Chapter 10: Parametric Equations and Polar Coordinates 10.2 Plane Curves and Parametric Equations A new way of graphing! EXAMPLE 1 Write x as a function of t. Write y as a function of t. t is called a parameter. The two equations are called parametric equations. x = t2 + t
2
y = 2t − 1 −1 ≤ t ≤ 1
-2
-4
Now create a table of values and plot the graph. Draw arrows on your graph to indicate how you plotted points as t increases. The direction is called the orientation. Taken together, the parametric equations and the graph are called a plane curve, denoted by C. t -­‐1 -­‐1/2 0 1/2 1 x y How does the graph change if you begin your graph at t = -­‐2 ? How might you eliminate the parameter and write these equations only in terms of x and y? Do you recognize how the new equation relates to the graph? EXAMPLE 2 Use your graphing calculator to sketch the parametric curve: Think about reasonable values for your window. 2
x = 5 cos t
y = 2 sin t 0 ≤ t ≤ 2π
Sketch the graph here. -5
Include the orientation arrows. How do you think you might change the orientation? Try it! -2
Now, eliminate the parameter by using the well known trig identity sin
How does the new equation relate to the graph? EXAMPLE 3 Use your graphing calculator to sketch the parametric curve: x = 5 cos 3t
y = 2 sin 3t 0 ≤ t ≤ 2π
2
t + cos 2 t = 1 2
Sketch the graph here. -5
Include the orientation arrows. -2
What is the interval of t that would create a single trace of the graph? EXAMPLE 4 Use your calculator to graph the curve represented by the parametric equations: 1
t +1
t
y=
t +1
t > −1
2
x=
5
-2
Eliminate the parameter and write the equation of the graph as y = f(x). Compare the graph of y = f(x) to the plane curve above. How must you adjust the domain of f(x) if it is to accurately represent the same set of points as the plane curve above? -4
-6
-8
Rectangular Coordinates to Parametric Equations Begin with an equation in rectangular form and write it in parametric form. EXAMPLE 1 Given: y
= 1 − x 2 Let x = t, then y = 1 − t . There are infinitely many different parametric equations that can represent a given 2
rectangular equation. Find another set of parametric equations to represent y = 1 − x EXAMPLE 2 Given: y = 2x − 5 and t = 0 at (3,1) Find a set of parametric equations that represents the same graph. EXAMPLE 3 Now check out the graph of this set of parametric equations which describes the path of a particle: 2
x = 3cos 2t
y = 1 + cos 2 (2t)
Describe the path of this particle completely. Determine the limits on x and y and the interval of t for which the path will be traced out exactly once. Homework 10.2 1. 2. Consider the parametric equations x = t and y = 3 − t . a. Construct a table of values for t = 0, 1, 2, 3, and 4 b. Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. c. Use a graphing utility to confirm your graph in part (b). d. Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. Consider the parametric equations x = 4 cos 2 θ and y = 2sin θ . π π π π
a. Construct a table of values for θ = − ,− ,0, , . 2 4
4 2
b. Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. c. Use a graphing utility to confirm your graph in part (b). d. Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. ⎡ π 3π ⎤
e. If values of θ were selected from the interval ⎢ , ⎥ for the table in ⎣2 2 ⎦
part (a), would the graph in part (b) be different? Explain. Sketch the curve represented by the parametric equations (indicate the orientation of the curve) and write the corresponding rectangular equation by eliminating the parameter. 3. x = t + 1 y = t 2 t2
2
x = t3
y=
x = 2t
y = t − 2 6. x = et
y = e 3t + 1 7. x = 8 cosθ
