Exercise. Exercise 1.1. MA112 Section : Prepared by Dr.Archara Pacheenburawana 1

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1 MA112 Section : Prepared by Dr.Archara Pacheenburawana 1 Exercise Exercise Find the vertex, focus, and directrix of the parabola and sketch its graph. 1. x = 2y y +x 2 = x 2 = y 4. y 2 = 12x 5. (x+2) 2 = 8(y 3) 6. x 1 = (y +5) 2 7. y 2 +2y +12x+25 = 0 8. y +12x 2x 2 = Find an equation of the parabola. Then find the focus and directrix. 9. y 10. y 2 1 x 1 2 x Find an equation for the parabola that satisfies the given conditions. 11. Vertex (0,0), focus (0, 2) 12. Vertex (0, 0), directrix x = Focus ( 4, 0), directrix x = Focus (3,6), vertex (3,2) 15. Vertex (0,0), axis y = 0, passing through (1, 4) 16. Vertical axis y = 0, passing through ( 2,3), (0,3), and (1,9)

2 MA112 Section : Prepared by Dr.Archara Pacheenburawana Find the vertices and foci of the ellipse and sketch its graph. x y2 5 = x y2 100 = x 2 +y 2 = x 2 +25y 2 = x 2 18x+4y 2 = x 2 +2y 2 6x+4y Find an equation of the ellipse. Then find its foci. 23. y 24. y x 1 2 x Find an equation for the ellipse that satisfies the given conditions. 25. Foci (±2,0), vertices (±5,0) 26. Foci (0,±5), vertices (0,±13) 27. Foci (0,2),(0,6), vertices (0,0),(0,8) 28. Foci (0, 1),(8, 1), vertex (9, 1) 29. Center (2,2), focus (0,2), vertex (5,2) 30. Foci (±2,0), passing through (2,1) 31. Ends of major axis (0,±6), pass through ( 3,2) 32. Foci ( 1,1) and (2, 3), minor axis of length 4

3 MA112 Section : Prepared by Dr.Archara Pacheenburawana Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 33. x y2 25 = y 2 16 x2 36 = y 2 x 2 = x 2 4y 2 = y 2 3x 2 4y +12x+8 = x 2 9y 2 +64x 90y = Find an equation for a hyperbola that satisfies the given conditions. 39. Foci (0,±3), vertices (0,±1) 40. Foci (±6,0), vertices (±4,0) 41. Foci (1, 3) and (7, 3), vertices (2, 3) and (7, 3) 42. Foci (2, 2) and (2, 8), vertex (2, 0) and (2, 6) 43. Vertices (±3, 0), asymptotes y = ±2x 44. Foci (2,2) and (6,2), asymptotes y = x 2 and y = 6 x 45. Vertices (0,6) and (6,6), foci 10 units apart 46. Asymptotes y = x 2 and y = x+4, pass through the origin Identify the type of conic section whose equation is given and find the vertices and foci. 47. x 2 = y x 2 = y x 2 = 4y 2y y 2 8y = 6x y 2 +2y = 4x x 2 +4x+y 2 = 0

4 MA112 Section : Prepared by Dr.Archara Pacheenburawana 4 Exercise Let an x y -coordinate system be obtained by rotating an xy-coordinate system through an angle of θ = 60. (a) Find the x y -coordinates of the point whose xy-coordinates are ( 2,6). (b) Find an equation of the curve 3xy +y 2 = 6 in x y -coordinates (c) Sketch the curve in part (b), showing both xy-axes and x y -axes. 2 6 Rotate the coordinate axes to remove the xy-term. Then identify the type of conic and sketch its graph. 2. xy = 9 3. x 2 +4xy 2y 2 6 = 0 4. x xy +3y x 2y = x 2 24xy +16y 2 80x 60y +100 = x 2 72xy +73y 2 +40x+30y 75 = 0 7. Let an x y -coordinate system be obtained by rotating an xy-coordinate system through an angle of θ = 45. Find an equation of the curve 3x 2 +y 2 = 6 in xy-coordinates. 8 9 Show that the graph of the given equation is a parabola. Find its vertex, focus, and directrix. 8. x 2 +2xy +y x 4 2y = x 2 24xy +16y 2 80x 60y +100 = Show that the graph of the given equation is an ellipse. Find its foci, vertices, and the ends of its minor axis x 2 168xy +337y = x xy +21y 2 32x+32 3y 80 = Show that the graph of the given equation is a hyperbola. Find its foci, vertices, and asymptotes. 12. x xy +11y = y 2 52xy 7x x 144 5y +900 = 0

