advertisement

BC Calculus Chapter 10 Review Section _______ Name ____________________ Date _____________________ Parametric Equations 1. Plot points to sketch the curve described by the parametric equations. Mark the orientation on the curve. y x t2 5 t t y x 2 x 3 t 2 y 2. Change the following to rectangular form by eliminating the parameter. Then graph. 1 t x and y , t 1 t 1 t 1 y x 3. Eliminate the parameter to sketch the curve. x cos and y 3sin , 0 2 y x 4. If x cos t and y 3sin t find dy d2y and . dx dx 2 dy dx dy dt dx dt d2y and dx 2 d dy dt dx dx dt 5. Find the slope and concavity of x t and y 4 t 2 1, t 0 at the point 2,3 . 1 1 6. Write an equation of a tangent line to the curve defined by x t 1 and y t 1 at the point when t 1. Arc Length: If a curve is smooth and does not intersect itself the length of an arc is given by arc length b a 2 2 dx dy dt dt dt 7. Using the parametric equations from example 6, find the arc length on the interval 1 t 3 . Polar Graphs Plotting points in polar form 8. Use the polar grid to plot these polar points. A C 3, 3, 6 5 6 B 4, D 2, Point conversions 9. Write the rectangular point a. r 0, 0 3 2 1, 3 in polar form such that: b. r 0, 0 c. r 0, 0 d. r 0, 0 10. Change the point 2, to rectangular form. 11. Change the point 3,3 to polar form. Equation conversions (rectangular to polar) 12. y 4 13. 3x y 2 0 14. x 2 y 2 2 x 0 Equation conversions (polar to rectangular) 15. r 2 16. r 3cos 17. r 2csc Sketching polar graphs: Circles: r d cos (x-axis symmetry) r d sin (y-axis symmetry) Rose petal curves: d is the diameter r a cos n (x-axis symmetry) r a sin n (y-axis symmetry) Limaçons: r a b cos r a b sin (x-axis symmetry) (y-axis symmetry) a is the maximum r , n petals if n is odd, 2n petals if n is even may have inner loop may or may not include the pole Sketching polar graphs (use a calculator only on those in bold) 18. Circles a. r 2cos 20. Limaçons a. r 2 3cos b. r 5sin 19. Rose petal curves a. r 3cos 2 b. r 3 3sin b. r 4sin 3 21. Lemniscate r 2 9sin 2 Tangent lines 22. Find an equation of the tangent line to the graph of r 2 1 sin at the point 2,0 . 23. Find the points at which the graph of r 2 2cos has horizontal tangents. Tangent lines at the pole 24. Find the equations of the lines tangent to r 4sin 3 at the pole. Polar Area 25. Find the area of one petal of the curve r 3cos 3 . Polar area = 1 2 r d 2 Intersections of Polar Graphs 26. Find the points of intersection of the graphs of r 1 2cos and r 1 . Area between two curves 27. Find the area of the region common to the two regions bounded by r 6cos and r 2 2cos . 28.. Find the area between the loops of r 2 1 2sin . 29. Find the length of the arc from 0 to 2 for the curve r 2 2cos . Arc Length r2 dr d 2 30. Given a position vector 3t 2 , t 3 3t 2 4 for a particle moving in the xy-plane find the following. a. graph the path of the particle on the interval 0 t 2 b. the velocity vector at time t 1 c. the speed of the particle at time t 1 y x d. the distance traveled between t 0 and t 3 e. the time(s) when the particle is at rest g. the direction of the particle at time t 1 and when t 2 f. the acceleration vector at time t 2 d