Chapter 10 Homework: Parametric Equations and Polar Coordinates

Transcription

1 Chapter 1 Homework: Parametric Equations and Polar Coordinates Name

2 Homework Consider the parametric equations x = t and y = 3 t. a. Construct a table of values for t =, 1, 2, 3, and 4 b. Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. c. Use a graphing utility to confirm your graph in part (b). d. Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. 2. Consider the parametric equations x = 4 cos 2 θ and y = 2sinθ. a. Construct a table of values for θ = π 2, π 4,, π 4, π 2. b. Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. c. Use a graphing utility to confirm your graph in part (b). d. Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. π e. If values of θ were selected from the interval 2, 3π 2 for the table in part (a), would the graph in part (b) be different? Explain.

3 Sketch the curve represented by the parametric equations (indicate the orientation of the curve) and write the corresponding rectangular equation by eliminating the parameter. 3. x = t +1 y = t 2 4. x = t 3 y = t x = 2t y = t 2 6. x = e t y = e 3t x = 8cosθ y = 8sinθ

4 Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. 8. x = cosθ y = 2sin2θ 9. x = 4secθ y = 3tanθ Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. 1. x = 2cosθ y = 2sinθ x = 4t 2 1 t y = 1/ t

5 Eliminate the parameter and obtain the standard form of the rectangular equation. 11. Circle: x = h + r cosθ y = k + rsinθ 12. Ellipse: x = h + acosθ y = k + bsinθ Find a set of parametric equations for the rectangular equation that satisfies the given equation. 13. y = 2x 5 t = at the point (3,1) 14. y = x 2 t = 4 at the point (4,16)

6 Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. 15. Cycloid: x = 2(θ sinθ) y = 2(1 cosθ) 16. Prolate cycloid: x = θ 3 2 sinθ y = cosθ 17. Hypocycloid: x = 3cos 3 θ y = 3sin 3 θ Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 18. The graph of the parametric equations x = t 2 and y = t 2 is the line y = x. 19. If y is a function of t and x is a function of t, then y is a function of x.

7 2. The curve represented by the parametric equations x = t and y = cost can be written as an equation of the form y = f (x). Answers: 1d. t = x 2 y = 3 x 2 x 2d. y x 4 = 1 3. y = (x 1) 2 4. y = x2/ y = x y = x 3 +1 x > 7. x 2 + y 2 = y 2 = 16x 2 (1 x 2 ) x y2 9 = 1 1. x 2 + y 2 = ( x h) + y k)2 = 1 r 2 r ( x h) + y k)2 = 1 a 2 b x = t + 3 y = 2t x = t y = t Not smooth at 2nπ = θ, n I. 16. Smooth everywhere. 17. Not smooth at nπ = θ, n I False. Only defined for non-negative values of x and y. 19. False. Consider x = 2cost, y = 2sint, for example. 2. True.

8 Homework 1.3 Find dy / dx. 1. x = t 2 y = 7 6t Find dy / dx and d 2 y / dx 2 and find the slope and concavity (if possible) at the given value of the parameter. 2. x = 4 cosθ y = 4sinθ θ = π 4 3. x = 2 + secθ y = 1+ 2 tanθ θ = π 6 Find the equation of the tangent lines at the point where the curve crosses itself. 4. x = 2sin2t y = 3sint

9 Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. 5. x = 3cosθ y = 3sinθ Determine the t intervals on which the curve is concave downward or concave upward. 6. x = 2t + lnt y = 2t lnt Find the arc length of the curve on the given interval. 7. x = 3t + 5 y = 7 2t 1 t 3

10 8. The path of a projectile is modeled by the parametric equations: x = (9cos 3! )t y = (9sin 3! )t 16t 2 where x and y are measured in feet. a. Use a graphing utility to graph the path of the projectile. b. Use a graphing utility to approximate the range of the projectile. c. Use the integration capabilities of a graphing utility to approximate the arc length of the path. Compare this result with the range of the projectile. Non-calculator Items 1.