5 MA112 Section : Prepared by Dr.Archara Pacheenburawana 5 Exercise Find the slope of the tangent line to the parametric curve x = t/2, y = t at t = 1 and at t = 1 without eliminating the parameter. 2. Find the slope of the tangent line to the parametric curve x = 3cost, y = 4sint at t = π/4 and at t = 7π/4 without eliminating the parameter. 3 8 Find dy/dx and d 2 y/dx 2 at the given point without eliminating the parameter. 3. x = t, y = 2t+4; t = 1 4. x = 1 2 t2 +1, y = 1 3 t3 t; t = 2 5. x = sect, y = tant; t = π/3 6. x = sinht, y = cosht; t = 0 7. x = θ+cosθ, y = 1+sinθ; θ = π/6 8. x = cosφ, y = 3sinφ; φ = 5π/6 9. (a) Find the equation of the tangent line to the curve x = e t, y = e t at t = 1 without eliminating the parameter. (b) Find the equation of the tangent line in part (a) by eliminating the parameter. 10. (a) Find the equation of the tangent line to the curve x = 2t+4, y = 8t 2 2t+4 at t = 1 without eliminating the parameter. (b) Find the equation of the tangent line in part (a) by eliminating the parameter Find all values of t at which the parametric curve has (a) a horizontal tangent line and (b) a vertical tangent line. 11. x = 2sint, y = 4cost (0 t 2π) 12. x = 2t 3 15t 2 +24t+7, y = t 2 +t+1

6 MA112 Section : Prepared by Dr.Archara Pacheenburawana 6 Exercise Plot the points in polar coordinates. 1. (a) (3,π/4) (b) (5,2π/3) (c) (1,π/2) (d) (4,7π/6) (e) ( 6, π) (f) ( 1,9π/4) 2. (a) (2, π/3) (b) (3/2, 7π/4) (c) ( 3,3π/2) (d) ( 5, π/6) (e) (2,4π/3) (f) (0,π) 3 4 Find the rectangular coordinates of the points whose polar coordinates are given. 3. (a) (6,π/6) (b) (7,2π/3) (c) ( 6, 5π/6) (d) (0, π) (e) (7,17π/6) (f) ( 5,0) 4. (a) ( 2,π/4) (b) (6, π/4) (c) (4,9π/4) (d) (3,0) (e) ( 4, 3π/2) (f) (0,3π) 5. In each part, a point is given in rectangular coordinates. Find two pairs of polar coordinates for the point, one pair satisfying r 0 and 0 θ < 2π, and the second pair satisfying r 0 and 2π < θ 0. (a) ( 5,0) (b) (2 3, 2) (c) (0, 2) (d) ( 8, 8) (e) ( 3,3 3) (f) (1,1) 6. In each part, find polar coordinates satisfying the stated conditions for the point whose rectangular coordinates are ( 3,1). (a) r 0 and 0 θ < 2π (b) r 0 and 0 θ < 2π (c) r 0 and 2π < θ 0 (d) r 0 and π < θ π 7 8 Identify the curve by transforming the given polar equation to rectangular coordinates. 7. (a) r = 2 (b) rsinθ = 4 (c) r = 3cosθ 6 (d) r = 3cosθ +2sinθ 8. (a) r = 5secθ (b) r = 2sinθ (c) r = 4cosθ +4sinθ (d) r = secθtanθ

7 MA112 Section : Prepared by Dr.Archara Pacheenburawana Express the given equations in polar coordinates. 9. (a) x = 3 (b) x 2 +y 2 = 7 (c) x 2 +y 2 +6y = 0 (d) 9xy = (a) y = 3 (b) x 2 +y 2 = 5 (c) x 2 +y 2 +4x = 0 (d) x 2 (x 2 +y 2 ) = y Use the method of Example2.13 to sketch the curve in polar coordinates. 11. r = 2(1+sinθ) 12. r = 1 cosθ Sketch the curve in polar coordinates. 13. θ = π θ = 3π r = r = 4cosθ 17. r = 6sinθ 18. r = 1+sinθ 19. 2r = cosθ 20. r 2 = 2cosθ 21. r = 3(1+sinθ) 22. r = 5 5sinθ 23. r = 4 4cosθ 24. r = 1+2sinθ 25. r = 1 cosθ 26. r = 4+3cosθ 27. r = 2+cosθ 28. r = 3 sinθ 29. r = 3+4cosθ 30. r 5 = 3sinθ 31. r = 5 2cosθ 32. r = 3 4sinθ 33. r 2 = cos2θ 34. r 2 = 9sin2θ 35. r 2 = 16sin2θ 36. r = 4θ (θ 0) 37. r = 4θ (θ 0) 38. r = 4θ 39. r = 2cos2θ 40. r = 3sin2θ 41. r = 9sin4θ 42. r = 2cos3θ 43. Find the highest point on the cardioid r = 1+cosθ. 44. Find the leftmost point on the upper half of the cardioid r = 1+cosθ. Exercise Find the slope of the tangent line to the polar curve for the given value of θ.

