Transcript AP Calculus BC – 3.6 Chain Rule 1

AP Calculus BC – Chapter 10
Parametric, Vector, and Polar Functions
10.5: Polar Coordinates and Polar Graphs
Goals: Graph polar equations and determine
the symmetry of polar graphs.
Convert Cartesian equations into polar form
and vice versa.
Polar Coordinates:
In polar coordinates, we identify the origin O as
the pole and the positive x-axis as the initial
ray of angles measured in the usual
trigonometric way. We can then identify
each point P in the plane by polar coordinates
(r, ), where r gives the directed distance
from O to P and  gives the directed angle
from the initial ray to the ray OP.
Polar Coordinates:
Example 1: (a) Find rectangular coordinates for
the points with given polar coordinates.
(i) (4, π/2) (ii) (-3, π) (iii) (16, 5 π /6) (iv) (-√2, - π /4)
(b) Find two different sets of polar
coordinates for the points with given
rectangular coordinates.
(i) (1, 0) (ii) (-3, 3) (iii) (0, -4) (iv) (1, √3)
Polar Coordinates:
Example 2: Graph all points in the plane that
satisfy the given polar equation.
(a) r=2
(b) r=-2
(c)  = π/6
Polar Coordinates:
Example 3: Find an appropriate graphing
window and produce a graph of the polar
curve.
(a) r=sin (b) r=1-2cos (c) r=4sin
Equations:
Polar-Rectangular Conversion Formulas:
x=r cos
r2 = x2 + y2
y= r sin
tan = y/x
Parametric Equations of Polar Curves:
The polar graph of r=f() is the curve defined
parametrically by:
x=r cos = f()cos
y=r sin = f()sin
Assignment and Notes:

HW 10.5: #1, 3, 5, 9, 15, 21, 27, 39, 43,
54, 55, 57, 67.
Test Friday, March 16.
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