SHOW ALL YOUR WORK ON A SEPARATE PAPER UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR THESE QUESTIONS

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Ada Knight
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1 AP Calculus BC CHAPTER 11 WORKSHEET PARAMETRIC EQUATIONS AND POLAR COORDINATES Name Seat # Date Curves Defined by Parametric Equations SHOW ALL YOUR WORK ON A SEPARATE PAPER UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR THESE QUESTIONS 1 Consider the parametric equations t and y 1 t a) Complete the table below and use your answers to make a sketch of the curve Make sure to indicate the orientation of the graph t y Use your calculator to verify your graph from part (a) is correct c) Eliminate the parameter to find the rectangular equation for the curve Consider the parametric equations 4cos θ and y sin θ a) Complete the table below and use your answers to make a sketch of the curve Make sure to indicate the orientation of the graph θ π π 4 0 π 4 π y Use your calculator to verify your graph from part (a) is correct c) Eliminate the parameter to find the rectangular equation for the curve In questions through, use your calculator to sketch the curve represented by the parametric equations (include the orientation of the graph) Then eliminate the parameter and write the corresponding Cartesian equation t t, y, for t 4 cosθ, y sin θ, for 0 θ π 4 cos θ, y 7sin θ, for 0 θ π SEE OTHER SIDE

2 6 Consider the two sets of parametric equations shown below I t, y t 1 II cos θ, y cosθ 1 for 6 t 6 for 0 θ π a) Determine whether each curve is smooth Show all your work Use your calculator to sketch each curve Make sure to indicate the orientation of the graphs c) Eliminate the parameter to find the rectangular equation for each curve What do you notice? d) How can two different sets of parametric equations have the same rectangular equation? Eplain True or False? Eplain 7 The graph of the parametric equations t and y t is the line y 8 The two sets of parametric equations t, y t 1 and t, y 9t 1 correspond to the same rectangular equation 9 If y is a function of t and is a function of t, then y is a function of 10 If f t 1 0 and g t 0, then the curve represented by the parametric equations y f t and gt passes through the origin In questions 11 and 1, find two different sets of parametric equations for each rectangular equation 11 y 1 y

3 AP Calculus BC CHAPTER 11 WORKSHEET PARAMETRIC EQUATIONS AND POLAR COORDINATES Curves Defined by Parametric Equations ANSWER KEY 1 t and y 1 t a) t y c) y 1 4cos θ and y sin θ a) θ π π 4 0 π 4 π y 0 c) y or 4 y t, t y y y 4 cos θ, y sin θ 1 or y y 4cos θ, y 7sin θ

4 6 I t, y t 1 II cos θ, y cosθ 1 a) Smooth? d 1 dt I dy dt d dy and do not equal 0 simultaneously, dt dt so the curve is smooth d sin θ dy II sin θ d dy and equal 0 when θ π, 0, π dtθ So the curve is not smooth at any of those values of θ c) Eliminate the parameter to find the rectangular equation for each curve What do you notice? I y 1 II y 1 Both sets of parametric equations have the same rectangular equation d) How can two different sets of parametric equations have the same rectangular equation? Eplain The parametric equations resulting in the same rectangular equation means that both sets of equations describe the same trajectory or path for the motion of an object However, the object traces this path in two very different manners depending on the parametric equation There are infinitely many parametric equations for y 1! 7 The graph of the parametric equations t and y t is the line y FALSE Even though the corresponding Cartesian equation is y, the parametric equations yield only positive values for both and y Therefore the graph is the ray with equation y, such that 0 8 The two sets of parametric equations t, y t 1 and t, y 9t 1 correspond to the same rectangular equation TRUE The rectangular equation is y 1 9 If y is a function of t and is a function of t, then y is a function of FALSE Consider this eample: t, y t Both and y are functions of t But the corresponding Cartesian equation, y is not a function 10 If f t 1 0 and g t 0, then the curve represented by the parametric equations y f t and gt passes through the origin FALSE f t 1 0 gt 0 t The y means that the y coordinate of the graph of the curve is equal to 0 for t 1 means that the coordinate of the graph of the curve is equal to 0 for graph of a set of parametric equations will pass through the origin if both and y are equal to 0 simultaneously, for the same value of the parameter This might not be the case Eample: t, y t 1

5 11 y t, y t or t, y t or cos θ, y cosθ 1 y t, y t or t, y 8t or sinθ, y sin θ

GRAPHING IN POLAR COORDINATES SYMMETRY

GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,