ࡱ> SUPQRjb jbjbj11 \SSBBBBBVVVV,TVi6666KKK=h?h?h?h?h?h?h$ jlbchiBK_GZKKKchBB66h^^^KB6B6=h^K=h^^^6PnnuM ^)hh0i^mWm^mB^ KK^KKKKKchchGZNKKKiKKKKmKKKKKKKKK : CALCULUS BC WORKSHEET ON PARAMETRIC EQUATIONS AND GRAPHING Work these on notebook paper. Make a table of values and sketch the curve, indicating the direction of your graph. Then eliminate the parameter. Do not use your calculator. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  Answers to Worksheet on Parametric Equations and Graphing 1.  EMBED Equation.DSMT4   t EMBED Equation.DSMT4  EMBED Equation.DSMT4 012x EMBED Equation.DSMT4 3 EMBED Equation.DSMT4 135y EMBED Equation.DSMT4 3  EMBED Equation.DSMT4 2 EMBED Equation.DSMT4 01 To eliminate the parameter, solve for EMBED Equation.DSMT4 . Substitute into ys equation to get EMBED Equation.DSMT4 . ______________________________________________________________________________________ 2.  EMBED Equation.DSMT4  t EMBED Equation.DSMT4 012x EMBED Equation.DSMT4 024y1014 To eliminate the parameter, solve for EMBED Equation.DSMT4 . Substitute into ys equation to get  EMBED Equation.DSMT4 . Note: The restriction on x is needed for the graph of  EMBED Equation.DSMT4  to match the parametric graph. _________________________________________________________________________________________ 3.  EMBED Equation.DSMT4  t EMBED Equation.DSMT4  EMBED Equation.DSMT4 012x EMBED Equation.DSMT4 121 EMBED Equation.DSMT4 y EMBED Equation.DSMT4  EMBED Equation.DSMT4 012 To eliminate the parameter, notice that t = y. Substitute into xs equation to get  EMBED Equation.DSMT4 . 4.  EMBED Equation.DSMT4  t EMBED Equation.DSMT4  EMBED Equation.DSMT4 27x0123y541 EMBED Equation.DSMT4  To eliminate the parameter, solve for  EMBED Equation.DSMT4 . Substitute into ys equation to get  EMBED Equation.DSMT4 . Note: The restriction on x is needed for the graph of  EMBED Equation.DSMT4  to match the parametric graph. _________________________________________________________________________________________ 5.  EMBED Equation.DSMT4  t0149x EMBED Equation.DSMT4  EMBED Equation.DSMT4 27y10 EMBED Equation.DSMT4  EMBED Equation.DSMT4  To eliminate the parameter, solve for EMBED Equation.DSMT4  (since  EMBED Equation.DSMT4 ). Substitute into ys equation to get  EMBED Equation.DSMT4 . ________________________________________________________________________________ _________ 6.  EMBED Equation.DSMT4  t EMBED Equation.DSMT4  EMBED Equation.DSMT4 0123x EMBED Equation.DSMT4  EMBED Equation.DSMT4 0246y321012 To eliminate the parameter, solve for  EMBED Equation.DSMT4 . Substitute into ys equation to get  EMBED Equation.DSMT4 . ________________________________________________________________________________________ 7.  EMBED Equation.DSMT4  t EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 1/201/212x EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 1/201/212y1/414und.411/4 To eliminate the parameter, notice that t = x. Substitute into y s equation to get EMBED Equation.DSMT4 . 8.  EMBED Equation.DSMT4  t0/23/22x1 EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 1y141 EMBED Equation.DSMT4 1 To eliminate the parameter, solve for  EMBED Equation.DSMT4  in xs equation and  EMBED Equation.DSMT4  in ys equation. Substitute into the trigonometric identity  EMBED Equation.DSMT4  to get  EMBED Equation.DSMT4 . _________________________________________________________________________________________ 9.  EMBED Equation.DSMT4  t0/23/22x EMBED Equation.DSMT4 1 EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 y32123 To eliminate the parameter, solve for  EMBED Equation.DSMT4  in y s equation and  EMBED Equation.DSMT4  in xs equation. Substitute into the trigonometric identity  EMBED Equation.DSMT4  to get  EMBED Equation.DSMT4 . _________________________________________________________________________________________ 10.  EMBED Equation.DSMT4  t0/4/23/45/43/27/42x1 EMBED Equation.DSMT4 und. EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 und. EMBED Equation.DSMT4 1y01und. EMBED Equation.DSMT4 01und. EMBED Equation.DSMT4 0 To eliminate the parameter, substitute into the trigonometric identity  EMBED Equation.DSMT4  to get  EMBED Equation.DSMT4 . CALCULUS BC WORKSHEET ON PARAMETRICS AND CALCULUS Work these on notebook paper. Do not use your calculator. On problems 1 5, find EMBED Equation.DSMT4 . 1.