I" Exercises 85 and 86, show that the graphs of the polar equations arc orthogonal at

In Kxercises 1 I, match the eqiiation with (he correct gruph. 1 1 he graphs are labeled (a), (b). (el. and ldl.|

In Kxercises 1 I, match the eqiiation with (he correct gruph. 1 1 he graphs are labeled (a), (b). (el. and ldl.|

In Kxercises 1 I, match the eqiiation with (he correct gruph. 1 1 he graphs are labeled (a), (b). (el. and ldl.|

In Kxercises 1 I, match the eqiiation with (he correct gruph. 1 1 he graphs are labeled (a), (b). (el. and ldl.|

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. 16.V- + 16y- - 16.V + 24y - .^ =

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. y-- I2y - 8.v + 20 =

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. 3.V- - 2y- + 24.V + 12\ + 24 =

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. .V- + y- - 16.V + 1.5 =

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. 3.V- + 2y= - 12.V + I2y + 29 =

In Exercises 5-10, analyze each equation and sketch its graph. Use a graphing utility to conlirni your results. 4.V- - 4\- - 4.V + 8^ - 1 1 =

In Kxercises 1 1 and 12, find an equation of the parabola. Vertex: (0.2); Directrix: \= -^

In Kxercises 1 1 and 12, find an equation of the parabola. Vertex: (4.2); Focus: (4.0)

In Exercises 13 and 14, find an equation of the ellipse. Vertices: (-3.0). (7.0); Foci: (0.0). (4.0)

In Exercises 13 and 14, find an equation of the ellipse.Center: (0. 0): Solution points: ( 1 . 2). (2.01

In Kxercises 15 and 16, find an equation of the hyperbola. Vertices: (4.0); Foci: (6.0)

In Kxercises 15 and 16, find an equation of the hyperbola. Foci: (0. S); Asymptotes: \' = 4v

In Kxercises 17 and IS. use a graphing utilil\ to approximate the perimeter of the ellipse.

In Kxercises 17 and IS. use a graphing utilil\ to approximate the perimeter of the ellipse.

A line is tangent to the parabola ) = .v- - 2v + 2 and perpendicular to the line y = ,v - 2, Find the equation of the line.

Satellite Antenna A cross section of a large parabolic antenna IS modeled by the graph of v = .v7200. - 100 < .v < I(.)0. The receiving and transmitting equipment is positioned at the focus. (a) Find (he coordinates of the focus. (b) Find the surface area of the antenna.

Consider a fire truck with a water tank Id led long whose vertical cross sections are ellipses modeled by the equation .V-/16 + i79 = I. (a) Find the volume of the tank. (b) Find the force on the end of the tank when it is full of water. (The density of water is 62.4 pounds per cubic foot. I (c) Find the depth of the water in the tank il it is j full (by volume) and the truck is on level ground. (d) Approximate the tank's surface area.

Consider the region hounded by tlie ellipse eecenlricity c = c/a. (a) Show that the area of the region is -mih. (b) Show that the solid (oblate spheroid) generated by revolv ing the region about the minor axis ol the ellipse has a \olume of I' = 4-irh/?i and a surface area of n=^ 34. S = lira- + m'AJ^^^' (c) Show that the solid (prolate spheroid) generated by revolv ing the region about the major axis ol the ellipse has a \olume of \' = 47r(//)-73 and a surface area of S = 2TTb- + 2Tr| lar

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter..V = I -I- 4(, y = 2 - 3r

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter..V = / -I- 4, y = t

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter.V = 6 cos 9, y = 6 sin H

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter.\ = 3 + 3 cos H. y = 2 + 5 sin

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter.A = 2 -I- sec 0. y = 3 + tan H

In Exercises 23-28, sketch the curve represented by the ametric equations (indicate the orientation of the curve). \\ri(e the correspondiii}; rectangular equation by eliminat the parameter.\ = 5 sin-' 0. V = 5 cos' II

In Kxercises 29-32. lind a parametric representation of the hne or conic. Lnic: Passes thioLigh ( - 2. h) and

In Kxercises 29-32. lind a parametric representation of the hne or conic. Lnic: Passes thioLigh ( - 2. h) and

In Kxercises 29-32. lind a parametric representation of the hne or conic. Ellipse: Center at (-3,4); Horizontal major axis of length iS and niHior axis of length h

In Kxercises 29-32. lind a parametric representation of the hne or conic. Hyperbola: Vertices at (0, 4); Foci at (f), 5)

Riitaiy Engine The rotary engine was developed by Felix Wankcl in the l'^)5()s (see Chapter 4). It features a rotor, which is a modified equilateral triangle. The mlur mo\es in a chLimber that, in two dimensions, is an epitrochoid. Use a graphing utility to graph the chamber modeled by the parametric equations. .V = cos 3fl + ,'i cos H and V = sin 3y + 5 sin 0.

A hypocycloid has the parametric equations Use a graphing utility to graph the hypocycloid for each of the following values of a and /'. ^ 35. (a) (( = 2. /) = 1 (b) a = 3. /) == 1 (c) a = 4. b = I (d) a = 10. /. = (e) a = 3. /) =- -) (f) a = 4. b = 3

Serpentine Curve Consider the parametric equations .V = 2 cot W and y = 4 sin $ cos H. < < n. (a) U.se a graphing utility to sketch the curve. (b) Eliminate the parameter to show that the rectangular equation of the seipentine curve is (4 + x-)y = 8.v.

