Lecture note (last updated on March 4, 2007)
Spring/2007
Dr. Firoz
Chapter 10 Parametric Equations and Polar Coordinates
Section 10.1 Parametric Curves (Page # 651). Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations is called parametric equations. Each value of t determines a point which can be ploted in a coordinate plane. As t varies, the point varies and traces out a curve C, called parametric curve.
Plot the points we have below.
1.
t / 0 1 2 3 4 5x / 1 2 1+ 1+ 3 1+
y / 0 -3 -4 -3 0 5
2.
t / 0x / 2 0 -2 0 2
y / -1 1+
6. a)
t / -2 -1 0 1 2 3x / -1 0 1 2 3 4
y / 9 7 5 3 1 -1
b) Eliminating t we have
8. a)
t / -3 -2 -1 0 1 2 3x / -8 -5 -2 1 4 7 10
y / -7 -2 1 2 1 -2 -7
b) Eliminating t we have
12. a) , which is an ellipse,
14. , which is a hyperbola,
18. , is a straight line.
20. , is a unit circle center at (2, 3). As t moves from 0 t0 , the particle makes one complete counterclockwise rotation around the circle, starting and ending at (3, 3).
24. a) with III, b) with I, c) with IV, d) with II.
Section 10.2 Calculus with Parametric Curves (Page # 660)
In this section we use
1. and
2. The equation of a tangent line at (a, b) to the graph of y = f(x) is y – b = m(x-a) where
3. Areas:
4. Arc length:
5. Surface Area: a)
b)
A homework problem hints:
4.
The tangent line at (19, 6) is .
6.
The tangent line at (1, 1) is .
8.
The tangent line at (1,) is .
18.
For horizontal tangents .
For vertical tangents
26. , equation of tangents at (0, 0) is
32.
42.
Section 10.3 Polar coordinates (Page # 669)
Review on polar coordinate system
1. In the rectangular coordinate system a point in the plane is specified by the order pair (x, y) that describes the distance for the x axis and y axis. The polar coordinate system represents a point in the plane using an ordered pair that describes a distance and direction from a fixed reference pint (see the figure).
Relationship between Cartesian and Polar Coordinates. (x, y)
By Pythagorean formula
r
and
Examples
1. Find rectangular coordinates of the point that has polar coordinates
. The point in Cartesian coordinate system is
2. Find the polar coordinates of the point with rectangular (Cartesian) coordinates (1, - 1).
. The point in polar coordinate system is . This point has other representations as
2. Graphs of polar equations
A polar equation is an equation involving the polar variables r and . The curve described by the equation is the collection of all points in the plane Pthat have at least one polar representation that satisfies the equation.
2. a) b) c)
4. a) b) c)
6. a) b)
8. Do yourself
14.
16. , a vertical line 20. , a parabola 26. , a hyperbola
34. is alimacon (read as lim-a-son), see Example 11 (Page # 677), in this case the axis will be x-axis.
56.
64. Horizontal at and vertical at
Section 10.4 Areas and Lengths in Polar coordinates (Page # 679)
5.
6.
7.
8.
19. ,
20. ,
24. ,
25. ,
26. is a circle with radius 3/2 has area , on the other hand the area inside the curve is
Make the graph and observe that the shaded area = 9
28.
29.
32. ,
35. ,