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  _________________________________________________________________________________________ 6. A curve C is defined by the parametric equations EMBED Equation.DSMT4 . (a) Find  EMBED Equation.DSMT4  in terms of t. (b) Find an equation of the tangent line to C at the point where t = 2. 7. A curve C is defined by the parametric equations EMBED Equation.DSMT4 . (a) Find  EMBED Equation.DSMT4  in terms of t. (b) Find an equation of the tangent line to C at the point where t = EMBED Equation.DSMT4 . __________________________________________________________________________________________ On problems 8 10, find: (a)  EMBED Equation.DSMT4 in terms of t. (b) all points of horizontal and vertical tangency 8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  __________________________________________________________________________________________ On problems 11 - 12, a curve C is defined by the parametric equations given. For each problem, write an integral expression that represents the length of the arc of the curve over the given interval. 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  Answers to Worksheet on Parametrics and Calculus  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  6. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  7. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  8. (a)  EMBED Equation.DSMT4  (b) Horiz. tangent at  EMBED Equation.DSMT4 . No point of Vert. tangency on this curve. 9. (a)  EMBED Equation.DSMT4  (b) Horiz. tangent at the points  EMBED Equation.DSMT4 . Vert. tangent at  EMBED Equation.DSMT4 . 10. (a)  EMBED Equation.DSMT4  (b) Vert. tangent at  EMBED Equation.DSMT4 . Horiz. tangent at  EMBED Equation.DSMT4 . 11.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  CALCULUS BC WORKSHEET 1 ON VECTORS Work the following on notebook paper. Use your calculator on problems 10 and 13c only. 1. If  EMBED Equation.DSMT4  2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is  EMBED Equation.DSMT4 , find its velocity vector at time t = 2. 3. A particle moves in the xy-plane so that at any time t, its coordinates are given by EMBED Equation.DSMT4  Find its acceleration vector at t = 1. 4. If a particle moves in the xy-plane so that at time t its position vector is EMBED Equation.DSMT4  find the velocity vector at time  EMBED Equation.DSMT4  5. A particle moves on the curve  EMBED Equation.DSMT4  so that its x-component has derivative  EMBED Equation.DSMT4  At time t = 0, the particle is at the point (1, 0). Find the position of the particle at time t = 1. 6. A particle moves in the xy-plane in such a way that its velocity vector is EMBED Equation.DSMT4  If the position vector at t = 0 is EMBED Equation.DSMT4 , find the position of the particle at t = 2. 7. A particle moves along the curve  EMBED Equation.DSMT4 ? 8. The position of a particle moving in the xy-plane is given by the parametric equations  EMBED Equation.DSMT4  For what value(s) of t is the particle at rest? 9. A curve C is defined by the parametric equations  EMBED Equation.DSMT4  Write the equation of the line tangent to the graph of C at the point  EMBED Equation.DSMT4  10. A particle moves in the xy-plane so that the position of the particle is given by  EMBED Equation.DSMT4   EMBED Equation.DSMT4 Find the velocity vector at the time when the particles horizontal position is x = 25. 11. The position of a particle at any time  EMBED Equation.DSMT4  is given by  EMBED Equation.DSMT4  (a) Find the magnitude of the velocity vector at time t = 5. (b) Find the total distance traveled by the particle from t = 0 to t = 5. (c) Find  EMBED Equation.DSMT4  as a function of x. 12. Point  EMBED Equation.DSMT4  moves in the xy-plane in such a way that  EMBED Equation.DSMT4  (a) Find the coordinates of P in terms of t given that t = 1,  EMBED Equation.DSMT4 , and y = 0. (b) Write an equation expressing y in terms of x. (c) Find the average rate of change of y with respect to x as t varies from 0 to 4. (d) Find the instantaneous rate of change of y with respect to x when t = 1. 13. Consider the curve C given by the parametric equations  EMBED Equation.DSMT4  (a) Find  EMBED Equation.DSMT4  as a function of t. (b) Find the equation of the tangent line at the point where  EMBED Equation.DSMT4  (c) The curve C intersects the y-axis twice. Approximate the length of the curve between the two y- intercepts. Answers to Worksheet 1 on Vectors 1. EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4. EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7.  