4. 5. y = 8sin θ Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. 8. x = cosθ y = 2sin 2θ 9. x = 4 secθ
y = 3tan θ Determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. 10. x = 2 cosθ
y = 2sin θ x = 4t 2 − 1 t
y = 1 / t Eliminate the parameter and obtain the standard form of the rectangular equation. x = h + r cosθ
11. Circle: y = k + r sin θ
12. Ellipse: x = h + a cosθ
y = k + bsin θ
Find a set of parametric equations for the rectangular equation that satisfies the given equation. y = 2x − 5
13. t = 0 at the point (3,1)
y = x2
14. t = 4 at the point (4,16)
Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. x = 2(θ − sin θ )
15. Cycloid: y = 2(1− cosθ )
16. Prolate cycloid: 3
x = θ − sin θ
2
3
y = 1− cosθ
2
17. Hypocycloid: x = 3cos 3 θ
y = 3sin 3 θ
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 18. The graph of the parametric equations x = t 2 and y = t 2 is the line y = x . 19. If y is a function of t and x is a function of t , then y is a function of x. 20. The curve represented by the parametric equations x = t and y = cost can be written as an equation of the form y = f (x) . Answers: 1d. t = x2
y = 3 − x2
x ≥ 0 2
2d. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. y
x
+ = 1 4 4
y = (x − 1)2 x 2/3
2
x+3
y=
x
x
y = − 2 2
y = x 3 + 1 x > 0 x 2 + y 2 = 64 y 2 = 16x 2 (1− x 2 ) y=
x 2 y2
− = 1 16 9
( x − h )2 + ( y − k )2 = 1 r2
r2
( x − h )2 + ( y − k )2 = 1 a2
b2
x=t+3
y = 2t + 1 x=t
15. 16. 17. 18. 19. 20. 21. y = t 2 Not smooth at 2nπ = θ , n ∈I . Smooth everywhere. nπ
Not smooth at = θ , n ∈I . 2
False. Only defined for non-­‐negative values of x and y. False. Consider x = 2 cost, y = 2sint , for example. True. 10.3 Parametric Equations and Calculus Parametric Form of the Derivative: For x = f (t) and y = g(t)
dy dy / dt
=
dx dx / dt
The derivative represents the slope of the curve at (x, y). Higher order derivatives are determined by: d ⎡ dy ⎤
d y dt ⎢⎣ dx ⎥⎦
=
dx 2
dx / dt
2
The second derivative can be used in the Second Derivative test for max or min values, and for determining concavity. Example 1: Examine the graph of the parametric equations: x = 2t − π sin t
y = 2 − π cos t
Graph with the following interval for t: −10 ≤ t ≤ 10
Locate the point on the graph that has two tangent lines. This happens when a plane curve loops around and crosses itself. Use the derivative and that point to determine the equations of the tangent lines at that point. Example 2: Examine the graph of the parametric equations: x = 3t 2
y = t3 − t
Compute the 2nd derivative and use it to determine the concavity of the graph. Check the graph using your calculator to confirm your findings regarding the concavity. Arc Length Parametric equations can be used to model the path of a particle or projectile. If you want to determine the distance traveled by that particle or projectile, you can use the formula for arc length for parametric equations. Arc Length in Parametric Form If a smooth curve C is given by x = f(t) and y = g(t) such that C does not intersect itself on the interval a ≤ t ≤ b (except possibly at the endpoints), then the arc length of C over the interval is given by b
s=
∫
a
2
2
⎛ dx ⎞
⎛ dy ⎞
⎜⎝ ⎟⎠ + ⎜⎝ ⎟⎠ dt dt
dt
**Note, be sure that the curve is only traced out one time over the interval of integration! Example 3: Find the distance traveled in one complete cycle for the parametric equations below. Use the graph to determine the limits of integration for one “cycle” of the graph. x = 5 cos t − cos 5t
y = 5 sin t − sin 5t
Homework 10.3 Find dy / dx . 1. x = t2
y = 7 − 6t Find dy / dx and d 2 y / dx 2 and find the slope and concavity (if possible) at the given value of the parameter. 2. x = 4 cosθ
