698 Chapter 11 Parametric Equations and Polar Coordinates

Size: px
Start display at page:

Download "698 Chapter 11 Parametric Equations and Polar Coordinates"

Transcription

1 698 Chapter Parametric Equations and Polar Coordinates

2 Chapter Practice Eercises (a Perihelion a ae a( e, Aphelion ea a a( e ( Planet Perihelion Aphelion Mercur AU AU Venus 0.78 AU 0.78 AU Earth 0.98 AU.067 AU Mars.87 AU.666 AU Jupiter.95 AU 5.58 AU Saturn 9.00 AU AU Uranus AU 0.06 AU Neptune 9.85 AU AU (0.87 a cos cos (0.7 a cos cos a cos cos (.5 a cos 0.09 cos (5.0 a cos 0.08 cos (9.59 a cos 0.05 cos (9.8 a cos cos (0.06 a cos cos 76. Mercur: r Venus: r Earth: r Mars: r Jupiter: r Saturn: r Uranus: r Neptune: r CHAPTER PRACTICE EXERCISES t. and t Ê t Ê. Èt and Èt Ê Ê Ê. tan t and sec t Ê tan t. cos t and sin t Ê cos t and and sec t tan t and sin t sec t Ê Ê

3 700 Chapter Parametric Equations and Polar Coordinates 5. cos t and cos t Ê ( 6. cos t and 9 sin t Ê 6 cos t and sin t Ê Ê Ê a and Ê cos t and sin t, 0 Ÿ t Ÿ 8. Ê cos t and sin t, 0 Ÿ t Ÿ 6 d d/dt sec t tan t tan t d È d d/dt sec t sec t d¹ tî È tan and È sec w d d/dt cos t d ; d d/dt cos t sec t d 9. tan t, sec t Ê sin t Ê sin ; t Ê Ê Ê cos ˆ ¹ tî d d/dt Š d 5 t t d d/dt d Š $ t t w d d/dt ˆ d $ $ d d/dt d ¹ Š t $ t t 0., Ê t Ê ¹ ( ; t Ê and Ê ; t Ê ( 6 È È Î 8. (a t, t Ê t Ê Š È ( cos t, tan t Ê sec t Ê tan t sec t Ê Ê. (a The line through a, with slope is 5 Ê t, t 5, _ t _ ( a a 9 Ê cost, sint Ê cost, sint, 0 Ÿ t Ÿ (c Ê t, t t, _ t _ 9 (d 9 6 Ê Ê cos t, sin t, 0 Ÿ t Ÿ d d $Î Î d Î Î d. Ê Ê Š ˆ Ê L É ˆ d Î Î Î Î Î $Î % ˆ 8 ˆ ˆ 0 Ê L É ˆ d É d ˆ a d 8 8 d d d d d 9 d 9 cî$ Î$ Î$. Ê Ê Š Ê L Ê Š d É Î$ d 8 8 È 9 Î$ d È9Î$ ˆ Î$ d; Î$ Î$ u 9 Ê du 6 d; Ê u, Î$ 0 %! 8 8 $ 7 Î 8 u 0 d L u du $Î Ê Ä u $Î $Î d d 5 Î& 5 %Î& d Î& Î& d Î& Î& 5. Ê Ê Š ˆ Î& Î& Î& Î& Î& Î& Ê L É a d Ê L É a d É a d

4 Chapter Practice Eercises 70 $ ( ˆ Î& Î& d 5 Î& 5 %Î& ˆ 5 5 % ˆ 5 5 ˆ 5 75 $ d d % d d Ê Ê Š Ê L % Ê Š d % % $ 6 % 8 7 É % d ÊŠ d Š d ˆ ˆ d d d d dt dt dt dt 7. 5 sin t 5 sin 5t and 5 cos t 5 cos 5t Êʈ Š Éa 5 sin t 5 sin 5t a5 cos t 5 cos 5t 5 È sin 5t sin t sin 5t sin t cos t cos t cos 5t cos 5t & È a sin t sin 5t cos t cos 5 t 5È a cos % t 5É% ˆ a cos % t! Èsin t!l sin tl! sin t (since! Ÿ t Ÿ Î! Ê Length! sin t dt c 5 cos td a & a a & a! Î! d d dt dt dt dt d d 8. t t and t t Ê Êˆ Š Éat t at t È88t 8t È kk t È6 t Ê Length È kk t È6 t dt È t È 6 t dt; u 6 t Ê du t dt!! È 7 È / 7 È / / a a 6 6 Ê du t dt; t 0 Ê u 6; t Ê u 7 ; Èu du u Š 7 6 È / / Š a7 6 ÈŠ a d d d d d d d d 9. $ sin and $ cos Êʈ Š Éa $ sin a$ cos È$ asin cos $ Ê Length $ d $ d $ ˆ! $ Î $ Î!! $ * t d d dt dt È È È t È $ 0. t and t, È Ÿ t Ÿ È Ê t and t Ê Length Éat at dt È È È Èt% t dt Èt% t dt t Éa dt at dt t È È È È t. and t, 0 Ÿ t Ÿ È d d 5 Ê t and Ê Surface Area (t È t Î dt u du * % dt dt È5 9 0 $Î u 76, where u t Ê du t dt; t 0 Ê u, t È5 Ê u 9 È d d t È dt t dt È t. t and t, Ÿ t Ÿ Ê t and ÎÈ t t Èt ÎÈ t t Ê Surface Area ˆ t ʈ t Š dt ˆ t Ɉ t dt ÎÈ t t ÎÈ 8 Î È È ˆ t ˆ t dt ˆ $ $ t t % dt t t t Š È È