EMBED Equation.DSMT4  8. t = 3 9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11. (a) EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c) EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  (d) 4 13. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  CALCULUS BC WORKSHEET 2 ON VECTORS Work the following on notebook paper. Use your calculator on problems 7 12 only. 1. If  EMBED Equation.DSMT4  in terms of t. 2. Write an integral expression to represent the length of the path described by the parametric equations  EMBED Equation.DSMT4  3. For what value(s) of t does the curve given by the parametric equations  EMBED Equation.DSMT4  have a vertical tangent? 4. For any time EMBED Equation.DSMT4 , if the position of a particle in the xy-plane is given by  EMBED Equation.DSMT4  find the acceleration vector. 5. Find the equation of the tangent line to the curve given by the parametric equations  EMBED Equation.DSMT4  at the point on the curve where t = 1. 6. If  EMBED Equation.DSMT4  are the equations of the path of a particle moving in the xy-plane, write an equation for the path of the particle in terms of x and y. 7. A particle moves in the xy-plane so that its position at any time t is given by  EMBED Equation.DSMT4  What is the speed of the particle when t = 2? 8. The position of a particle at time  EMBED Equation.DSMT4  is given by the parametric equations  EMBED Equation.DSMT4 . (a) Find the magnitude of the velocity vector at t = 1. (b) Find the total distance traveled by the particle from t = 0 to t = 1. (c) When is the particle at rest? What is its position at that time? 9. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time with  EMBED Equation.DSMT4 . Find the acceleration vector and the speed of the object when t = 5. 10. A particle moves in the xy-plane so that the position of the particle is given by  EMBED Equation.DSMT4   EMBED Equation.DSMT4  EMBED Equation.DSMT4  Find the velocity vector when the particles vertical position is y = 5. 11. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  At time t = 1, the object is at the position (3, 4). (a) Write an equation for the line tangent to the curve at (3, 4). (b) Find the speed of the object at time t = 2. (c) Find the total distance traveled by the object over the time interval  EMBED Equation.DSMT4  (d) Find the position of the object at time t = 2. 12. A particle moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  At time t = 1, the particle is at the position (5, 6). (a) Find the speed of the object at time t = 2. (b) Find the total distance traveled by the object over the time interval  EMBED Equation.DSMT4  (c) Find  EMBED Equation.DSMT4 . (d) For  EMBED Equation.DSMT4 , there is a point on the curve where the line tangent to the curve has slope 8. At what time t,  EMBED Equation.DSMT4 , is the particle at this point? Find the acceleration vector at this point. Answers to Worksheet 2 on Vectors 1. EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. EMBED Equation.DSMT4  7. 12.304 8. (a)  EMBED Equation.DSMT4  (b) 3.816 (c) At rest when t = 2. Position = (4, 0) 9. EMBED Equation.DSMT4 , speed = 28.083  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (b) 2.084 (c) 1.126 (d) (3.436, 3.557) 12. (a) 1.975 (b) 1.683 (c) 7.661 (d)  EMBED Equation.DSMT4  CALCULUS BC WORKSHEET 3 ON VECTORS Work the following on notebook paper. Use your calculator only on problems 3 7. 1. The position of a particle at any time t EMBED Equation.DSMT4 0 is given by  EMBED Equation.DSMT4  (a) Find the magnitude of the velocity vector at t = 2. (b) Set up an integral expression to find the total distance traveled by the particle from t = 0 to t = 4. (c) Find  EMBED Equation.DSMT4  as a function of x. (d) At what time t is the particle on the y-axis? Find the acceleration vector at this time. 2. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with the velocity vector  EMBED Equation.DSMT4  At time t = 1, the object is at (ln 2, 4). (a) Find the position vector. (b) Write an equation for the line tangent to the curve when t = 1. (c) Find the magnitude of the velocity vector when t = 1. (d) At what time t > 0 does the line tangent to the particle at  EMBED Equation.DSMT4  have a slope of 12? 3. A particle moving along a curve in the xy-plane has position EMBED Equation.DSMT4 , with EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (a) Is the particle moving to the left or to the right when t = 2.4? Explain your answer. (b) Find the velocity vector at the time when the particles vertical position is y = 7. 