y = 4 sin θ
θ=
π
4
3. x = 2 + secθ
y = 1+ 2 tan θ
θ=
π
6
Find the equation of the tangent lines at the point where the curve crosses itself. 4. x = 2sin 2t
y = 3sint Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. 5. x = 3cosθ
y = 3sin θ Determine the t intervals on which the curve is concave downward or concave upward. 6. x = 2t + lnt
y = 2t − lnt Find the arc length of the curve on the given interval. 7. x = 3t + 5
y = 7 − 2t
−1 ≤ t ≤ 3
8. The path of a projectile is modeled by the parametric equations: x = (90 cos 30! )t
y = (90sin 30! )t
where x and y are measured in feet. a. Use a graphing utility to graph the path of the projectile. b. Use a graphing utility to approximate the range of the projectile. c. Use the integration capabilities of a graphing utility to approximate the arc length of the path. Compare this result with the range of the projectile. Answers: dy
3
1. = − dx
t
2. slope = -­‐1, concave down 3. slope = 4, concave down 3
3
4. At t = 0, y = x , t = π , y = − x 4
4
π 3π
5. θ= ,
2 2
6. Concave up (0,∞) 7. 8b. 4 13 approximately 219.213 feet, 8c. approximately 230.845 feet 10.4 Polar Coordinates and Polar Graphs Defining Polar Coordinates and Creating Graphs We can plot points using polar coordinates OR rectangular coordinates. Use the following relationships to convert back and forth between the two forms. tan θ =
Polar to Rectangular: Use ⎛
⎝
Convert the point ⎜ 3,
x = r cos θ
y = r sin θ
π⎞
⎟ to rectangular coordinates. 6⎠
y
y
sin θ =
x
r
cos θ =
x
r
Rectangular to Polar: Use tan θ
=
y
2
2
r = x + y
x
⎛ −1 −1 ⎞
to polar coordinates. ,
⎝ 2 2 ⎟⎠
Convert the point ⎜
Note: In rectangular coordinates there is one way to name any given point. In polar coordinates there are infinitely any ways to name a given point. Homework Examples: 1. Plot the point in polar coordinates and find the corresponding rectangular ⎛
⎝
coordinates for the point. ⎜ −3,
2. Plot the point in rectangular coordinates and find two sets of polar coordinates for the point for 0
3. 5π ⎞
⎟ 3⎠
(
≤ θ ≤ 2π . −6 3, 6
) Convert the rectangular equation to polar form and sketch its graph. x 2 + y 2 = 25 x = 3 6x − x 2 = y 2 4. Convert the polar equation to rectangular form and sketch its graph. r = 3sec θ r = −2 sin θ r sin 2θ
5. Use a graphing utility to graph the polar equation. Find an interval for θ over which the graph is traced only once. 2
r = 4 + 3cos θ r = sin
5θ
2
= 2 Graphing polar equations Use your graphing calculator or Geometer’s Sketchpad to create and compare graphs of these interesting equations. Use Radian mode with θ min = 0 , θ max = 2π , and θ step = .125π . Use a variety of values for the constants a and b. r = a cosθ
r = asin θ
r = aθ
r = a + a cosθ
r = a + asin θ
r = a + b cosθ
r = a + bsin θ
r = asin(nθ ) for n an even integer
r = asin(nθ ) for n an odd integer
r = a cos(nθ ) for n an even integer
r = a cos(nθ ) for n an odd integer
Note the direction in which the graph develops. Use the trace feature on your calculator to slowly observe the development of the graph. Note whether or not a particular curve is “traced” once or twice, etc. as you graph from 0 to 2π. Calculus with Polar Graphs Slope in Polar Form If f is a differentiable function of θ then the slope of the tangent line to the graph of r = f (θ ) at the point ( r, θ ) is given by dy dy / dθ
f (θ )cos θ + f '(θ )sin θ
=
=
dx dx / dθ − f (θ )sin θ + f '(θ )cos θ x = r cosθ = f (θ )cosθ
y = r sin θ = f (θ )sin θ
Solutions to Solutions to If dy
dx
= 0 yield horizontal tangents, provided ≠ 0.
dθ
dθ
dx
dy
= 0 yield vertical tangents, provided ≠ 0.