5 70 Chapter Parametric Equations and Polar Coordinates. r cos ˆ È r ˆ Ê cos cos sin sin È È È Ê r cos r sin Ê r cos È r sin È Ê È È È Ê È È È È È. r cos ˆ r ˆ Ê cos cos sin sin Ê r cos r sin Ê Ê cos 5. r sec Ê r Ê r cos Ê 6. r È sec Ê r cos È Ê È 7. r csc Ê r sin Ê

6 Chapter Practice Eercises r È csc Ê r sin È Ê È 9. r sin Ê r r sin Ê 0 Ê ( ; circle with center (!ß and radius. 0. r È sin Ê r È r sin Ê È 7 0 Ê Š ;!ß È È circle with center Š and radius È. r È cos Ê r È r cos Ê È 0 Ê Š È ; circle with center Š Èß 0 and radius È. r 6 cos Ê r 6r cos Ê 6 0 Ê ( 9; circle with center ( ß0 and radius

7 70 Chapter Parametric Equations and Polar Coordinates. 5 0 Ê ˆ 5 5 Ê C ˆ 5!ß 5 and a ; r 5r sin 0 Ê r 5 sin. 0 Ê ( Ê C (!ß and a ; r r sin 0 Ê r sin 5. 0 Ê ˆ 9 Ê C ˆ ß! and a ; r r cos 0 Ê r cos 6. 0 Ê ( Ê C ( ß0 and a ; r r cos 0 Ê r cos d 0. e. l. f. k. h 5. i 6. j

8 Chapter Practice Eercises A r d ( cos d cos cos d cos d 0 a ˆ cos ˆ 9 cos cos d 9 sin sin 9 0! Î Î cos 6 Î$ A asin d ˆ d sin 6! Î Î 0 0 Î ˆ cos sin Î% ˆ 0 8! 8 9. r cos and r Ê cos Ê 0 cos Ê Ê ; therefore A c( cos dd a cos cos d cos d sin The circle lies interior to the cardioid. Thus, Î A [( sin ] d (the integral is the area of the cardioid minus the area of the circle cî Î Î a sin sin d (6 8 sin cos d c6 8 cos sin d cî cî c ( d 5 Î Î dr 5. r cos Ê sin ; Length È( cos ( sin d È cos d d 0 0 ( cos É d sin d cos ( ( ( ( 8 0 0! d d Î Î È È ˆ È 0! dr 5. r sin cos, 0 Ÿ Ÿ Ê cos sin ; r ˆ dr ( sin cos ( cos sin 8asin cos 8 Ê L È 8 d d d Î Î Î % % ˆ É ˆ ˆ cos ˆ $ 5. r 8 sin ˆ dr, 0 Ÿ Ÿ Ê 8 sin ˆ cos ˆ $ ; r ˆ dr 8 sin ˆ 8 sin ˆ cos ˆ 6 sin Ê L 6 sin d 8 sin d 8 d cos ˆ d 6 sin ˆ ˆ 6 sin ˆ 0 0 Î Î%! 6 dr sin dr sin d È cos d cos dr sin cos cos sin d cos cos cos Î cos cos L È d È ˆ cî È 5. r È Î cos Ê ( cos ( sin Ê ˆ Ê r ˆ ( cos sin cos Ê 55. Ê Ê p Ê p ; 56. Ê Ê p Ê p ; therefore Focus is (0ß, Directri is therefore Focus is ˆ!ß ; Directri is

9 706 Chapter Parametric Equations and Polar Coordinates 8 8 ˆ 8 ˆ ˆ ß ß! 57. Ê Ê p Ê p ; 58. Ê Ê p Ê p ; therefore Focus is 0, Directri is therefore Focus is, Directri is 7 6 c c È c c ; e c È a ; e a Ê 60. Ê Ê c Ê Ê Ê 5 c c a a È5 6. Ê Ê c Ê Ê c 5 9 Ê c ; e ; the asmptotes are Ê c, e ; the asmptotes are È 6. Ê Ê p Ê p Ê focus is (!ß, directri is, verte is (0 ß 0; therefore new verte is (ß, new focus is (ß0, new directri is 6, and the new equation is ( ( 6. 0 Ê Ê p 0 Ê p 5 Ê focus is ˆ 5 ß 0, directri is 5 0, verte is (0 ß 0; therefore new verte is ˆ ß, new focus is (ß, new directri is, and the new equation is ( 0 ˆ 65. Ê a 5 and Ê c È5 9 Ê foci are a!ß, vertices are a!ß 5, center is 9 5 (0ß 0; therefore the new center is ( $ß 5, new foci are ( ß and ( ß 9, new vertices are ( $ß 0 and ( ( ( $ß 0, and the new equation is

