a) (c) e e cos e, r e., r e., r cos parabola 9,,. cos cos,. cos cos ellipse hperbola e =. e =. e =
Section. Polar Equations of Conics and Kepler s Laws 7. r (a) (c) e e sin e, r, parabola 9 9 sin e., r e., r,. sin sin,. sin sin ellipse hperbola e =. e =. e =. 8. r e e sin 9 (a) e, r e., r, sin parabola,. sin sin ellipse 9 e =. e =. e = 9 (c) e., r,. sin sin hperbola. r e sin (a) e =. e =. e =. e =.7 e =.9 The conic is an ellipse. As e, the ellipse becomes more elliptical, and as e, it becomes more circular. (c) 9 8 e =. e =. e = 9 The conic is a hperbola. As e, the hperbolas opens more slowl, and as e, the open more rapidl. The conic is a parabola.. r. cos (a) Because e. <, the conic is an ellipse with vertical directri to the left of the pole. (c) 7 9 r. cos 8 8 The ellipse is shifted to the left. The vertical directri is to the right of the pole 7 r.. sin The ellipse has a horizontal directri below the pole. 7. Parabola; Matches (c) 8. Ellipse; Matches (f) 9. Hperbola; Matches (a). Parabola; Matches (e). Ellipse; Matches. Hperbola; Matches (d)
8 Chapter Conics, Parametric Equations, and Polar Coordinates. r sin. r cos Parabola because e ; d d Distance from pole to directri: Parabola because e ; d Directri: Directri: Distance from pole to directri: d Verte: r,, Verte: r,, 8 8. r cos cos. r sin sin Ellipse because e ; d Ellipse because e < ; d Directri: Distance from pole to directri: Vertices: r,,,, d Directri: Distance from pole to directri: Vertices: r, 8,,, d 7. r sin 8. r cos r sin sin Ellipse because e ; d r cos cos Ellipse because e < ; d Directri: Directri: Distance from pole to directri: Vertices: r,,,, d Distance from pole to drectri: Vertices: r,,,, d
Section. Polar Equations of Conics and Kepler s Laws 9 9. r cos cos. r 7 sin 7 sin Hperbola because e > ; d Hperbola because e 7 > ; d 7 Directri: Distance from pole to directri: Vertices: d r,,,, Directri: Distance from pole to directri: Vertices: 7 d 7 r,,,, 8. r sin sin. r cos Hperbola because e > ; d Hperbola because e > ; d Directri: Distance from pole to directri: Vertices: d r, 8,,, Directri: Distance from pole to directri: Vertices: r,,,, d. r sin. r sin. r cos. r sin Ellipse Hperbola Parabola Hperbola 7. r sin 8. r cos Rotate the graph of Rotate the graph of r sin r cos counterclockwise through the angle. counterclockwise through the angle.
Chapter Conics, Parametric Equations, and Polar Coordinates 9. r cos Rotate the graph of r cos clockwise through the angle. 8. r 7 sin Rotate graph of r 7 sin clockwise through angle of.. Change to r : cos. Change to r : sin. Parabola e,, d r ed e cos cos. Parabola e,, d r ed e sin sin. Ellipse e,, d r ed e sin sin sin. Ellipse e,, d r ed e sin sin sin 7. Hperbola e,, d r ed e cos cos 8. Hperbola e,, d r ed e cos cos cos 9. Parabola Verte:, e, d, r sin. Parabola Verte: e, d r, ed e cos cos. Ellipse Vertices:,, 8, e, d r ed e cos cos cos
Section. Polar Equations of Conics and Kepler s Laws. Ellipse Vertices: e, d 8 r, ed e sin 8 sin 8 sin,,. Hperbola Vertices: e, d 9 r,, 9, ed e sin 9 sin 9 sin. Hperbola Vertices:,,, e, d r ed e cos cos cos. Ellipse if < e <, parabola if e, hperbola if e >.. r is a parabola with horizontal directri above the pole. sin (a) Parabola with vertical directri to left of pole. Parabola with horizontal directri below pole. (c) Parabola with vertical directri to right of pole. (d) Parabola rotated counterclockwise. 7. (a) Hperbola e > 8. If the foci are fied and e, then d. To see this, Ellipse e compare the ellipses < r, e, d cos (c) Parabola e (d) Rotated hperbola e r, e, d. cos 9. r b cos a cos a b r a b a a b b r cos a r sin a b r a cos b a a b a b a b a cos a b a c cos b ca cos b e cos b. r b r cos a r sin a b r b cos a cos a b r a cos a b a b a b a c cos b c a cos b e cos a b b a a b. a, c, e, b. a, c, b, e. a, b, c, e. r 9 cos r 9 cos a, b, c, e. r cos A 9 r cos d cos d.88 9 cos