Polar (BC Only) They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative
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1 Polar (BC Only) Polar coordinates are another way of expressing points in a plane. Instead of being centered at an origin and moving horizontally or vertically, polar coordinates are centered at the pole and measure a radius out from the pole at a given angle. The beginning angle of zero radians corresponds to the positive x-axis. Although polar functions are differentiated in r and, the coordinates and slope of a line tangent to a polar curve are given in rectangular coordinates. Tangent lines Remember the following conversions: x rcos y rsin They are necessary to find the derivative of a polar curve in x- and y-coordinates. The derivative dy dy is done parametrically so that d. Remember, when differentiating in x or y, use the dx dx d product rule formula. Find the slope of the rose curve r sin at the point where and use it to write an equation for the line tangent to the graph. Solution First, find the rectangular coordinates of the point 9 x rcos x sin cos 4 y rsin y sin sin 4 Next, find the derivative and evaluate it at the given angle dy dy d sincos cossin dx dx sin sin cos cos d 5 sincos cossin 4 5 sinsin coscos Tangent line is y x 4 4. Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
2 Area enclosed by polar curves Polar (BC only) The area enclosed by a polar curve is calculated by using the formula, Remember that d refers to a thin pie-shaped sector.. A r d Find the area enclosed by the cardioid r cos from. Solution A cos d Area between two polar curves The area enclosed between two polar curves is given by A router rinner d. Notice that the integrand is the difference of the squared radii, not the square of the difference of the radii. (Like the washer method) Find the area of the region that lies inside the circle r and outside the cardioid r cos. Solution The outer curve is the circle r and the inner curve is the cardioid r cos. The points of intersection are and. A cos d.5. Alternate approach: If the region has symmetry, determine the area of one segment and then multiply it by the number of symmetric segments. In this case, integrate from to and double the result. A () ( cos ) d.5 Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
3 Polar (BC only) Multiple Choice. (calculator not allowed) Which of the following is equal to the area of the region inside the polar curve r cos and outside the polar curve r cos? (A) (B) (C) cos d cos d cos (D) cos d (E) cos d d. (calculator not allowed) The area of the region inside the polar curve given by r 4sin and outside the polar curve r is (A) 4sin 4 (B) 4sin 4 5 d d (C) 4sin d 5 (D) sin 4 d (E) sin 4 d Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
4 . (calculator not allowed) Polar (BC only) Which of the following expressions gives the total area enclosed by the polar curve r sin shown in the figure above? (A) (B) (C) (D) sin sin d sin 4 sin 4 d (E) d d sin 4 d Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
5 Polar (BC only) 4. (calculator not allowed) (AP AB Calculus Course Description sample questions) What is the area of the region enclosed by the lemniscate r 8cos( ) shown in the figure below? (A) 9 (B) 9 (C) 8 (D) 4 (E) 5. (calculator not allowed) Which of the following gives the area of the region enclosed by the loop of the graph of the r 4cos shown in the figure above? polar curve (A) cos( ) d (B) 8 cos( ) d (C) (D) (E) 8 cos ( ) d cos ( ) d 8 cos ( ) d Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
6 Polar (BC only). (calculator not allowed) The area of the region enclosed by the polar curve r sin( ) for is (A) (B) (C) (D) 8 (E) 4 7. (calculator not allowed) Determine the slope of the line tangent to the polar curve r sin at. (A) (B) (C) (D) (E) 8. (calculator not allowed) Which of the following represents the graph of the polar curve r sec? Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
7 Free Response Polar (BC only) 9. (calculator allowed) C P O S 5 The figure above shows the graphs of the line x y and the curve C given by x y. Let S be the shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P. (a) Find the coordinates of point P and the value of dx for the curve C at point P. dy (c) Curve C is a part of the curve x y. Show that polar equation r. cos sin x y can be written as the (d) Use the polar equation from part (c) to setup an integral expression with respect to the polar angle that represents the area of S. Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
8 . (calculator allowed) Polar (BC only) y r ΘsinΘ O x The curve above is drawn in the xy-plane and is described by the equation in polar r sin for, where r is measured in meters and is measured coordinates dr cos. d in radians. The derivative of r with respect to is given by (a) Find the area bounded by the curve and the x-axis (b) Find the angle that corresponds to the point on the curve with x-coordinate -. Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
9 Polar (BC only) dr (c) For, is negative. What does this fact say about r? What does this fact d say about the curve? (d) Find the value of in the interval that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer. Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
10 . (calculator allowed) y Polar (BC only) O R x The graphs of the polar curves r and r cos are shown in the figure above. 4 The curves intersect when and. (a) Let R be the region that is inside the graph of r and also inside the graph of r cos, as shaded in the figure above. Find the area of R. (b) A particle moving with nonzero velocity along the polar curve r cos has position xt, yt at time t, with when t. This particle moves along the dr dr curve so that. Find the value of dr at and interpret your answer in dt d dt terms of the motion of the particle. dy dy (c) For the particle described in part (b),. Find the value of dy dt d dt interpret your answer in terms of the motion of the particle. at and Copyright 4 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at
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