10 Chapter Practice Eercises Ê a and Ê c È69 5 Ê foci are a 5ß 0, vertices are a ß0, center is (0ß0; therefore the new center is (5ß, new foci are (0ß and (0ß, new vertices are (8ß and ( 5 ( 69 ( 8ß, and the new equation is 67. Ê a È and È Ê c È8 È0 Ê foci are Š 0ß È0, vertices are 8 Š 0ß È, center is (0ß0, and the asmptotes are ; therefore the new center is Š ßÈ, new foci are Š ß È È0, new vertices are Š ß È and (ß0, the new asmptotes are È and Š È 8 È ( ; the new equation is 68. Ê a 6 and 8 Ê c È6 6 0 Ê foci are a 0ß 0, vertices are a 6ß0, the center is (0ß0 and the asmptotes are or ; therefore the new center is ( 0ß, the new foci are ( 0ß and (0ß, the new vertices are ( 6ß and ( ß, the new asmptotes are and 9 ( 0 ( 6 6 ; the new equation is ( Ê Ê ( Ê, a hperola; a and Ê c È È5 ; the center is (ß 0, the vertices are (!ß 0 and (ß 0; the foci are Š È5 ß 0 and the asmptotes are ( Ê Ê ( Ê, a hperola; a and Ê c È È5 ; the center is (!ß, the vertices are (ß and ( ß, the foci are Š È5ß and the asmptotes are Ê 6 8 Ê ( 6(, a paraola; the verte is ( $ß; p 6 Ê p Ê the focus is ( 7ß and the directri is Ê 8 6 Ê ( 8(, a paraola; the verte is (ß ; p 8 Ê p Ê the focus is (ß and the directri is Ê 9 a 6 6 a Ê 9 a a Ê ( ( 9( 6( Ê 6 9, an ellipse; the center is ( ß; a and Ê c È6 9 È7 ; the foci are Š $ È7ß ; the vertices are (ß and ( 7ß Ê 5 a 9 a 6 Ê 5 a 9 a ( ( Ê, an ellipse; the center is (ß ; a 5 and Ê c È ; the foci are (ß and (ß 7; the vertices are (ß and (ß Ê Ê ( (, a circle with center (ß and radius È 76. Ê 6 Ê ( ( 6, a circle with center ( ß and radius È 6

11 708 Chapter Parametric Equations and Polar Coordinates cos 77. r Ê e Ê paraola with verte at (ß0 8 cos ˆ cos 78. r Ê r Ê e Ê ellipse; a a e ˆ ke Ê k Ê k 8; k ea Ê 8 a 6 Ê a Ê ea ˆ ˆ 6 8 ; therefore the center is ˆ 8 ; vertices are ( and ˆ 8 ß ß ß0 6 cos 79. r Ê e Ê hperola; ke 6 Ê k 6 Ê k Ê vertices are ( ß and (6 ß sin ˆ sin 80. r Ê r Ê e ; ke Ê k Ê k ; a a e Ê a ˆ 9 Ê a Ê ea ˆ ˆ 9 ; therefore the center is ˆ ; vertices are ˆ and ˆ ß ß 6ß ke (( e cos cos 8. e and r cos Ê is directri Ê k ; the conic is a hperola; r Ê r Ê r cos ke (( e cos cos 8. e and r cos Ê is directri Ê k ; the conic is a paraola; r Ê r Ê r cos ke ( ˆ e sin ˆ sin 8. e and r sin Ê is directri Ê k ; the conic is an ellipse; r Ê r Ê r sin ke (6 ˆ e sin ˆ sin 8. e and r sin 6 Ê 6 is directri Ê k 6; the conic is an ellipse; r Ê r Ê r 6 sin (a Around the -ais: 9 6 Ê 9 Ê É9 and we use the positive root: $ 0 0! 9 V Š É9 d ˆ 9 9 d 9

12 Chapter Additional and Advanced Eercises ( Around the -ais: 9 6 Ê Ê É and we use the positive root: $ $ ! V Š É d ˆ d , Ê Ê È ; V È 9 Š d a d ˆ 6 ˆ 8 ˆ ( $ % k 87. (a r Ê r er cos k Ê e k Ê e cos È È k e Ê k ke e Ê e ke k 0 Ê a e ke k 0 ( e 0 Ê k 0 Ê k Ê circle; 0 e Ê e Ê e 0 Ê B AC 0 a e ( ae 0 Ê ellipse; e Ê B AC 0 (0( 0 Ê paraola; e Ê e Ê B AC 0 a e ( e 0 Ê hperola 88. Let (r ß e a point on the graph where r a. Let (r ß e on the graph where r a and. Then r and r lie on the same ra on consecutive turns of the spiral and the distance etween the two points is r r a a a( a, which is constant. CHAPTER ADDITIONAL AND ADVANCED EXERCISES. Directri and focus (ß0 Ê verte is ˆ 7 ß! 7 Ê p Ê the equation is ( Ê 6 9 Ê Ê verte is (ß0 and p Ê focus is (ß and the directri is. Ê verte is (!ß 0 and p Ê focus is (!ß ; thus the distance from P(ß to the verte is È and the distance from P to the focus is È ( Ê È È ( Ê c ( d Ê 8 Ê 8 0, which is a circle. Let the segment a intersect the -ais in point A and intersect the -ais in point B so that PB and PA a (see figure. Draw the horizontal line through P and let it intersect the -ais in point C. Let npbo Ê napc. Then sin and cos a Ê cos sin. a c c a 5. Vertices are a!ß Ê a ; e Ê 0.5 Ê c Ê foci are a0ß