8 MA112 Section : Prepared by Dr.Archara Pacheenburawana 8 1. r = 2sinθ; θ = π/6 2. r = 1+cosθ; θ = π/2 3. r = 1/θ; θ = 2 4. r = asec2θ; θ = π/6 5. r = sin3θ; θ = π/4 6. r = 4 3sinθ; θ = π 6 7 Find polar coordinates of all points at which the polar curve has a horizontal or a vertical tangent line. 6. r = a(1+cosθ) 7. r = asinθ 8 13 Use Formula (2.8) to calculate the arc length of the polar curve. 8. The entire circle r = a 9. The entire circle r = 2acosθ 10. The entire cardioid r = a(1 cos θ) 11. r = sin 2 (θ/2) from θ = 0 to θ = π 12. r = e 3θ from θ = 0 to θ = r = sin 3 (θ/3) from θ = 0 to θ = π/2 14. In each part, find the area of the circle by integration. (a) r = 2asinθ (b) r = 2acosθ Find the area of the region described. 15. The region that is enclosed by the cardioid r = 2+2sinθ. 16. The region in the first quadrant within the cardioid r = 1+cosθ. 17. The region enclosed by the rose r = 4cos3θ. 18. The region enclosed by the rose r = 2sin2θ. 19. The region inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ. 20. The region outside the cardioid r = 2 2 cos θ and inside the circle r = The region inside the cardioid r = cos θ and outside the circle r = The region inside the rose r = 2acos2θ and outside the circle r = a 2.

9 MA112 Section : Prepared by Dr.Archara Pacheenburawana 9 Exercise A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and give the coordinates of the corners. 2. Suppose that a box has its faces parallel to the coordinate planes and the points (4,2, 2) and ( 6,1,1) are endpoints of a diagonal. Sketch the box and give the coordinates of the remaining six corners. 3. Interpret the graph of x = 1 in the contexts of (a) a number line (b) 2-space (c) 3-space 4. Find the center and radius of the sphere that has (1, 2,4) and (3,4, 12) as endpoints of a diameter. 5. Show that (4,5,2), (1,7,3), and (2,4,5) are vertices of an equilateral triangle. 6. (a) Show that (2,1,6), (4,7,9), and (8,5, 6) are the vertices of a right triangle. (b) Which vertex is at the 90 angle? (c) Find the area of the triangle. 7. Find equations of two spheres that are centered at the origin and are tangent to the sphere of radius 1 centered at (3, 2,4) Describe the surface whose equation is given. 8. x 2 +y 2 +z 2 +10x+4y +2z 19 = 0 9. x 2 +y 2 +z 2 y = x 2 +2y 2 +2z 2 2x 3y +5z 2 = x 2 +y 2 +z 2 +2x 2y +2z +3 = x 2 +y 2 +z 2 3x+4y 8z +25 = x 2 +y 2 +z 2 2x 6y 8z +1 = In each part, sketch the portion of the surface that lies in the first octant. (a) y = x (b) y = z (c) x = z

10 MA112 Section : Prepared by Dr.Archara Pacheenburawana In each part, sketch the graph of the equation in 3-space. (a) x = 1 (b) y = 1 (c) z = In each part, sketch the graph of the equation in 3-space. (a) x 2 +y 2 = 25 (b) y 2 +z 2 = 25 (c) x 2 +z 2 = In each part, sketch the graph of the equation in 3-space. (a) x = y 2 (b) z = x 2 (c) y = z Sketch the surface in 3-space. 18. y = sinx 19. y = e x 20. z = 1 y z = cosx 22. 2x+z = x+3y = x 2 +9z 2 = z = 3 x 26. y 2 4z 2 = yz = If a bug walks on the sphere x 2 +y 2 +z 2 +2x 2y 4z 3 = 0 how close and how far can it get from the origin? 29. Describe the set of all points in 3-space whose coordinates satisfy the inequality x 2 +y 2 +z 2 2x+8z Describe the set of all points in 3-space whose coordinates satisfy the inequality y 2 +z 2 +6y 4z > The distance between a point P(x,y,z) and the point A(1, 2,0) is twice the distance between P and the point B(0,1,1). Show that the set of all such points is a sphere, and find the center and radius of the sphere.