4. A particle moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with EMBED Equation.DSMT4 . The derivative  EMBED Equation.DSMT4  is not explicitly given. At time t = 2, the object is at position  EMBED Equation.DSMT4 . (a) Find the x-coordinate of the position at time t = 3. (b) For any t EMBED Equation.DSMT4 0, the line tangent to the curve at  EMBED Equation.DSMT4  has a slope of t + 3. Find the acceleration vector of the object at time t = 2. 5. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  At time t = 3, the object is at the point (1, 4). (a) Find the equation of the tangent line to the curve at the point where t = 3. (b) Find the speed of the object at t = 3. (c) Find the total distance traveled by the object over the time interval  EMBED Equation.DSMT4  (d) Find the position of the object at time t = 2. TURN->>> 6. A particle moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  At time t = 2, the particle is at the point (5, 3). (a) Find the acceleration vector for the particle at t = 2. (b) Find the equation of the tangent line to the curve at the point where t = 2. (c) Find the magnitude of the velocity vector at t = 2. (d) Find the position of the particle at time t = 1. 7. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  The derivative  EMBED Equation.DSMT4  is not explicitly given. At t = 3, the object is at the point (4, 5). (a) Find the y-coordinate of the position at time t = 1. (b) At time t = 3, the value of  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4  Find the value of  EMBED Equation.DSMT4  when t = 3. (c) Find the speed of the object at time t = 3. Answers to Worksheet 3 on Vectors 1. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  2. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c)  EMBED Equation.DSMT4  (d)  EMBED Equation.DSMT4  3. (a) The particle is moving to the left because dx/dt <0. (b) EMBED Equation.DSMT4  4. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  5. (a)  EMBED Equation.DSMT4  (b) 0.555 (c) 0.878 (d) (0.529, 4.031) 6. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  (c) 3.750 (d) (2.239, 1.664) 7. (a) 1.269 (b)  EMBED Equation.DSMT4  (c) 3.368 POLAR GRAPHS Put your graphing calculator in POLAR mode and RADIAN mode. Graph the following equations on your calculator, sketch the graphs on this sheet, and answer the questions. 1.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   What do you notice about these graphs? _______________________________________________________________________________ 2.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   Which graphs go through the origin? Which ones do not go through the origin? Which ones have an inner loop? 3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   What do you notice about these graphs? __________________________________________________________________________________ 4.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   What do you notice about these graphs? ________________________________________________________________________________ 5.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   What do you notice about these graphs? CALCULUS BC WORKSHEET 1 ON POLAR Work the following on notebook paper. Do NOT use your calculator. Convert the following equations to polar form. 1. y = 4 2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  __________________________________________________________________________________________ Convert the following equations to rectangular form. 4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  __________________________________________________________________________________________ For the following, find  EMBED Equation.DSMT4  for the given value of  EMBED Equation.DSMT4 . 7.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  __________________________________________________________________________________________ 11. Find the points of horizontal and vertical tangency for  EMBED Equation.DSMT4  Give your answers in polar form,  EMBED Equation.DSMT4  __________________________________________________________________________________________ Make a table, tell what type of graph (circle, cardioid, limacon, lemniscate, rose), and sketch the graph. 12.  EMBED Equation.DSMT4  15.  EMBED Equation.DSMT4  18.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  19.