11 2. 3. Which of the following gives the length of the path described by the parametric equations and y = e 5t from t = to t = π? (A) (B) π ( ) + e 1t sin 2 t 3 dt (C) 9t 4 cos 2 t 3 π ( ) + e 1t cos 2 t 3 dt (D) 3t 2 cos t 3 π π ( ) + 25e 1t ( ) + 5e 5t dt dt x = sin( t 3 ) 4. In the xy-plane, a particle moves along the parabola y = x 2 x with a constant speed of 2 1 units dx per second. If, what is the value of when the particle is at the point (2, 2)? dt > dy dt 2 1 (A) 2/3 (B) (C) 3 (D) 6 (E) For t 13, an object travels along an elliptical path given by the parametric equations x = 3cost and y = 4sint. At the point where t = 13, the object leaves the path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels? (A) -4/3 (B) -3/4 (C) 4 tan (D) (E) 3 3tan13 4 tan13

12 6. The position of a particle moving in the xy-plane is given by the parametric equations x(t) = t 3 3t 2 y(t) = 2t 3 3t 2 12t For what values of t is the particle at rest? (A) -1 only (B) only (C) 2 only (D) -1 and 2 only (E) -1, and 2 only 7. A curve C is defined by the parametric equations x(t) = t 2 4t +1 y(t) = t 3 Which of the following is an equation of the line tangent to the graph of C at the point (-3, 8)? (A) x = -3 (B) x = 2 (C) y = 8 (D) y = 27 ( 1 x + 3) + 8 (E) y = 12(x + 3) Which of the following gives the length of the path described by the parametric equations x(t) = 2 + 3t and y(t) = 1+ t 2 from t = to t = 1? 1 (A) 1+ 4t 2 9 dt 1 (B) 1+ 4t 2 dt (C) (D) t + t 2 dt 9 + 4t 2 dt ( ) 2 dt (E) (2 + 3t) t 2

13 9. If x = t 2 1 and y = lnt, what is d 2 y in terms of t? 2 dx (A) 1 2t 4 (B) 1 2t 4 (C) 1 t 3 (D) 1 2t 2 (E) 1 2t 2 1. A particle moves in the xy-plane with position given by ( x(t), y(t) ) = ( 5 2t,t 2 3) at time t. In which direction is the particle moving as it passes through the point (3, -2)? (A) up and to the left (B) down and to the left (C) up and to the right (D) down and to the right (E) straight up 11. If x(t) = t and y(t) = t 4 + 3, for t >, then in term of t, d 2 y dx = 2 (A) 1/2 (B) 2 (C) 4t (D) 6t 2 (E) 12t 2 Calculator Active Items 1. A particle moves in the xy-plane so that its position at any time t is given by x(t) = t 2 and y(t) = sin(4t). What is the speed of the particle when t = 3? (A) 2.99 (B) 3.62 (C) (D) 9.16 (E) The position of a particle moving in the xy-plane is given by the parametric functions x(t) and y(t) for which x'(t) = t sint and y'(t) = 5e 3t + 2. What is the slope of the line tangent to the path of the particle at the point at which t = 2? (A).94 (B) 1.17 (C) (D) 2.12 (E) 3.66

14 3. For t, the components of the velocity of a particle moving in the xy-plane are given by the parametric equations x'(t) = 1 t +1 and y'(t) = kekt, where k is a positive constant. The line y = 4x + 3 is parallel to the line tangent to the path of the particle at the point where t = 2. What is the value of k? (A).72 (B).433 (C).495 (D).83 (E)_.828 Answers: 1. dy dx = 3 t 2. slope = -1, concave down 3. slope = 4, concave down 4. At t =, y = 3 4 x, t = π, y = 3 4 x 5. θ = π 2, 3π 2 6. Concave up (, ) b. approximately feet, 8c. approximately feet No Calculator Items 1b, 2a, 3c, 4d, 5d, 6c, 7a, 8d, 9a, 1, 11 Calculator Active Items 1c, 2b, 3c

15 Homework 1.4 Plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. 1. 8, π , 3π 4 The rectangular coordinates of a point are given. Plot the point the find two sets of polar coordinates for the point for θ < 2π. 3. ( 1, 3) Convert the rectangular equation to polar form and sketch its graph. 4. x 2 + y 2 = 9 5. y = x y + 2 =

16 Convert the polar equation to rectangular form and sketch its graph. 7. r = 4 8. r = secθ tanθ Use a graphing utility to graph the polar equation. Find an interval for θ over which the graph is traced only once. 9. r = 2 5cosθ 1. r = 2 1+ cosθ 11. r = 2cos 3θ 2 Find dy / dx and the slopes of the tangent lines at the points indicated on the graph of the polar equation. 12. r = 2 + 3sinθ

