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

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1 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 not on the line. Use the definition of a parabola and the graph provided to derive the equation of the parabola in standard form with vertex h, k ( ) and directrix y = k - p. What is the length of the focal diameter? 2. (a) Graph ( y - 2) 2 = -4 ( x -1). (b) Find the length of the curve on [ 0, 2]. 3. Ellipse An ellipse is the set of all points x, y ( ) the sum of whose distances from two fixed distinct points is constant. Use the definition of an ellipse and the graph provided to derive the equation of the ellipse x2 a + y2 =1 where 2 2 b a > b.

2 4. (a) Write the equation 25x 2 + 9y x + 36y - 89 = 0in standard form and graph it. (b) Find dy dx. 5. (a) Graph C with equation ( x - 2) 2-16y 2 = 16. (b) R is the region enclosed by C and the vertical line x = 7. Find the volume of the solid generated when R is revolved about the y - axis Plane curves and Parametric Equations 1. Consider the cubic equation x = f (t) and the quadratic equation y = g( t). (a) Determine the interval(s) where the dy dx > 0, dy dx = undefined, and dy dx < 0. (b) Where is d 2 y dx 2 > 0. (c) On the given Cartesian plane sketch the rectangular equation which contains x and y. Make a table of values to come up with your sketch.

3 2. Eliminate the parameter and state the restrictions for t, x, and y. (a) x =1+t, y = 1 t (b) x = t, y = t - 2 (c) x = tant, y = 2sec 2 t (d) x = sin2t, y = cos 2 t -sin 2 t (e) x = cotq, y = 2sin2q (f) x = -sint, y = tant 3. Parameterize the following equations. There are multiple correct answers. (a) ( x - 2) 2 + y2 4 =1 (b) x2 - y 2 = 9 4. Parameterize the line segment that is at ( 1, 2) when t =1 and at ( 4, - 2) when t = Parameterize the curve that starts at ( 1, 0) and follows the path y = x 2-1 and terminates at (-3, 8).

4 6. Graph the parametric equations and label the orientation. (a) x = e -t, y = e -2t +1 (b) x = sint, y = -1+ 2sint (c) x = cotq, y = cscq 10.3 Derivatives of Parametric Equations 1. Let x = f (t) and y = g( t). Find an expression for dy dx, d 2 y dx, and d 3 y 2 dx Let the curve C be defined y = t 3-6t 2-15t and x = 2 3 t 3 +t 2. (a) Determine the value(s) of t for which the curve C has horizontal tangent line(s). (b) Find the equation of the tangent line to C when t = -2. (c) On what interval(s) is C concave up? 3. Find the equation of the tangent line(s) to the curve defined by x = t 3 - t and y = 2t 2 at the point ( 0, 2). ( ) y = tanq 4. A curve C is defined by x = sin 2q and. (a) Find the domain and the range of C. (b) Determine the values of q for which the curve C has vertical tangent lines. (c) Find the value of q for which x is reaches its absolute maximum. 5. Find the d 2 y dx 2. (a) x = e -2t -1 and y = 2t. (b) x = lnt t and y = ( lnt ) 2.

5 If s t t ( ) = position, then s ( t) = v( t) 2 = velocity, speed = v( t) and distance = v( t) dt ò t 1. For parametric equations, and the speed and the distance traveled is the length of the curve [ ]2 b L = 1+ éë f ( x) ù û 2 b dy / dt ò dx = ò 1+ dx = a a dx / dt [ ] 2 ò b a 2 æ dx ö æ ç + dy ö ç è dt ø è dt ø 2 ( dx) ( dt / dx) = ò b a 2 ædx ö æ ç + dy ö ç dt è dt ø è dt ø 2 6. An object travels along a curve C defined by x ( t) =1+ t 2 and y( t) =1+t. (a) Determine the speed of the object when t = 1 2. é (b) Find the distance traveled by the object on the interval 0, ë ê 1ù 2û ú. ( ) = t cost and y( t) = t sint. 7. An object travels along a curve C defined by x t (a) Determine the speed of the object when t =1 (b) Find the distance traveled by the object on the interval é ë0, 3ù û. 8. An object travels along a curve such that x t GRAPHING calculator. (a) If dy dx = when t = p 6 find y æ p ö ç è 6. ø (b) Find the speed of the object when t = p 6. ( ) = 2t 2 + ln( t) where y( t) is not explicitly defined. USE A

6 10.4 Polar Coordinates 1. Plot the following point. Find two additional points that are equivalent to the given point. æ (a) -2, 2p ö ç è 3 ø (b) æ 3, - p ö ç è 4 ø 2. Convert each equation from rectangular coordinates to polar coordinates. (a) x = 2 (b) x 2 - y 2 = 4 (c) xy 3 - y = 4 3. Convert each equation from polar to rectangular coordinates. (a) r 2 = sin ( 2q ) (b) q = p 4 (c) r = 3tanq 4. Graph the each equation and label the orientation where 0 q 2p. Label the point where t = 0. (a) r =1+ 2sinq (b) r = 2 + 2cosq

7 (c) r = 2cscq (d) r = 3sin( 2q ) (e) r = 2cos( 2q ) (f) q = p 4

8 5. Let r = 2sinq + 3cos 2 q, 0 q p. (a) Find the slope of the tangent line to r = f ( q ) when t = p 2. (b) Find dr dq when t = p and explain the significance of your answer. 6 (c) Find the values of q for which the distance from the pole to the curve increases. (d) Find the rectangular coordinates ( x, y) when t = 2p Let r =q + sinq, 0 q p. (a) Find the equation of the tangent line to r = f q (b) Find dr dq. ( ) at the rectangular point ç 1, 3 è 2 æ ö. ø 7. Let r =q, 0 q p. é (a) Find the length of the curve on the interval 0, p ë ê 4. ù û ú (b) Find the slope of the tangent line to r = f ( q ) when t = p Area in Polar Coordinates 1. Find the area enclosed by r = 2 + 2cosq in quadrant II. Graph r.

9 2. (a) Find the area enclosed by the inner loop of r = 2-4sinq. Graph r. (b) Find the area between the inner and outer loops of r = 2-4sinq. 3. Find the area of the enclosed by r = 3cos 3q. Graph r.

10 4. Find the area in common to r = 2 and r = 4 cosq. Graph the equations. 5. Find the area in common to r = 2 and r = 4sin ( 2q ). A graph has been provided.