AP Calculus BC Ch. 10B - Polar Functions - Lesson Plans
#1
Precalculus Polar Stuff Needed for Calculus (No Show & Tell for HW #1)
(No calculator – Graph by hand. Draw helping graphs if needed.)
p. 722 Sec. 10.4 #2, 3, 5, 7, 11, 13, 23, 27, 29, 31, 33, 35, 36, 38, 39, 40, 43, 51,
77, 79,80, 81, 82, 87, 88, 96abcd, 97abc
#2 Polar – Derivatives, horizontal & vertical tangents and tangent line to polar curve
(No calculator – Graph by hand. Draw helping graphs if needed.)
p. 722 Sec. 10.4 #59, 60, 61, 62, 63, 64, 676, 68, 69, 70,71, 72, 73, 75, 76,
p. AP10-2 #6bc, 7bc, 10
Critical Thinking Journal (CTJ 15 pts. Due __________) p. 722 Sec. 10.4 #93, 94, 95
#3 Area Inside Polar Curves
p. 731 Sec. 10.5 #1, 2, 3, 4, 7, 9, 11, 16, 17, 20, 21, 23, 29, 33, 35, 41
p. AP10-2 #6a, 7a, 8abc
#4 Arc Length and Surface Area of Polar Curves
p. 731 Sec. 10.5 #51 & 53 (Graph by hand and use Geometry formulas.),
#55, 59, 63, 67 (Graph by hand, set up integral, then evaluate integral on calculator.)
p. AP10-2 #9abcd. Polar Problem Set below is part of HW.
(1) The area of one loop of the graph of the polar equation r = 2 sin 3 is (graphing calculator allowed, but none
of you need it!!!)


3
3
( A ) 4  sin 2 3 d
0
( D ) 2  sin 2 3 d
0
0
2
3
( E ) 2  sin 3 d
0


3
6


3

6
( D ) 16  cos 2 3 d

(4)
3
( C ) 2  sin 2 3 d
Which of the following gives the area of the region enclosed by the loop of the graph of the
polar curve r = 4cos 3 shown in the figure. (noncalculator)
( A ) 16  cos 3 d
(3)
( B ) 2  sin 3 d
0
2
3
(2)


6

( B ) 8  cos 3 d


3
( C ) 8  cos 2 3 d

6

3

6
( E ) 8  cos 2 3 d


6
Find the number of points of intersection of the polar curves r = 2 cos Θ and
r = 1 + 2 sin Θ. (noncalculator) (A) 1, (B) 2, (C) 3, (D) 4, (E) 5
Find the area of the right half of the polar curve r = 1 + sin Θ. (noncalculator)
(A) ½ , (B) 3/4, (C) , (D) 3/2, (E) 2
(5) Free Response Question: Consider the polar curve r = 2 sin 3 for 0 <  < .
(a) Sketch the curve in the xy-plane.
(b) Find the area of the region inside the curve.
(c) Find the slope of the curve at the point where  = /4.
Ans: (1) C, (2) E, (3) D (4) B, (5) (a) 3 leaves, (b)  (c) ½
TEST I: All formulas below, (no calculator) HW Bonus based on HW #14 and Problem Sets
1. Change polar coordinates to rectangle
coordinates and vice versa
2. Change polar equations to rectangle
equations and vice versa
3. Symmetry rules
4. Finding relative extrema of r = f(  )
5.
6.
7.
8.
9.
10.
How to find slope of a curve, H & V tangents
How to find tangent line at pole (radial line)
Area enclosed by a polar curve
arc length of a polar curve
surface area of solid of revolution
matching polar graphs
TEST II: Problems using the formulas.
Little Green Book of Calculus BC Properties
Ch. 10B – Polar Functions
10B-1
Polar Coordinate System (parts of polar graph, polar axis, locating points, discuss
+ and – r) (ex. locate (3, 60) and (3, 60), find several names for the same point)
10B-2 Changing Polar Points and Equations to Rectangular Points and Equations (long
way and calculator way, ex.)
10B-3 Changing Rectangular Points & Equations to Polar Points & Equations (ex. each)
10B-4 Finding Excluded Values of  (How and why, give ex.)
10B-5 Radial Lines (definition and how to find)
10B-6 Symmetry of Polar Graphs (how to determine, give ex.)
10B-7 Using Rectangular Graph to Aid in Sketching Polar Graph (discuss similarities,
ex. graph y = 3 sin 4x and r = 3sin 4)
10B-8 Spiral of Archimedes (polar equation, t-up, and graph)
10B-9 Circles in Polar Plane (polar equations and graphs of r = k, r = a sin , r = a cos , using a)
10B-10 Lines (polar equations and graphs of r = a sec , r = a csc , using a)
10B-11 Cardioid and Limacons (polar equations and graphs of r = a + b sin , r = a + b cos
, using a = b, a < b, a > b, a, b)
10B-12 Lemniscates (polar equations and graphs of r2 = a sin 2, r2 = a cos 2, using a)
10B-13 Roses (polar equations and graphs of r = a cos n, r = a sin n, using a and even and odd n)
10B-14 Tangent lines to polar curves (Finding equations of horizontal and vertical lines
tangent to the polar curve. finding slope of lines to polar curves and equations lines
at any point on the polar curve, finding tangent lines at the pole)
10B-15 Relative Extrema of r (how to find, discuss dr  slope of tangent line)
dt
10B-16 Geometry Formulas for Area of a Sector and Length of an Arc using Radians
(ex: If the radius = 6 and angle = 1.5 radians, find area of sector and arc length.)
10B-17 Area Bounded by One Polar Curve (formula and ex.)
10B-18 Area Between Two Polar Curves (formula and ex.)
10B-19 Arc length of a Polar Curve (formula and ex.)
10B-20 Surface Area of Solid of Revolution of a Polar Curve (formula and ex.)
10B-21 Disher Philosophy “Do not complain about something unless you are willing to work to
change it.” (Comment on how you see it in my life and how you can use it in your life.)