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  17.  EMBED Equation.DSMT4  Answers 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. r = 5 4. x = 3 5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. 0 8. 1 9.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  11. Horiz:  EMBED Equation.DSMT4  Vert.:  EMBED Equation.DSMT4  12. circle centered on the x-axis with diameter 3 13. circle centered on the y-axis with diameter 2 14. cardioid with y-axis symmetry 15. limacon without a loop with x-axis symmetry 16. lemniscate 17. limacon with a loop with y-axis symmetry 18. rose with 4 petals 19. rose with 3 petals CALCULUS BC WORKSHEET 2 ON POLAR Work the following on notebook paper. On problems 1 5, sketch a graph, shade the region, set up the integrals needed, and then find the area. Do not use your calculator. 1. Area of one petal of  EMBED Equation.DSMT4  4. Area of the interior of  EMBED Equation.DSMT4  2. Area of one petal of  EMBED Equation.DSMT4  5. Area of the interior of  EMBED Equation.DSMT4  3. Area of the interior of  EMBED Equation.DSMT4  __________________________________________________________________________________________ On problems 6 7, sketch a graph, shade the region, set up the integrals needed, and then use your calculator to evaluate. 6. Area of the inner loop of  EMBED Equation.DSMT4  7. Area between the loops of  EMBED Equation.DSMT4  Answers to Worksheet 2 on Polar 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. Top half:  EMBED Equation.DSMT4  Between the loops:  EMBED Equation.DSMT4  OR  EMBED Equation.DSMT4  CALCULUS BC WORKSHEET 3 ON POLAR Work the following on notebook paper. On problems 1 2, sketch a graph, shade the region, set up the integrals needed, and then find the area. Do not use your calculator. 1. Area inside  EMBED Equation.DSMT4  and outside  EMBED Equation.DSMT4  2. Area of the common interior of  EMBED Equation.DSMT4  __________________________________________________________________________________________ On problems 3 5, sketch a graph, shade the region, set up the integrals needed, and then use your calculator to evaluate. 3. Area inside  EMBED Equation.DSMT4  and outside  EMBED Equation.DSMT4  4. Area of the common interior of  EMBED Equation.DSMT4  5. Area of the common interior of  EMBED Equation.DSMT4  __________________________________________________________________________________________ Do not use your calculator on problem 6. 6. Given  EMBED Equation.DSMT4  find EMBED Equation.DSMT4 . __________________________________________________________________________________________ Use your calculator on problem 7. 7. A particle moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4 . The derivative  EMBED Equation.DSMT4  is not explicitly given. At time t = 3, the object is at position  EMBED Equation.DSMT4 . (a) Find the y-coordinate of the position at time t = 1. (b) For t = 3, the line tangent to the curve at  EMBED Equation.DSMT4  has a slope of 1.8. Find the value of  EMBED Equation.DSMT4  when t = 3. (c) Find the speed of the particle when t = 3. Answers to Worksheet 3 on Polar 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  Total area =  EMBED Equation.DSMT4  or 9.827  EMBED Equation.DSMT4  7. (a) 0.269 (b)  EMBED Equation.DSMT4  1.636 (c) 3.368 CALCULUS BC WORKSHEET 4 ON POLAR Work the following on notebook paper. Do not use your calculator on problems 1, 2, and 5. 1. Sketch a graph, shade the region, and find the area inside  EMBED Equation.DSMT4  and outside  EMBED Equation.DSMT4 . __________________________________________________________________________________________ 2. Given  EMBED Equation.DSMT4  __________________________________________________________________________________________ You may use your calculator on problems 3 and 4. 