POLAR FUNCTIONS From the AP Calculus BC Course Description, students in Calculus BC are required to know: The analsis of planar curves, including those given in polar form Derivatives of polar functions Finding the area of a region, including a region bounded b polar curves In Precalculus students should have learned to: r,. Plot points in polar form, Graph polar curves: circles, limacons, cardiods, roses, lemniscates---without a graphing calculator. Convert points from rectangular coordinates to polar coordinates and vice versa. Convert equations from rectangular form to polar form and vice versa. The website for polar graph paper that produced this polar graph is listed below. http://www.incompetech.com/beta/plaingraphpaper/polar/ With the website ou can choose how man spokes ou want and what tpe of labels
DISCOVERY LESSON ON POLAR GRAPHS Put our graphing calculator in POLAR mode and RADIAN mode. Graph the following equations on our calculator, sketch the graphs on this sheet, and answer the questions. Use WINDOW settings: [0, :[0,, /4], :[ 6,6,1], :[ 4,4,1] 1. r cos r cos r cos r cos r sin r sin r sin r sin What do ou notice about these graphs? _. r cos r 1 cos r cos r sin r 1 sin r sin Which graphs go through the pole? Which ones do not go through the pole? Which ones have an inner loop? What causes these things to happen?
. r cos r cos5 r 4cos7 r sin r sin5 r 4sin7 What do ou notice about these graphs? 4. r cos r cos4 r 4cos6 r sin r sin 4 r 4sin6 What do ou notice about these graphs? _ 5. r 4cos r 4sin What do ou notice about these graphs?
Polar Coordinates and Polar Graphs Rectangular coordinates are in the form,. Polar coordinates are in the form r,. E. 1 Graph the following polar coordinates: P 4, 5 Q, 7 6 5 R, 4 S 1, _ In Precalculus ou learned that: cos so = sin so = tan so r = 5 E. Convert, to rectangular coordinates. 6 E. Convert, to polar coordinates.
E. Convert the following equations to polar form. (a) = 4 (b) 5 E. Convert the following equations to rectangular form, and sketch the graph. (a) r sin (b) r cos (c) To find the slope of a tangent line to a polar graph r f, we can use the facts that r cos and r sin, together with the product rule: d d d d d d E. Find d and the slope of the graph of the polar curve at the given value of. d r sin, 6
The area of a polar region is based on the area of a sector of a circle. Sector Area Circle Area Central Angle So Total of Angles Sector Area Therfore, Sector Area = If we take the function r f( ) on [, ] and partition it into equal subintervals, then the radius of the kth sector = f ( k ) and the central angle of the kth sector = n n 1 and the Area of the sector [ f ( k )]. k1 Taking the limit as n, Sector Area = n 1 lim [ f ( k )] n k 1 If f is continuous and nonnegative on the interval [, ], then the area of the region bounded b the graph of r f( ), then the area of the region bounded b the graph of r f( ) and the radial lines and is given b the formula: Polar Area = Area between two polar curves: Shaded Area =
The area bounded b the polar curve r f between and is given b the formula: 1 A r d, E. Find the area bounded b the graph of r sin. Make a table of values and sketch the graph. E. Set up an integral to find the area of one petal of r sin. Sketch the graph. E. Set up an integral to find the area of one petal of r 4cos. Sketch the graph.
For each of the following, make a table of values, and sketch the graph. E. Set up an integral to find the area inside r sin and outside r sin. E. Set up an integral to find the area of the common interior of r cos and r1 cos.
Polar Graphs with the Graphing Calculator E. A curve is drawn in the -plane and is described b the equation in polar coordinates r sin for 0, where r is measured in meters and is measured in radians. (a) Sketch the graph of the curve. (b) Find the area bounded b the curve and the -ais. (c) Find the angle that corresponds to the point on the curve with -coordinate 1.
CALCULUS BC WORKSHEET 1 ON POLAR Work the following on notebook paper. Convert the following equations to polar form. 1. = 4. 5 0. 5 Convert the following equations to rectangular form. 5 4. r sec 5. r sin 6. 6 For the following, find d for the given value of. d 7. r sin, 9. r 4sin, 8. r 1 cos, 10. r sin, 4 11. Find the points of horizontal and vertical tangenc for r 1 sin. Give our answers in polar form, r,. Make a table, tell what tpe of graph it is, and sketch the graph on polar paper. 1. r cos 1. r sin 14. r sin 15. r cos r 4sin 16. On problems 17-19, make a table, tell what tpe of graph it is, sketch the graph on polar paper, and answer the questions asked. 17. r 1 sin Notice that this graph has an inner loop. At what value of does the loop begin? At what value of does the loop end? What do ou notice about the values of r for the points that are on the loop? 18. r 4cos Name the values of where the petals begin and end. What is the maimum value of r on our graph? Name the values of that give a maimum value for r. 19. r 6sin Name the values of where the petals begin and end. What is the maimum value of r on our graph? Name the values of that give a maimum value for r.
CALCULUS BC WORKSHEET ON POLAR For each of the following, sketch a graph, shade the region, and find the area. Do not use our calculator. r cos 1. one petal of. one petal of r 4sin. interior of r cos 4. interior of r sin
5. interior of r 4sin 6. inner loop of r 1 cos 7. between the loops of r 1 cos
CALCULUS BC WORKSHEET ON POLAR For each of the following, sketch a graph, shade the region, and find the area. Do not use our calculator. 1. inside r cos and outside r cos 4. common interior of r cos and r1 cos. common interior of r 4sin and r 5. common interior of r 4sin and r. inside r sin and outside r 1 sin
Work the following on notebook paper. Do not use our calculator ecept on problem 10. r sin at the point,. 1. Find the slope of the curve. Find the equation of the tangent line to the curve r sin at the point where. On problems 5, set up an integral to find the area described below. Do not evaluate.. r 4. r 1 sin 5. r sin 4 Find area in QI and QII. Find area in QII Find area of ONE petal. 6. Sketch the polar region described b the following integral epression for area: 1 sin d 0 7. (a) In polar coordinates, write equations for the line = 1 and the circle of radius centered at the origin. (b) Write the integral in polar coordinates representing the area of the region to the right of = 1 and inside the circle. (c) Evaluate the integral. r 4cos and outside the 8. (a) Sketch the bounded region inside the lemniscate circle r. (b) Compute the area of the region described in part (a). 9. Find the area between the two spirals r and r for 0. Use our calculator on problem 10. 10. Given the polar curve r sin for 0 (a) Sketch the graph of the curve. (b) Find the angle that corresponds to the point(s) on the curve where 1. (c) Find the angle that corresponds to the point(s) on the curve where.
Answers to Worksheet 1 on Polar 1. r 4csc 10. 1 1 7 1 11. r 11. Horiz:,,,,, cos 5sin 6 6. r = 5 5 4. = Vert.:,,, 6 6 5. 1. (b).786 6. (c) 0.661 and. 7. 0 (d) 0 and. 8. 1 r is increasing. The curve is getting farther 9. from the origin. Answers to Worksheet on Polar 1.. 6 5. 4 7.. 4. 9 6. Answers to Worksheet on Polar 1.. 5. 16 4. 8 5 4. 4 1. 9. 4 5 4 1. d 0 6. 1 petal of r sin 9. 4 7. (a) r sec, r 10. (a) graph (b) sec 1 d (c) 4 1 4 sin 4 d 8. (a) graph 0 (b) 4. 1 sin 5. 1 d (b) 1.89, 4.95 (c) 0.91,.56