dθ
dθ
dy
dx
and dθ
dθ
both equal 0, no conclusion can be drawn about tangent lines. Homework Examples: 1. Evaluate the derivative of r = 3 − 2 cos θ at θ
graph to see if your answer makes sense. dy dy / dθ
f (θ )cos θ + f '(θ )sin θ dx
=
dx / dθ
=
= 0 and at θ = π / 2 . Check the − f (θ )sin θ + f '(θ )cos θ
You don’t have to memorize the formula above if you simply write your equation in parametric form, r = 3 − 2 cos θ , in parametric form: x = r cosθ = (3− 2cosθ )cosθ
y = rsinθ = (3− 2cosθ )sinθ . Now you can differentiate to find dx / dθ and dy / dθ . 2. Find any points on the graph having horizontal or vertical tangents. r = 3sin θ First, write the equation in parametric form. Then set each of the derivatives equal to 0. 3. Determine any tangents at the pole for the polar equation: r
Graph the equation in your graphing calculator and see if your solution(s) make sense. = − sin 5θ f
(
θ
)
=
0
and
f
'(
θ
)
≠
0
Tangents at the poll occur when . Homework 10.4 Plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. ⎛ π⎞
1. ⎜⎝ 8, ⎟⎠
2 3π ⎞
⎛
2. ⎜⎝ −4,− ⎟⎠
4 The rectangular coordinates of a point are given. Plot the point the find two sets of polar coordinates for the point for 0 ≤ θ < 2π . 3. −1,− 3
Convert the rectangular equation to polar form and sketch its graph. 4. x 2 + y 2 = 9 5. y = 8 6. 3x − y + 2 = 0 (
)
Convert the polar equation to rectangular form and sketch its graph. r = 4 7. 8. r = secθ tan θ Use a graphing utility to graph the polar equation. Find an interval for θ over which the graph is traced only once. 9. r = 2 − 5 cosθ 2
10. r=
1+ cosθ ⎛ 3θ ⎞
r = 2 cos ⎜ ⎟
11. ⎝ 2⎠
Find dy / dx and the slopes of the tangent lines at the points indicated on the graph of the polar equation. 12. r = 2 + 3sin θ Use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of θ , and (c) find dy / dx at the given value of θ . (Hint: let the increment between the values of θ equal π / 24 . 13. r = 3(1− cosθ ), θ = π / 2 Find the points of vertical and horizontal tangency (if any) to the polar curve. 14. r = 2 cscθ + 3 Sketch a graph of the polar equation and find the tangents at the pole. 15. r = 5sin θ 16. r = 4 cos 3θ Answers: 1. (0, 8) 2. 2 2,2 2 (
)
π⎞
⎛ 4π ⎞ ⎛
3. ⎜ 2, ⎟ , ⎜ −2, ⎟ ⎝ 3 ⎠ ⎝
3⎠
4. r = 3 8
5. r =
sin θ
−2
6. r =
3cosθ − sin θ
7. x 2 + y 2 = 16 8. x 2 = y 9. [0,2π ) 10. (−π , π ) 11. [0, 4π ) 3π ⎞
⎛ π⎞
⎛
12. At ⎜ 5, ⎟ dy / dx = 0, at ( 2, π ) dy / dx = −2 / 3, at ⎜ −1, ⎟ dy / dx = 0. ⎝ 2⎠
⎝
2 ⎠
13. θ = π / 2, dy / dx = −1. 3π π 5π
7π 11π
14. Horizontal tangents at θ =
, Vertical tangents at θ =
, ,
,
2 6 6
6 6
15. θ = 0, π ,2π π π 5π
16. θ = , ,
6 2 6
10.5 Area in Polar Coordinates Suppose we want the area of a region of a graph in polar form. We will use what we already know about the area of a circular sector to sum up the areas of infinitely many very small partitions of the area we are trying to determine using an integral. From Larson Text, 2010 edition. More on Area and Polar Graphs from Paul Dawkins on-­‐line Math notes http://tutorial.math.lamar.edu/terms.apx Homework 10.5 Find the area of the region. 1. Interior of r = 6sin θ 2. One petal of r = 2 cos 3θ 3. Interior of r = 1− sin θ Use a graphing utility to graph the polar equation and find the area of the given region. 4. Inner loop of r = 1+ 2 cosθ 5. 6. Between the loops of r = 1+ 2 cosθ Between the loops of r = 3 − 6sin θ Optional problems: Find the points of intersection of the graphs of the equations. r = 3(1+ sin θ )
r = 1+ cosθ
7. 8. r = 3(1− sin θ )
r = 1− sin θ
Use a graphing utility to graph the polar equations and find the area of the given region. 9. Common interior of r = 4 sin 2θ and r = 2 . 10. Common interior of r = 4 sin θ and r = 2 . Answers: 1. 28.274 2. 1.047 3. 4.712 4. 0.544 5. 8.337 6. 75.040 Optional answers: 7. (x,y): (3,0) (3, π ) (0,0) (r, θ ): (3, 0) (0, π /2) (0, 3 π /2) 8. (x,y): (-­‐0.207, 0.207) (1.207, -­‐1.207) (0,0) (r, θ ): (0.292, 3 π /4) (1.707, 7 π /4) (0,0) 9. 9.827 10. 4.913