13 70 Chapter Parametric Equations and Polar Coordinates a a 6. Let the center of the ellipse e (ß0; directri, focus (ß0, and e Ê e c Ê e c Ê a ( c. Also c ae a Ê a ˆ a Ê a a Ê 5 a Ê a ; a e Ê ˆ ˆ 8 8 Ê Ê the center is ˆ 8ß0 8 8 ; c Ê c so that c a ˆ 8 5ˆ ; therefore the equation is ˆ ˆ 80 or ˆ ˆ 7. Let the center of the hperola e (0ß. (a Directri, focus (0ß 7 and e Ê a c 6 Ê a c 6 Ê a c. Also c ae a e e a e ( c a 6 6 8; therefore the equation is 6 8 a a 5 e e c 5 a ˆ 5 ; ( the center is ˆ e 5 ; c a c a ˆ 6ˆ 6 6 ; therefore the equation is ˆ 5 ˆ 75 or Ê a (a Ê a Ê c 8; ( Ê Ê the center is (0ß; c a Ê ( e 5 Ê c 6 Ê c 6 Ê a 5c 0. Also, c ae 5a Ê a 5(5a 0 Ê a 0 Ê a Ê Ê Ê!ß Ê 9 a a 9 % % a a a 8. The center is (0ß0 and c Ê a Ê a. The equation is Ê Ê Ê 9 a a a a a a Ê 96 9a a a a Ê a 97a 96 0 Ê aa 96aa 0 Ê a or a ; a Ê ( 0 which is impossile; a Ê ; therefore the equation is d d a a 9. a a Ê ; at ( ß the tangent line is Š ( Ê a a a Ê a a 0 d d a a 0. a a Ê ; at ( ß the tangent line is Š ( Ê a a a Ê a a 0....

14 Chapter Additional and Advanced Eercises 7 5. a9 6a 9 6 Ÿ 0 Ê 9 6 Ÿ 0 and or and 9 6 Ÿ 0 6. a9 6a 9 6 0, which is the complement of the set in Eercise 5 7. (a e t cos t and e t sin t Ê e t cos t e t sin t e t. Also e t cos t tan t Ê t tan ˆ c % tan aî Ê e is the Cartesian equation. Since r and tan ˆ, the polar equation is r e or r e for r 0 ( ds r d dr ; r e Ê dr e d Ê ds r d ˆ e d ˆ e d e d 5e d Ê ds È5 e d Ê L È5 e d È 5e È5! ae 0 t e sin t $ 8. r sin ˆ Ê dr sin ˆ cos ˆ $ d Ê ds r d dr sin ˆ d sin ˆ cos ˆ d sin ˆ d sin % ˆ cos ˆ d sin % ˆ sin ˆ cos ˆ d sin % ˆ d ˆ ˆ ˆ ˆ $ 0 0! Ê ds sin d. Then L sin d cos d sin 9. e and r cos Ê is the directri Ê k ; the conic is a hperola with r ke e cos (( cos cos Ê r 0. e and r cos Ê is the directri Ê k ; the conic is a paraola with r ke e cos (( cos cos Ê r ke e sin ˆ r ˆ sin sin. e and r sin Ê is the directri Ê k ; the conic is an ellipse with r Ê ke e sin 6 ˆ 6 r ˆ sin sin. e and r sin 6 Ê 6 is the directri Ê k 6; the conic is an ellipse with r Ê

15 7 Chapter Parametric Equations and Polar Coordinates. Arc PF Arc AF since each is the distance rolled; Arc PF Arc AF npcf Ê Arc PF ( npcf; a Ê Arc AF a Ê a ( npcf Ê npcf ˆ a ; nocb and nocb npcf npce npcf ˆ ˆ ˆ! a! Ê ˆ a ˆ Ê ˆ a!! Ê! ˆ a Ê! ˆ a. Now OB BD OB EP (a cos cos! (a cos cos ˆ ˆ a (a cos cos cos ˆ ˆ a sin sin ˆ ˆ a (a cos cos ˆ ˆ a and PD CB CE (a sin sin! (a sin sin ˆ ˆ a (a sin sin cos ˆ ˆ a cos sin ˆ ˆ a (a sin sin ˆ ˆ a ; therefore (a cos cos ˆ ˆ a and (a sin sin ˆ ˆ a d d. a(t sin t Ê a( cos t and let $ Ê dm da d ˆ dt dt dt 0 sin t 0 0 a( cos t a ( cos t dt a ( cos t dt; then A a ( cos t dt a a cos t cos t dt a ˆ cos t cos t dt a t sin t a ; µ = a(t sin t and µ = a( cos t Ê M µ dm µ $ da $ $ a a( cos t a ( cos t dt a ( cos t dt $ $ a cos t cos t cos t dt a $ cos t cos t sin t (cos t dt a $ 5 t sin t sin t sin t sin $ a t 0 5a$ $ 5 a M Š 5. Therefore a. Also, M µ dm µ da M a 6 $ $ 0 0 $ $ t t cos t 8! $ M $ a 5 M a ˆ 6 a(t sin t a ( cos t dt a at t cos t t cos t sin t sin t cos t sin t cos t dt a cos t t sin t t cos t sin t cos t sin t a. Thus a Ê aß a is the center of mass. tan < tan < 5. < < Ê tan tan ( < < tan < tan < ; the curves will e orthogonal when tan is undefined, or r when tan < Ê w w w Ê r f (g( tan < g ( r w f(!! sin ˆ d sin $ ˆ cos ˆ % % 6. r sin ˆ dr $ Ê sin ˆ cos ˆ Ê tan tan ˆ < dr r a sin d ˆ dr 6a cos 6 7. r a sin Ê 6a cos Ê tan < tan ; when, tan < tan Ê < d

16 Chapter Additional and Advanced Eercises 7 8. (a dr ( r Ê r Ê d Ê tan < k c c Ê lim tan < _ Ä_ Ê < Ä from the right as the spiral winds in around the origin. È È sin È cos cos sin 9. tan < cot is at ; tan < tan is È at ; since the product of these slopes is, the tangents are perpendicular r a( cos ˆ dr a sin 0. tan < is at Ê < d