3. The figure shows the graphs of the line  EMBED Equation.DSMT4  and the curve C given by  EMBED Equation.DSMT4 . Let S be the region P in the first quadrant bounded by the two graphs and the S x-axis. The line and the curve intersect at point P. (a) Find a polar equation to represent curve C. (b) Find the polar coordinates of point P. (c) Find the value of  EMBED Equation.DSMT4  at point P. What does your answer tell you about r? What does it tell you about the curve? (a) Use the polar equation found in (c) to set up and evaluate an integral expression with respect to the polar angle  EMBED Equation.DSMT4  that gives the area of S. __________________________________________________________________________________________ 4. A curve is drawn in the xy-plane and is described by the equation in polar coordinates  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4 , where r is measured in meters and  EMBED Equation.DSMT4  is measured in radians. (a) Find the area bounded by the curve and the y-axis. (b) Find the angle  EMBED Equation.DSMT4  that corresponds to the point on the curve with y-coordinate  EMBED Equation.DSMT4 . (c) For what values of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , is  EMBED Equation.DSMT4  positive? What does this say about r? What does it say about the curve? (d) Find the value of  EMBED Equation.DSMT4  on the interval  EMBED Equation.DSMT4  that corresponds to the point on the curve with the greatest distance from the origin. What is the greatest distance? Justify your answer. (e) A particle is traveling along the polar curve given by  EMBED Equation.DSMT4 so that its position at time t is  EMBED Equation.DSMT4  and such that  EMBED Equation.DSMT4  Find the value of  EMBED Equation.DSMT4  at the instant that  EMBED Equation.DSMT4 and interpret the meaning of your answer in the context of the problem. TURN->>> Do not use your calculator on problem 5. 5. The graph of the polar curve  EMBED Equation.DSMT4  is shown on the right. Let S be the shaded region in the fourth quadrant bounded by the curve and the x-axis. (a) Write an expression for  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  (b) A particle is traveling along the polar curve given by  EMBED Equation.DSMT4 so that its position at time t is  EMBED Equation.DSMT4  and such that  EMBED Equation.DSMT4  Find the value of  EMBED Equation.DSMT4  at the instant that  EMBED Equation.DSMT4 and interpret the meaning of your answer in the context of the problem. __________________________________________________________________________________________ Use your calculator on problem 6. 6. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  At time t = 1, the object is at the point (3, 4). (a) Find the equation of the tangent line to the curve at the point where t = 1. (b) Find the speed of the object at t = 2. (c) Find the total distance traveled by the object over the time interval  EMBED Equation.DSMT4  (d) Find the position of the object at time t = 2. __________________________________________________________________________________________ Answers to Worksheet 4 on Polar 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. (a)  EMBED Equation.DSMT4  (b) (1.442, 0.588) (c)  EMBED Equation.DSMT4  so r is decreasing, and the curve is moving closer to the origin. (d) 0.927 4. (a) 19.675 (b) 3.485 (c)  EMBED Equation.DSMT4 . This means that the r is getting larger, and the curve is getting farther from the origin.  (d)  EMBED Equation.DSMT4  r  EMBED Equation.DSMT4  4.302 5.245  EMBED Equation.DSMT4  4.712 The greatest distance is 5.245 when  EMBED Equation.DSMT4  = 4.302. (e)  EMBED Equation.DSMT4 . This means that the y-coordinate is decreasing at a rate of 10.348. 5. (a)  EMBED Equation.DSMT4  (b)  EMBED Equation.DSMT4  the y-coordinate is increasing at a rate of 2. 6. (a)  EMBED Equation.DSMT4  (b) 2.084 (c) 1.126 (d)  EMBED Equation.DSMT4  AP CALCULUS BC REVIEW SHEET FOR TEST ON PARAMETRICS, VECTORS, POLAR, & AP REVIEW Use your calculator on problems 2 3 and 9. Show supporting work, and give decimal answers correct to three decimal places. 1. Find EMBED Equation.DSMT4  given  EMBED Equation.DSMT4 . __________________________________________________________________________________________ 2. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  At time t = 2, the object is at the position (7, 4). (a) Write an equation for the line tangent to the curve at the point where t = 2. (b) Find the speed of the object at time t = 2. (c) Find the total distance traveled by the object over the time interval  EMBED Equation.DSMT4  (d) For what value of t,  EMBED Equation.DSMT4  does the tangent line to the curve have a slope of 4? Find the acceleration vector at this time. (e) Find the position of the object at time t = 1. __________________________________________________________________________________________ 3. An object moving along a curve in the xy-plane has position  EMBED Equation.DSMT4  at time t with  EMBED Equation.DSMT4  The derivative  EMBED Equation.DSMT4  is not explicitly given. At t = 2, the object is at the point  EMBED Equation.DSMT4  (a) Find the x-coordinate of the position at time t = 3. (b) For any  EMBED Equation.DSMT4  the line tangent to the curve at  EMBED Equation.DSMT4 has a slope of  EMBED Equation.DSMT4  Find the acceleration vector of the object at time t = 2. __________________________________________________________________________________________ No calculator. 