11 MA112 Section : Prepared by Dr.Archara Pacheenburawana 11 Exercise Sketch the vectors with their initial points at the origin. 1. (a) 2,5 (b) 5, 4 (c) 2,0 (d) 5i + 3j (e) 3i 2j (f) 6j 2. (a) 3,7 (b) 6, 2 (c) 0, 8 (d) 4i + 2j (e) 2i j (f) 4i 3. (a) 1, 2,2 (b) 2,2, 1 (c) i+2j+3k (d) 2i+3j k 4. (a) 1,3,2 (b) 3,4,2 (c) 2j k (d) i j+2k 5 6 Find the components of the vector P 1 P (a) P 1 (3,5), P 2 (2,8) (b) P 1 (7, 2), P 2 (0,0) (c) P 1 (5, 2,1), P 2 (2,4,2) 6. (a) P 1 ( 6, 2), P 2 ( 4, 1) (b) P 1 (0,0,0), P 2 ( 1,6,1) (c) P 1 (4,1, 3), P 2 (9,1, 3) 7. (a) Find the terminal point of v = 3i 2j if the initial point is (1, 2). (b) Find the terminal point of v = 3,1,2 if the initial point is (5,0, 1). 8. (a) Find the terminal point of v = 7,6 if the initial point is (2, 1). (b) Find the terminal point of v = i+2j 3k if the initial point is ( 2,1,4) Perform the stated operations on the vectors u, v, and w. 9. u = 3i k, v = i j+2k, w = 3j (a) w v (b) 6u+4w (c) v 2w (d) 4(3u+v) (e) 8(v+w)+2u (f) 3w (v w)

12 MA112 Section : Prepared by Dr.Archara Pacheenburawana u = 2, 1,3, v = 4,0, 2, w = 1,1,3 (a) u w (b) 7v+3w (c) w+v (d) 3(u 7v) (e) 3v 8w (f) 2v (u+w) Find the norm of v. 11. (a) v = 1, 1 (b) v = i+7j (c) v = 1,2,4 (d) v = 3i+2j+k 12. (a) v = 3,4 (b) v = 2i 7j (c) v = 0, 3,0 (d) v = i+j+k 13. Let u = i 3j+2k, v = i+j, and w = 2i+2j 4k. Find (a) u+v (b) u + v (c) 2u +2 v 1 (d) 3u 5v+w (e) w w (f) 1 w w Find the unit vectors that satisfy the stated conditions. 14. (a) Same direction as i + 4j. (b) Oppositely directed to 6i 4j+2k. (c) Same direction as the vector from the point A( 1,0,2) to the point B(3,1,1). 15. (a) Oppositely directed to 3i 4j. (b) Same direction as 2i j 2k. (c) Same direction as the vector from the point A( 3,2) to the point B(1, 1) Find the vectors that satisfy the stated conditions. 16. (a) Oppositely directed to v = 3, 4 and half the length of v. (b) Length 17 and same direction as v = 7,0, (a) Same direction as v = 2i+3j and three times the length of v. (b) Length 2 and oppositely directed to v = 3i + 4j + k. 18. In each part, find the component form of the vector v in 2-space that has the stated length and makes the stated angle θ with the positive x-axis. (a) v = 3; θ = π/4 (b) v = 2; θ = 90 (c) v = 5; θ = 120 (d) v = 1; θ = π

13 MA112 Section : Prepared by Dr.Archara Pacheenburawana Find the component form of v + w and v w in 2-space, given that v = 1, w = 1, v makes an angle of π/6 with the positive x-axis, and w makes an angle of 3π/4 with the positive x-axis. 20. Let u = 1,3, v = 2,1, and w = 4, 1. Find the vector x that satisfies 2u v+x = 7x+w. 21. Let u = 1,1, v = 0,1, and w = 3,4. Find the vector x that satisfies u 2x = x w+3v. 22. Find u and v if u+2v = 3i k and 3u v = i+j+k. 23. Find u and v if u+v = 2, 3 and 3u+2v = 1, In each part, find two unit vectors in 2-space that satisfy the stated condition. (a) Parallel to the line y = 3x + 2 (b) Parallel to the line x+y = 4 (c) Perpendicular to the line y = 5x + 1 Exercise In each part, find the dot product of the vectors and the cosine of the angle between them. (a) u = i+2j, v = 6i 8j (b) u = 7, 3, v = 0,1 (c) u = i 3j+7k, v = 8i 2j 2k (d) u = 3,1,2, v = 4,2, 5 2. In each part use the given information to find u v. (a) u = 1, v = 2, the angle between u and v is π/6. (b) u = 2, v = 3, the angle between u and v is In each part, determine whether u and v make an acute angle, an obtuse angle, or are orthogonal. (a) u = 7i+3j+5k, v = 8i+4j+2k