17 7 Chapter Parametric Equations and Polar Coordinates NOTES:

668 Chapter 11 Parametric Equatins and Polar Coordinates

668 Chapter 11 Parametric Equatins and Polar Coordinates 668 Chapter Parametric Equatins and Polar Coordinates 5. sin ( sin Á r and sin ( sin Á r Ê not symmetric about the x-axis; sin ( sin r Ê symmetric about the y-axis; therefore not symmetric about the origin

More information

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems

Edexcel GCE A Level Maths. Further Maths 3 Coordinate Systems Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse

More information

CHAPTER 11 Vector-Valued Functions

CHAPTER 11 Vector-Valued Functions CHAPTER Vector-Valued Functions Section. Vector-Valued Functions...................... 9 Section. Differentiation and Integration of Vector-Valued Functions.... Section. Velocit and Acceleration.....................

More information

REVIEW OF CONIC SECTIONS

REVIEW OF CONIC SECTIONS REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

More information

by Abhijit Kumar Jha

by Abhijit Kumar Jha SET I. If the locus of the point of intersection of perpendicular tangents to the ellipse x a circle with centre at (0, 0), then the radius of the circle would e a + a /a ( a ). There are exactl two points

More information

1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if

1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if . Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (

More information

Objective Mathematics

Objective Mathematics Chapter No - ( Area Bounded by Curves ). Normal at (, ) is given by : y y. f ( ) or f ( ). Area d ()() 7 Square units. Area (8)() 6 dy. ( ) d y c or f ( ) c f () c f ( ) As shown in figure, point P is

More information

Not for reproduction

Not for reproduction REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

More information

+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h

+ 4 Ex: y = v = (1, 4) x = 1 Focus: (h, k + ) = (1, 6) L.R. = 8 units We can have parabolas that open sideways too (inverses) x = a (y k) 2 + h Unit 7 Notes Parabolas: E: reflectors, microphones, (football game), (Davinci) satellites. Light placed where ras will reflect parallel. This point is the focus. Parabola set of all points in a plane that

More information

The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse

The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse 72 Chapter 10 Topics in Analtic Geometr 10.3 ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems.

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 10.1 Parabolas with Verte at the Origin Answers 1. up. left 3. down 4.focus: (0, -0.5), directri: = 0.5 5.focus: (0.065, 0), directri: = -0.065 6.focus: (-1.5, 0), directri: = 1.5 7.focus: (0, ), directri:

More information

Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas

Math 180 Chapter 10 Lecture Notes. Professor Miguel Ornelas Math 180 Chapter 10 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 180 Lecture Notes Section 10.1 Section 10.1 Parabolas Definition of a Parabola A parabola is the set of all points in a plane

More information

Review Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.

Review Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4. Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:

More information

Conic Sections Session 2: Ellipse

Conic Sections Session 2: Ellipse Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct 2017 1 / 24 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2

More information

CURVATURE AND RADIUS OF CURVATURE

CURVATURE AND RADIUS OF CURVATURE CHAPTER 5 CURVATURE AND RADIUS OF CURVATURE 5.1 Introduction: Curvature is a numerical measure of bending of the curve. At a particular point on the curve, a tangent can be drawn. Let this line makes an

More information

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)

More information

660 Chapter 10 Conic Sections and Polar Coordinates

660 Chapter 10 Conic Sections and Polar Coordinates Chpter Conic Sections nd Polr Coordintes 8. ( (b (c (d (e r r Ê r ; therefore cos Ê Ê ( ß is point of intersection ˆ ˆ Ê Ê Ê ß ß ˆ ß 9. ( r cos Ê cos ; r cos Ê r Š Ê r r Ê (r (b r Ê cos Ê cos Ê, Ê ß or

More information

104Math. Find the equation of the parabola and sketch it in the exercises 10-18:

104Math. Find the equation of the parabola and sketch it in the exercises 10-18: KING SAUD UNIVERSITY COLEGE OF SCIENCE DEPARTMENT OF MATHEMATICS Math Prof Messaoud Bounkhel List of Eercises: Chapter [Parabola] Find the elements of the parabola and sketch it in the eercises -9: ( )

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, )

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) D: (-, 0) (0, ) Midterm Practice Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the domain and graph the function. ) G(t) = t - 3 ) 3 - -3 - - 3 - - -3

More information

Problems with an # after the number are the only ones that a calculator is required for in the solving process.

Problems with an # after the number are the only ones that a calculator is required for in the solving process. Instructions: Make sure all problems are numbered in order. (Level : If the problem had an *please skip that number) All work is in pencil, and is shown completely. Graphs are drawn out by hand. If you

More information

DIFFERENTIAL EQUATION

DIFFERENTIAL EQUATION MDE DIFFERENTIAL EQUATION NCERT Solved eamples upto the section 9. (Introduction) and 9. (Basic Concepts) : Eample : Find the order and degree, if defined, of each of the following differential equations

More information

Chapter 2 Section 3. Partial Derivatives

Chapter 2 Section 3. Partial Derivatives Chapter Section 3 Partial Derivatives Deinition. Let be a unction o two variables and. The partial derivative o with respect to is the unction, denoted b D1 1 such that its value at an point (,) in the

More information

The details of the derivation of the equations of conics are com-

The details of the derivation of the equations of conics are com- Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles

More information

Find the rectangular coordinates for each of the following polar coordinates:

Find the rectangular coordinates for each of the following polar coordinates: WORKSHEET 13.1 1. Plot the following: 7 3 A. 6, B. 3, 6 4 5 8 D. 6, 3 C., 11 2 E. 5, F. 4, 6 3 Find the rectangular coordinates for each of the following polar coordinates: 5 2 2. 4, 3. 8, 6 3 Given the

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3 2, 5 2 C) - 5 2 Test Review (chap 0) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ) Find the point on the curve x = sin t, y = cos t, -

More information

Lecture for Week 6 (Secs ) Derivative Miscellany I

Lecture for Week 6 (Secs ) Derivative Miscellany I Lecture for Week 6 (Secs. 3.6 9) Derivative Miscellany I 1 Implicit differentiation We want to answer questions like this: 1. What is the derivative of tan 1 x? 2. What is dy dx if x 3 + y 3 + xy 2 + x

More information

Algebra 2 Unit 9 (Chapter 9)

Algebra 2 Unit 9 (Chapter 9) Algebra Unit 9 (Chapter 9) 0. Spiral Review Worksheet 0. Find verte, line of symmetry, focus and directri of a parabola. (Section 9.) Worksheet 5. Find the center and radius of a circle. (Section 9.3)

More information

C H A P T E R 9 Topics in Analytic Geometry

C H A P T E R 9 Topics in Analytic Geometry C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas.................... 77 Section. Ellipses........................... 7 Section. Hperbolas......................... 7 Section. Rotation

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,

More information

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist. . The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

More information

WBJEEM Answer Keys by Aakash Institute, Kolkata Centre MATHEMATICS

WBJEEM Answer Keys by Aakash Institute, Kolkata Centre MATHEMATICS WBJEEM - 05 Answer Keys by, Kolkata Centre MATHEMATICS Q.No. μ β γ δ 0 B A A D 0 B A C A 0 B C A * 04 C B B C 05 D D B A 06 A A B C 07 A * C A 08 D C D A 09 C C A * 0 C B D D B C A A D A A B A C A B 4

More information

D In, RS=10, sin R

D In, RS=10, sin R FEBRUARY 7, 08 Invitational Sickles The abbreviation NOTA means None of These Answers and should be chosen if choices A, B, C and D are not correct Solve for over the Real Numbers: ln( ) ln() The trigonometric

More information

PRACTICE PAPER 6 SOLUTIONS

PRACTICE PAPER 6 SOLUTIONS PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6

More information

Summary, Review, and Test

Summary, Review, and Test 944 Chapter 9 Conic Sections and Analtic Geometr 45. Use the polar equation for planetar orbits, to find the polar equation of the orbit for Mercur and Earth. Mercur: e = 0.056 and a = 36.0 * 10 6 miles

More information

Trigonometric Functions. Section 1.6

Trigonometric Functions. Section 1.6 Trigonometric Functions Section 1.6 Quick Review Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Radian

More information

Problems with an # after the number are the only ones that a calculator is required for in the solving process.

Problems with an # after the number are the only ones that a calculator is required for in the solving process. Instructions: Make sure all problems are numbered in order. All work is in pencil, and is shown completely. Graphs are drawn out by hand. If you use your calculator for some steps, intermediate work should

More information

Conic Sections Session 3: Hyperbola

Conic Sections Session 3: Hyperbola Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct 2017 1 / 16 Problem 3.1 1 Recall that an ellipse is defined as the locus of points P such that

More information

Page 1 MATHEMATICS

Page 1 MATHEMATICS PREPARED BY :S.MANIKANDAN., VICE PRINCIPAL., JOTHI VIDHYALAYA MHSS., ELAMPILLAI., SALEM., 94798 Page + MATHEMATICS PREPARED BY :S.MANIKANDAN., VICE PRINCIPAL., JOTHI VIDHYALAYA MHSS., ELAMPILLAI., SALEM.,

More information

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3

1 (C) 1 e. Q.3 The angle between the tangent lines to the graph of the function f (x) = ( 2t 5)dt at the points where (C) (A) 0 (B) 1/2 (C) 1 (D) 3 [STRAIGHT OBJECTIVE TYPE] Q. Point 'A' lies on the curve y e and has the coordinate (, ) where > 0. Point B has the coordinates (, 0). If 'O' is the origin then the maimum area of the triangle AOB is (A)

More information

a Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).

a Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8). Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates

More information

SECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z

SECTION 8-7 De Moivre s Theorem. De Moivre s Theorem, n a Natural Number nth-roots of z 8-7 De Moivre s Theorem 635 B eactl; compute the modulus and argument for part C to two decimal places. 9. (A) 3 i (B) 1 i (C) 5 6i 10. (A) 1 i 3 (B) 3i (C) 7 4i 11. (A) i 3 (B) 3 i (C) 8 5i 12. (A) 3

More information

Objective Mathematics

Objective Mathematics . A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four

More information

236 Chapter 4 Applications of Derivatives

236 Chapter 4 Applications of Derivatives 26 Chapter Applications of Derivatives Î$ &Î$ Î$ 5 Î$ 0 "Î$ 5( 2) $È 26. (a) g() œ ( 5) œ 5 Ê g () œ œ Ê critical points at œ 2 and œ 0 Ê g œ ± )(, increasing on ( _ß 2) and (!ß _), decreasing on ( 2 ß!)!