Involute of a Circle The involute of a circle is described by the endpoint P of a string that is held taut as it is unwound from a spool that does not turn (see figure). Show that a parametric representalion of (he involute is .V = j-(cos H + H sm H) and \' = r(sin - 6 con 0).

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 37-46. (a) find dy/dx and all points of horizontal tangency. (b) eliminate the parameter where possi ble, and (c) sketch the curve represented by the parametric equations.

In Exercises 47 and 48. (a) use a graphing utility to sketch the curve represented by the parametric equations, (b) use a graphing utility to find dx/dO. dyjdtt. and dyjdx for = ttJ(\. and (c) use a graphing utility to graph the tangent line to the curve when = 7t/6. .V = cot V = sin 20

In Exercises 47 and 48. (a) use a graphing utility to sketch the curve represented by the parametric equations, (b) use a graphing utility to find dx/dO. dyjdtt. and dyjdx for = ttJ(\. and (c) use a graphing utility to graph the tangent line to the curve when = 7t/6. A y 20 - sin 2 - cos

In Exercises 49 and 50. find the length of the curve represented by the parametric equations over the given interval. V = r(cos e + (9 sin $) V = /'(sin - 9 cos 0) < e < IT

In Exercises 49 and 50. find the length of the curve represented by the parametric equations over the given interval. a = 6 cos H y = 6 sin < e < TT

In Exercises 51 and 52. the rectangular coordinates of a point are given. Plot the point and find two sets of polar coor dinates for the point for < < 2tt. (4. -4)

In Exercises 51 and 52. the rectangular coordinates of a point are given. Plot the point and find two sets of polar coor dinates for the point for < < 2tt. (-1.3)

In P^xercises 5.^-60, convert the polar equation to rectangular form. ( = 3 cos d

In P^xercises 5.^-60, convert the polar equation to rectangular form. ) = 10

In P^xercises 5.^-60, convert the polar equation to rectangular form. /- = -2(1 + cos d)

In P^xercises 5.^-60, convert the polar equation to rectangular form. / cos

In P^xercises 5.^-60, convert the polar equation to rectangular form. /- = cos2ff

In P^xercises 5.^-60, convert the polar equation to rectangular form. ) = 4 sec| ^

In P^xercises 5.^-60, convert the polar equation to rectangular form. ; = 4 cos 2 sec

In P^xercises 5.^-60, convert the polar equation to rectangular form. =

In Exercises 61-64, convert the rectangular equation to polar form. (a- + y-)- = a.x-y

In Exercises 61-64, convert the rectangular equation to polar form. A- + \- - 4 V = A'

In Exercises 61-64, convert the rectangular equation to polar form. A- + V- = (V- arctan ' .

In Exercises 61-64, convert the rectangular equation to polar form. (a- + V-) arclan

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 65-76, sketch a graph of the polar e(|uation.

In Exercises 77-80, use a graphing utility to graph the polar equation.

In Exercises 77-80, use a graphing utility to graph the polar equation.

In Exercises 77-80, use a graphing utility to graph the polar equation.

In Exercises 77-80, use a graphing utility to graph the polar equation.

In Exercises 81 and 82, (a) find the tangents at the pole, (b) find all points of horizontal and vertical tangency. and (c) use a graphing utility to graph the polar equation and draw a tangent line to the graph for = tjJ6. )- = 1 - 2 cos

In Exercises 81 and 82, (a) find the tangents at the pole, (b) find all points of horizontal and vertical tangency. and (c) use a graphing utility to graph the polar equation and draw a tangent line to the graph for = tjJ6. ;- = 4 sin 20

Pmd the angle between the circle r = 3 sm and the liniac^on I- = 4 - 5 sin at tlie punit of intersection (3/2. Tr/6).

True or False? There is a unique polar coordinate represen lalion tor each point in the plane. Hxpiain,

I" Exercises 85 and 86, show that the graphs of the polar equations arc orthogonal at the points of intersection. Use a graphing utility to confirm your results graphically. ; = 1 + cos r = \ - cos

I" Exercises 85 and 86, show that the graphs of the polar equations arc orthogonal at the points of intersection. Use a graphing utility to confirm your results graphically. a sin CI cos

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places.

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places.

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places.

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places.

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places. Interior of r- = 4 sin 20

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places. Common interior of r = 3 and r- = IS sin 2(*

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places. Common interior of / = 4 cos Wand / - 2

In Exercises S7-94. use a graphing utility to graph the polar equation. Set up an integral for finding the area of the indicated region and use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places. Reiiion bounded by the polar axis and c = c" lor < < it

In Exercises 95 and 96. find the perimeter of the curve.

In Exercises 95 and 96. find the perimeter of the curve.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 97-102. sketch and identify the graph. Use a graphing utilit\ to confirm your results.

In Exercises 10.3-108, find a polar equation for the line or conic. Circle Circle Solution point: (0. 0)

In Exercises 10.3-108, find a polar equation for the line or conic. 1 ine Solution point; (0,0) Slope; y3

In Exercises 10.3-108, find a polar equation for the line or conic. Parabola Vertex; (2. n) Focus; (0, 0)

In Exercises 10.3-108, find a polar equation for the line or conic. Parabola Vertex: (2. tt/2) Focus; (0. 0)

In Exercises 10.3-108, find a polar equation for the line or conic. Ellipse Vertices; (5.0). (I.ttOne focus: (0. 0)

In Exercises 10.3-108, find a polar equation for the line or conic. Hyperbola Vertices: (1.0). (7.0) One focus; (0, 0)