4. Find  EMBED Equation.DSMT4  for the given value of  EMBED Equation.DSMT4  given  EMBED Equation.DSMT4 . __________________________________________________________________________________________ No calculator. 5. Find the area of the interior of  EMBED Equation.DSMT4 . 6. Find the area of one petal of  EMBED Equation.DSMT4  7. Set up the integral(s) needed to find the area inside  EMBED Equation.DSMT4  and outside  EMBED Equation.DSMT4 . Do not evaluate. 8. Set up the integral(s) needed to find the area of the common interior of  EMBED Equation.DSMT4 . Do not evaluate. __________________________________________________________________________________________ Use your calculator. 9. 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Equation.DSMT4 , where r is measured in meters and  EMBED Equation.DSMT4  is measured in radians. (a) Find the area bounded by the curve and the y-axis. (b) Find the angle  EMBED Equation.DSMT4  that corresponds to the point on the curve with y-coordinate  EMBED Equation.DSMT4 . Answers  EMBED Equation.DSMT4  2. (a)  EMBED Equation.DSMT4  (b) 1.186 (c) 0.976 (d) t = 0.6164,  EMBED Equation.DSMT4  (e)  EMBED Equation.DSMT4  3. (a) 3.996 (b)  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. Top half doubled:  EMBED Equation.DSMT4  8. Right side doubled:  EMBED Equation.DSMT4  9. 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@ A B C F I J K L M N O P Q R U X Y Z [ \ ] ^ _ ` a d g h i j k l m n o p q r s t w z { | }   dydx and   d 2 ydx 2 FMathType 6.0 Equation MathType EFEquation.DSMT49q.d-PRD-DSMT6WinAllBasicCodePagesObjInfo Equation Native  J_1390636184FJuJuOle  Times New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   dydt==2++sine t () FMathType 6.0 Equation MathType EFEquation.DSMT49qCompObj iObjInfo Equation Native  _1390636253 FJuJu, t-PRD-DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   `dxdt FMathType 6.0 Equation MathTyOle  CompObj iObjInfo Equation Native  pe EFEquation.DSMT49qd-PRD-DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   5,4()_1391834667`FJuJuOle  CompObj iObjInfo"  FMathType 6.0 Equation MathType EFEquation.DSMT49qL-PRD-DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!ED/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A   dxdtEquation Native # _1391185959FJuJuOle ' CompObj( i FMathType 6.0 Equation MathType EFEquation.DSMT49qsIOT DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/GD/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Area==13cosq() 2 ObjInfo* Equation Native + e_1391185987FJuJuOle 5 dq"-12"-cosq() 2 dq== 0 p3 +" ...== 0 p3 +" 3 3  FMathType 6.0 Equation MathType EFEquation.DSMT49qCompObj6 iObjInfo8 Equation Native 9 _1391281310FJuJusvOTlDSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/GD/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Area==14sinq() 2 dq++12() 2 dq==  p6 p2 +" ...== 0 p6 +"  8p3"-2 3  FMathType 6.0 Equation MathType EFEquation.DSMT49qOle D CompObjE iObjInfoG Equation Native H tLOT DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/GD/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Area== 1213sinq() 2 dq"- 1211++sinq() 2 dq==  p6 5p6 +" p  p6 5p6 +"  FMathType 6.0 Equation MathType EFEquation.DSMT49qsoOT DSMT6WinAllBasicCodePages_1391186168FJuJuOle S CompObjT iObjInfoV Equation Native W _1391186305FJuJuOle b CompObjc iTimes New RomanSymbolCourier NewMT Extra!/GD/APG_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Area==11++cosq() 2 dq++13cosq() 2 dq==  p3 p2 +"  5p4 0 p3 +"  or 3.927 FMathType 6.0 Equation MathType EFEquation.DSMT49qsfOT DSMT6WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/GD/APObjInfoe Equation Native f _1391186507FJuJuOle u G_APAPAE%B_AC_AE*_HA@AHA*_D_E_E_A  Area in Quad. 1== 1214sin2q()() 2 dq++ 1212() 2 dq++ 1214sin2q()() 2 dq  5p12 p2 +"   p12 5p12 +" 0 p12 +"  FMathType 6.0 Equation MathType EFEquation.DSMT49qfkV?8|w~WJY!n 쭲,;-% q]L·R|238IA1>;D:ʝ˘&8Oqܣ3Gj*};$1lHp[@#N\Jּy;mSn>\dU*}i[4"5՝c"ҪZY뾦zǵ/:4,Űk0gr>oT/]/Έ8SAiYWD전Pؾg˗˖ׇS>J3!|} f=FG}aJ` ϸNDd <b  c $A? 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