More information

Notes 10-3: Ellipses

Notes 10-3: Ellipses Notes 10-3: Ellipses I. Ellipse- Definition and Vocab An ellipse is the set of points P(x, y) in a plane such that the sum of the distances from any point P on the ellipse to two fixed points F 1 and F

More information

Chapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages

Chapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages Trigonometric Identities and Equations 7- Basic Trigonometric Identities Pages 47 430. Sample answer: 45 3. tan, cot, cot tan cos cot, cot csc 5. Rosalinda is correct; there may be other values for which

More information

Chapter 7 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 7. to accompany. College Algebra by Julie Miller

Chapter 7 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 7. to accompany. College Algebra by Julie Miller Chapter 7 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 7 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal

More information

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i}

Chapter 8. Complex Numbers, Polar Equations, and Parametric Equations. Section 8.1: Complex Numbers. 26. { ± 6i} Chapter 8 Complex Numbers, Polar Equations, and Parametric Equations 6. { ± 6i} Section 8.1: Complex Numbers 1. true. true. true 4. true 5. false (Every real number is a complex number. 6. true 7. 4 is

More information

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET

SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.

More information

Transition to College Math

Transition to College Math Transition to College Math Date: Unit 3: Trigonometr Lesson 2: Angles of Rotation Name Period Essential Question: What is the reference angle for an angle of 15? Standard: F-TF.2 Learning Target: Eplain

More information

CHAPTER 3 Applications of Differentiation

CHAPTER 3 Applications of Differentiation CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 78 Section. Rolle s Theorem and the Mean Value Theorem. 8 Section. Increasing and Decreasing Functions and the First

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 2

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 2 Cal II- Final Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Epress the following logarithm as specified. ) ln 4. in terms of ln and

More information

Parametric Equations and Polar Coordinates

Parametric Equations and Polar Coordinates Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another

More information

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone 3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong

More information

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1

1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1 Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares

More information

Time : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A

Time : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new

More information

Solutions to the Exercises of Chapter 4

Solutions to the Exercises of Chapter 4 Solutions to the Eercises of Chapter 4 4A. Basic Analtic Geometr. The distance between (, ) and (4, 5) is ( 4) +( 5) = 9+6 = 5 and that from (, 6) to (, ) is ( ( )) +( 6 ( )) = ( + )=.. i. AB = (6 ) +(

More information

Mathematics Extension 1

Mathematics Extension 1 NORTH SYDNEY GIRLS HIGH SCHOOL 05 TRIAL HSC EXAMINATION Mathematics Etension General Instructions Reading Time 5 minutes Working Time hours Write using black or blue pen Black pen is preferred Board approved

More information

CHAPTER 2 LIMITS AND CONTINUITY

CHAPTER 2 LIMITS AND CONTINUITY CHAPTER LIMITS AND CONTINUITY RATES OF CHANGE AND LIMITS (a) Does not eist As approaches from the right, g() approaches 0 As approaches from the left, g() approaches There is no single number L that all

More information

Parametric Curves You Should Know

Parametric Curves You Should Know Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b d) with slope c/a (i.e.

More information

APPM 1360 Final Exam Spring 2016

APPM 1360 Final Exam Spring 2016 APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan

More information

" $ CALCULUS 2 WORKSHEET #21. t, y = t + 1. are A) x = 0, y = 0 B) x = 0 only C) x = 1, y = 0 D) x = 1 only E) x= 0, y = 1

 $ CALCULUS 2 WORKSHEET #21. t, y = t + 1. are A) x = 0, y = 0 B) x = 0 only C) x = 1, y = 0 D) x = 1 only E) x= 0, y = 1 CALCULUS 2 WORKSHEET #2. The asymptotes of the graph of the parametric equations x = t t, y = t + are A) x = 0, y = 0 B) x = 0 only C) x =, y = 0 D) x = only E) x= 0, y = 2. What are the coordinates of

More information

CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES

CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ $ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin (

More information

Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Math 101 chapter six practice exam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Math 1 chapter si practice eam MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Which equation matches the given calculator-generated graph and description?

More information

DIFFERENTIAL EQUATION. Contents. Theory Exercise Exercise Exercise Exercise

DIFFERENTIAL EQUATION. Contents. Theory Exercise Exercise Exercise Exercise DIFFERENTIAL EQUATION Contents Topic Page No. Theor 0-0 Eercise - 04-0 Eercise - - Eercise - - 7 Eercise - 4 8-9 Answer Ke 0 - Sllabus Formation of ordinar differential equations, solution of homogeneous

More information

Add Math (4047) Paper 2

Add Math (4047) Paper 2 1. Solve the simultaneous equations 5 and 1. [5]. (i) Sketch the graph of, showing the coordinates of the points where our graph meets the coordinate aes. [] Solve the equation 10, giving our answer correct

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,

More information

TMTA Calculus and Advanced Topics Test 2010

TMTA Calculus and Advanced Topics Test 2010 . Evaluate lim Does not eist - - 0 TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {}

More information

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A

3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.

More information

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions

CHAPTER 5 Logarithmic, Exponential, and Other Transcendental Functions CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. The Natural Logarithmic Function: Differentiation.... 9 Section 5. The Natural Logarithmic Function: Integration...... 98

More information

Geometry and Motion, MA 134 Week 1

Geometry and Motion, MA 134 Week 1 Geometry and Motion, MA 134 Week 1 Mario J. Micallef Spring, 2007 Warning. These handouts are not intended to be complete lecture notes. They should be supplemented by your own notes and, importantly,

More information

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436) HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum

More information

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science

Calculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,

More information

Ready To Go On? Skills Intervention 10-1 Introduction to Conic Sections

Ready To Go On? Skills Intervention 10-1 Introduction to Conic Sections Find this vocabular word in Lesson 10-1 and the Multilingual Glossar. Graphing Parabolas and Hperbolas on a Calculator A is a single curve, whereas a has two congruent branches. Identif and describe each

More information

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths

Topic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus

Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point

More information

Coordinate goemetry in the (x, y) plane

Coordinate goemetry in the (x, y) plane Coordinate goemetr in the (x, ) plane In this chapter ou will learn how to solve problems involving parametric equations.. You can define the coordinates of a point on a curve using parametric equations.

