HW - Chapter 10 - Parametric Equations and Polar Coordinates

Transcription

1 Berkeley City College Due: HW - Chapter 0 - Parametric Equations and Polar Coordinates Name Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion. ) x = cos t, y = sin t,! t! y x ) x = sin t, y = cos t, 0 t! y x Instructor K. Pernell

2 ) x = t, y = t, - t y x Obtain the Cartesian equation of the curve by eliminating the parameter. ) x = t, y = t + 8 ) x = t - 7, y = t 6) x = t - t, y = t -

3 7) x = 9 sin t, y = 9 cos t; 0 t! Find dy/dx without eliminating the parameter. 8) x = t, y = t ; t 0 9) x = 6 tan t -, y = sec t +, t (n+)! 0) x = ln(t), y = e t ) x = /t 6, y = - + ln t Find d y/dx without eliminating the parameter. ) x = t, y = 0 t ; t 0

4 ) x = - sin t, y = + 7 cos t, t n! ) x = ln(t), y = ln(8t), t > 0 ) x = t + 8t, y = t - t, t -8 Find an equation for the line tangent to the curve at the point defined by the given value of t. 6) x = sin t, y = 6 sin t, t =! Find the length of the parametric curve defined over the given interval. 7) x = 6t - 6, y = t +, 0 t

5 Find the area of the surface generated by revolving the curves about the indicated axis. 8) x = sin t, y = + cos t, 0 t!; x-axis 9) x = t + 6, y = t + 6t, - 6 t 6; y-axis Change the given polar coordinates (r, θ) to rectangular coordinates (x, y). 0) (,!/6) ) (-, -!/) Find a set of polar coordinates for the point for which the rectangular coordinates are given. ) (-, ) A) 0,! B) 0,! 6 C),! 6 D),!

6 For the given rectangular equation, write an equivalent polar equation. ) x + y = 6 ) x + y - 0x = 0 For the given polar equation, write an equivalent rectangular equation. ) r cos θ = 6) r = -9 csc θ 7) r = 7 cos θ - sin θ 8) r sin θ = 6

7 Find the area of the specified region. 9) Inside one leaf of the four-leaved rose r = sin θ Find the length of the curve. 0) The spiral r = θ, 0 θ Graph the polar equation. ) r = - sin θ r Find an equation for the line tangent to the curve at the point defined by the given value of t. ) x = 6t -, y = t, t = 7

8 Obtain the Cartesian equation of the curve by eliminating the parameter. ) x = cos θ, y = - sin θ ) x = 9 sec t, y = 7 tan t Find the length of the curve. ) The spiral r = e θ, 0 θ " 6) The circle r = 7 cos θ 8

9 Answer Key Testname: MATHB_HWCH0 ) x + y = 6 y x Counterclockwise from (-, 0) to (, 0) Objective: (0.) Graph Parametric Equations and Eliminate the Parameter ) x + y 9 = y x Counterclockwise from (0, ) to (0, ), one rotation Objective: (0.) Graph Parametric Equations and Eliminate the Parameter 9

10 Answer Key Testname: MATHB_HWCH0 ) x = y y x Entire parabola, bottom to top (from fourth quadrant to origin to first quadrant) Objective: (0.) Graph Parametric Equations and Eliminate the Parameter ) y = x + 8 Objective: (0.) Convert Parametric Equations to Cartesian Equation I ) x 9 + y 8 = Objective: (0.) Convert Parametric Equations to Cartesian Equation I 6) x = y + y Objective: (0.) Convert Parametric Equations to Cartesian Equation I 7) x + y = 9 Objective: (0.) Convert Parametric Equations to Cartesian Equation II 8) 0 t Objective: (0.) Find dy/dx Given Parametric Equations 9) sin t Objective: (0.) Find dy/dx Given Parametric Equations 0) te t Objective: (0.) Find dy/dx Given Parametric Equations ) - t6 6 Objective: (0.) Find dy/dx Given Parametric Equations ) 0 00t Objective: (0.) Find dy/dx Given Parametric Equations 0

11 Answer Key Testname: MATHB_HWCH0 ) - 7 sec t ) 0 ) Objective: (0.) Find dy/dx Given Parametric Equations Objective: (0.) Find dy/dx Given Parametric Equations 0 (t + 8) Objective: (0.) Find dy/dx Given Parametric Equations 6) y = 6x Objective: (0.) Find Equation of Tangent Given Parametric Equations 7) 6 Objective: (0.) Find Length of Parametric Curve I 8) " Objective: (0.) Find Surface Area of Revolution 9) 8 " Objective: (0.) Find Surface Area of Revolution 0), Objective: (0.) Convert From Polar to Cartesian Coordinates ) (-, ) Objective: (0.) Convert From Polar to Cartesian Coordinates ) B Objective: (0.) Convert From Cartesian to Polar Coordinates ) r = 8 Objective: (0.) Convert Cartesian Equation to Polar Form ) r = 0 cos θ Objective: (0.) Convert Cartesian Equation to Polar Form ) x = Objective: (0.) Convert Polar Equation to Cartesian Form 6) y = -9 Objective: (0.) Convert Polar Equation to Cartesian Form 7) 7x - y = Objective: (0.) Convert Polar Equation to Cartesian Form 8) y = x Objective: (0.) Convert Polar Equation to Cartesian Form 9) 9" 8 Objective: (0.) Find Area of Region Inside Polar Curve

12 Answer Key Testname: MATHB_HWCH0 0) 80 ) Objective: (0.) Find Length of Polar Curve r Objective: (0.) Graph Polar Equation II ) y = x - Objective: (0.) Find Equation of Tangent Given Parametric Equations ) y = -6x (- x ) Objective: (0.) Convert Parametric Equations to Cartesian Equation II ) ) x 8 - y 9 = Objective: (0.) Convert Parametric Equations to Cartesian Equation II 7 (e" - ) Objective: (0.) Find Length of Polar Curve 6) 7" Objective: (0.) Find Length of Polar Curve