14 MA112 Section : Prepared by Dr.Archara Pacheenburawana 14 (b) u = 6i+j+3k, v = 4i 6k (c) u = 1,1,1, v = 1,0,0 (d) u = 4,1,6, v = 3,0,2 4. Does the triangle in 3-space with vertices ( 1,2,3), (2, 2,0), and (3,1, 4) have an obtuse angle? Justify your answer. 5. The accompanying figure shows eight vectors that are equally spaced around a circle of radius 1. Find the dot product of v 0 with each of the other seven vectors. v 2 v 3 v 1 v 4 v 0 v 5 v 6 v 7 6. The accompanying figure shows six vectors that are equally spaced around a circle of radius 5. Find the dot product of v 0 with each of the other five vectors. v 2 v 1 v 3 v 0 v 4 v 5 7. (a) Use vectors to show that A(2, 1,1), B(3,2, 1), and C(7,0, 2) are vertices of the right triangle. At which vertex is the right angle? (b) Use vectors to find the interior angles of the triangle with vertices ( 1,0), (2, 1), and (1,4). 8. (a) Show that if v = ai+bj is a vector in 2-space, then the vectors v 1 = bi+aj and v 2 = bi aj are both orthogonal to v. (b) Use the result in part (a) to find two unit vectors that are orthogonal to the vector v = 3i 2j. Sketch the vectors v, v 1, and v 2.

15 MA112 Section : Prepared by Dr.Archara Pacheenburawana Explain why each of the following expressions makes no sense. (a) u (v w) (c) u v (b) (u v)+w (d) k (u+v) 10. True or false? If u v = u w and if u 0, then v = w. Justify your conclusion. 11. Verify part (b) and (c) of Theorem12.7 for the vectors u = 6i j+2k, v = 2i+7j+4k, w = i+j 3k and k = Let u = 1,2, v = 4, 2, and w = 6,0. Find (a) u (7v+w) (c) u (v w) (b) (u w)w (d) ( u v) w 13. Find r so that the vector from the point A(1, 1, 3) to the point B(3, 0, 5) is orthogonal to the vector from A to the point P(r,r,r). 14. Find two unit vectors in 2-space that make an angle of 45 with 4i+3j Find the direction cosines of v. 15. (a) v = i+j k (b) v = 2i 2j+k 16. (a) v = 3i 2j 6k (b) v = 3i 4k 17. In each part, find the vector component of v along b and the vector component of v orthogonal to b. (a) v = 2i j, b = 3i+4j (b) v = 4,5, b = 1, 2 (c) v = 3i 2j, v = 2i+j 18. In each part, find the vector component of v along b and the vector component of v orthogonal to b. (a) v = 2i j+3k, b = i+2j+2k (b) v = 4, 1,7, b = 2,3, 6 (c) v = 3i 2j, v = 2i+j Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b.

16 MA112 Section : Prepared by Dr.Archara Pacheenburawana (a) v = 2i 4j, b = i+j (b) v = 3i+j 2k, b = 2i k (c) v = 4i 2j+6k, b = 2i+j 3k 20. (a) v = 3,5, b = 1,1 (b) v = 2,1,6, b = 0, 2,1 (c) v = 1,4,1, b = 3, 2,5 21. Find the work done by a force F = 3j (pounds) applied to a point that moves on the line from (1,3) to (4,7), Assume that distance is measured in feet. 22. A force F = 4i 6j+k newtons is applied to a point that moves a distance of 15 meters in the direction of the vector i+j+k. How much work is done? 23. A boat travels 100 meters due north while the wind that applies a force of 500 newtons toward the northwest. How much work does the wind do? 24. A box is dragged along the floor by a rope that applies a force of 50 lb at an angle of 60 with the floor. How much work is done moving the box 15 ft?