More information

610 Chapter 9 Further Applications of Integration

610 Chapter 9 Further Applications of Integration 6 Chapter 9 Further Applications of Integration 4. So for the orthogonals: 4. c Ê c Ê Ê Ê Ê Ê C Ê É C, C ab ˆ Ê % Ê ˆ ˆ ab Ê ˆ 5. k Ê k Ê k a bab. So for the orthogonals: Ê Ê ln C 4 Ê Ê ln / ln ln C /

More information

AP Calculus BC : The Fundamental Theorem of Calculus

AP Calculus BC : The Fundamental Theorem of Calculus AP Calculus BC 415 5.3: The Fundamental Theorem of Calculus Tuesday, November 5, 008 Homework Answers 6. (a) approimately 0.5 (b) approimately 1 (c) approimately 1.75 38. 4 40. 5 50. 17 Introduction In

More information

Trigonometric Functions

Trigonometric Functions Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle

More information

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola) QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents

More information

CALCULUS II MATH Dr. Hyunju Ban

CALCULUS II MATH Dr. Hyunju Ban CALCULUS II MATH 2414 Dr. Hyunju Ban Introduction Syllabus Chapter 5.1 5.4 Chapters To Be Covered: Chap 5: Logarithmic, Exponential, and Other Transcendental Functions (2 week) Chap 7: Applications of

More information

6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary

6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary 6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3

More information

Not for reproduction

Not for reproduction ROTATION OF AES For a discussion of conic sections, see Review of Conic Sections In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the

More information

1. Solve for x and express your answers on a number line and in the indicated notation: 2

1. Solve for x and express your answers on a number line and in the indicated notation: 2 PreCalculus Honors Final Eam Review Packet June 08 This acket rovides a selection of review roblems to hel reare you for the final eam. In addition to the roblems in this acket, you should also redo all

More information

Identifying second degree equations

Identifying second degree equations Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +

More information

Chapter 11 Parametric Equations, Polar Curves, and Conic Sections

Chapter 11 Parametric Equations, Polar Curves, and Conic Sections Chapter 11 Parametric Equations, Polar Curves, and Conic Sections ü 11.1 Parametric Equations Students should read Sections 11.1-11. of Rogawski's Calculus [1] for a detailed discussion of the material

More information

11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS

11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS. Parametric Equations Preliminar Questions. Describe the shape of the curve = cos t, = sin t. For all t, + = cos t + sin t = 9cos t + sin t =

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t;

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. B) = 2t + 1; D) = 2 - t; Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculate the derivative of the function. Then find the value of the derivative as specified.

More information

CHAPTER 6 Applications of Integration

CHAPTER 6 Applications of Integration PART II CHAPTER Applications of Integration Section. Area of a Region Between Two Curves.......... Section. Volume: The Disk Method................. 7 Section. Volume: The Shell Method................

More information

January 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.

January 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola. January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the

More information

IAS 3.1 Conic Sections

IAS 3.1 Conic Sections Year 13 Mathematics IAS 3.1 Conic Sections Robert Lakeland & Carl Nugent Contents Achievement Standard.................................................. The Straight Line.......................................................

More information

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Section 10.1 Geometry of Parabola, Ellipse, Hyperbola a. Geometric Definition b. Parabola c. Ellipse d. Hyperbola e. Translations f.

More information

Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane

Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane Arc-length of a curve on the plane (Sect. 11.2) Review: Parametric curves on the plane. The slope of tangent lines to curves. The arc-length of a curve. The arc-length function and differential. Review:

More information

Chapter 8 Analytic Geometry in Two and Three Dimensions

Chapter 8 Analytic Geometry in Two and Three Dimensions Section 8. Conic Sections and Parabolas Chapter 8 Analtic Geometr in Two and Three Dimensions Section 8. Conic Sections and Parabolas Eploration. From Figure 8., we see that the ais of the parabola is

More information

Review Exercises for Chapter 2

Review Exercises for Chapter 2 Review Eercises for Chapter 367 Review Eercises for Chapter. f 1 1 f f f lim lim 1 1 1 1 lim 1 1 1 1 lim 1 1 lim lim 1 1 1 1 1 1 1 1 1 4. 8. f f f f lim lim lim lim lim f 4, 1 4, if < if (a) Nonremovable

More information

10.1 Review of Parametric Equations

10.1 Review of Parametric Equations 10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations

More information

REVIEW. cos 4. x x x on (0, x y x y. 1, if x 2

REVIEW. cos 4. x x x on (0, x y x y. 1, if x 2 Math ` Part I: Problems REVIEW Simplif (without the use of calculators). log. ln e. cos. sin (cos ). sin arccos( ). k 7. k log (sec ) 8. cos( )cos 9. ( ) 0. log (log) Solve the following equations/inequalities.

More information

REVIEW, pages

REVIEW, pages REVIEW, pages 5 5.. Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. a) tan (5 ) b) cos c) sec ( ) cos º cos ( ) cos

More information