17 Use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of θ, and (c) find dy / dx at the given value of θ. (Hint: let the increment between the values of θ equal π / r = 3(1 cosθ), θ = π / 2 Find the points of vertical and horizontal tangency (if any) to the polar curve. 14. r = 1 sinθ Sketch a graph of the polar equation and find the tangents at the pole. 15. r = 5sinθ

18 16. r = 4 cos 3θ No Calculator Items What is the slope of the line tangent to the polar curve r = cos θ at the point where = ' (? (A) 3 (B) +, (C) +,, (D) - (E) 3

19 Answers: 1. (, 8) ( ) , , 4π 3 4. r = 3 5. r = 8 sinθ 6. r =, 2, π 3 2 3cosθ sinθ 7. x 2 + y 2 = x 2 = y 9. [,2π ) 1. ( π,π ) 11. [,4π ) 12. At 5, π 2 dy / dx =, at 2,π 13. θ = π / 2, dy / dx = 1. ( ) dy / dx = 2 / 3, at 1, 3π 2 dy / dx =. 14. Horizontal tangents at θ = 3π 2, π 6, 5π 6, Vertical tangents at θ = 7π 6,11π At θ =, dy / dx =, at θ = π, dy / dx = At θ = π 6, dy / dx = 1 3, at θ = π 2, dy / dx = 3, at θ = 5π 6, dy / dx = 1 3 No Calculator Items: 1b, 2b, 3b

20 Homework 1.5 Find the area of the region. 1. Interior of r = 6sinθ 2. One petal of r = 2cos 3θ 3. Interior of r = 1 sinθ Use a graphing utility to graph the polar equation and find the area of the given region. 4. Inner loop of r = 1+ 2cosθ

21 5. Between the loops of r = 1+ 2cosθ 6. Between the loops of r = 3 6sinθ

22 1 No Calculator Items 1. f( θ) = sin( θ) Which of the following expressions gives the total area enclosed by the polar curve figure above? 1 (A) (B) (C) 2 sin2 θ dθ sin 2 θ dθ (D) π π sin 4 θ dθ (D) π π 2 sin 4 θ dθ π 1 2 sin4 θ dθ r = sin 2 θ shown in the 2. Which of the following integrals represents the area enclosed by the smaller loop of the graph of r = 1+ 2sinθ? 11π /6 1 (A) ( 1+ 2sinθ ) 2 dθ 2 7π /6 (B) r = 2θ 1 2 (C) (D) (E) 1 2 7π /6 π /6 7π /6 π /6 π /6 7π /6 11π /6 7π /6 ( 1+ 2sinθ ) dθ ( 1+ 2sinθ ) 2 dθ ( 1+ 2sinθ ) 2 dθ ( 1+ 2sinθ ) dθ

23 3. Let R be the region in the first quadrant that is bounded by the polar curves r = Ɵ and Ɵ = k, where k is a constant < k < π/2, as shown in the figure above. What is the area of R in terms of k? (A) k 3 /6 (B) k 3 /3 (C) k 3 /2 (D) k 2 /4 (E) k 2 /2 4. Which of the following integrals gives the area of the region that is bounded by the graphs of the polar equations θ =, θ = π 4, and r = 2 cosθ + sinθ? (A) (B) (C) (D) (E) π /4 π /4 π /4 π /4 π /4 1 dθ cosθ + sinθ 2 dθ cosθ + sinθ 2 dθ ( cosθ + sinθ ) 2 4 dθ ( cosθ + sinθ ) 2 ( ) 2 ( cosθ + sinθ ) 4 2 cosθ sinθ dθ

24 Calculator Active Items 1. 2.

25 3. Let R be the region in the first quadrant that is bounded above by the polar curve r = 4 cosθ and below by the line θ = 1, as shown in the figure above. What is the area of R? (A).317 (B).465 (C).929 (D) (E) Answers: No Calculator Items: 1d, 2a, 3a, 4 Calculator Active Items: 1d, 2d, 3b

26 Optional problems: Find the points of intersection of the graphs of the equations. 1. r = 3(1+ sinθ) r = 3(1 sinθ) 2. r = 1+ cosθ r = 1 sinθ

27 Use a graphing utility to graph the polar equations and find the area of the given region. 3. Common interior of r = 4sin2θ and r = Common interior of r = 4sinθ and r = 2. Optional answers: 1. (x,y): (3,) (3,π ) (,) (r, θ ): (3, ) (, π /2) (, 3π /2) 2. (x,y): (-.27,.27) (1.27, -1.27) (,) (r, θ ): (.292, 3π /4) (1.77, 7π /4) (,)