Chapter 11: Parametric and Polar Equations
11.1
p. 692 1 – 15 odds
11.1
p. 693 24 – 26, 28
11.2
p. 702 1-7 odds, 17
11.2
p. 702 11,13, 19, 31, 33
11.2
P. 702 37 – 43 odds
11.3
p. 713 1 – 5 odds, 9, 13
11.3
p.713 15 – 23 odds, 29, 57, 61
11.4
p. 719 1,3,5,7, 11, 15, 17
11.4
p. 719 11, 15,17,25,31
TEST – sections 11.1-11.4
[sug rev: p. 733 1-37 odd ]
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11.1
Curves Defined by Parametric Equations (Day 1)
Example 1: a. Complete the table below for each equation given that −2 ≤ t ≤ 3 .
b. Sketch and identify the ‘curve’.
c. Indicate with an arrow the direction in which the curve is traced at t increases.
t
y=
2
x=
t2 − 4
t
−2
−1
0
1
2
3
x
y
2
Example 2: a. Complete the table below for each equation given that 0 ≤ θ ≤
9π
.
4
b. Sketch, identify and write the equation of the ‘curve’.
c. Indicate with an arrow the direction in which the curve is traced as θ increases.
=
θ
cos
y sin θ
x
θ
0
π
4
π
2
3π
4
π
5π
4
3π
2
7π
4
2π
9π
4
x
y
Example 3: a. Complete the table below for each equation given that t ≥ 0 .
b. Sketch and identify the ‘curve’.
c. Indicate with an arrow the direction in which the curve is traced at t increases.
x
1
t
=
y
t +1
t +1
3
Section 11.1 (Day 2)
Example #4:
a. Use the graphs of the parametric equations x = f(t) and y = g(t) below to sketch the parametric
curve in terms of x and y. Indicate with arrows the direction in which the curve is traced as t
increases.
b. Write the equations of the parametric equations and graph then in your calculator. (Tmin= - 30
Tstep = 0.1)
x= f(t)
t
−2
y = f(t)
−1
0
1
2
y = f(x)
3
x
y
The process of graph an x and y location with respect to time can be tedious. This process can
sometimes be simplified by finding a rectangular equation (in x and y only- no t) that has the same
graph. This process is called eliminating the parameter.
Eliminating the parameter:
Step 1: Solve for t in one of the equations.
Step 2. Substitute the ‘modified’ equation in step 1 into the other equation.
Step 3: Simplify the equation.
Practice #1: Write the opening problem as a rectangular equation/eliminate the parameter. Use your
graph from the opening problem to identify the domain and range
x=
t2 − 4
t
y=
2
4
Practice #2: Write the parametric equations below as a rectangular equation (eliminate the parameter).
Sketch the ‘curve’ using your rectangular equation. Identify the domain and range.
x=
3 − 2t
y=
2 + 3t
Practice #3: Write the parametric equations below as a rectangular equation (eliminate the parameter).
Sketch the ‘curve’ using your rectangular equation. Identify the domain and range.
x
1
t
y
=
t +1
t +1
5
Practice #4: a. Use your calculator to graph the following parametric equations. Make sure your
calculator is in radian mode because t represents an angle in radians, and adjust your window
accordingly.
b. Identify the curve and find its center.
Make sure your calculator is in radian mode because t represents an angle in radians, and adjust
your window accordingly.
x =2 +
3
cos t
y =−1 + 4 tan t
Practice #5 - 7: a. Eliminate the parameter (write the equations in rectangular form – without t).
b. State the domain and range.
Practice #5:
x=
t −1
y=
t+2
6
t
=
x e=
y e−t
Practice #6:
=
x ln=
t
y
Practice #7:
t
,
t ≥1
7
11.2 Arc Length with Parametrics (Day 1)
8
Example #1:
a. Sketch the graph on the xy plane and include the directional arrows.
b.
Find dy/dx.
c. Find the equation of the tangent line at t = 0.
d. Find the equation of the tangent line(s) at the point (0,4).
Example #2:
a.
x = tet , y = t + et
Find dy/dx.
b. Find the equation of the tangent line at t = 0.
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Example #3: Sketch the graph and then determine the x-y coordinates of the point where the following
parametric equations will have horizontal or vertical tangents.
x=
t 3 − 3t
y=
3t 2 − 9
10
Example #4: Find the points on the curve where the tangent is horizontal or vertical. Sketch the graph
on your calculator to check your work.
=
x cos
=
3θ , y 2sin θ
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11.2 Concavity with Parametrics (Day 2)
Opening problem:
Use the following parametric equations
a.
x= t 2 , y= t 3 − 3t .
Complete the table below.
t
x
y
−2
−
3
−1
0
1
3
b.
Sketch the graph below and include directional arrows.
c.
Find the equation of the tangent line(s) at (3,0).
d.
Find the point(s) where the tangent is horizontal.
e.
Find the point(s) where the tangent is vertical.
2
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Example #1: Using the information from the opening problem, find the values of t that the curve is
concave upward and downward.
13
Example #2: Find dy/dx and d 2 y / dx 2 . Find the values of t that the curve is concave upward and
downward. With the use of your calculator, sketch the graph.
x=
t 3 − 12t
y=
t 2 −1
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11.2 Area with Parametrics (Day 3)
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Example #3: Determine the area under the parametric curve given by the following parametric
equations: x = 6θ − 6sin θ
y = 6 − 6 cos θ
0 ≤ θ ≤ 2π
Precalculus Tip of the Day:
cos 2 θ=
1 + cos 2θ 1 1
=
+ cos 2θ
2
2 2
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Example #4: Find the area under one arch of the cycloid
x =θ − sin θ
y =1 − cos θ
0 ≤ θ ≤ 2π
Example #5: Determine the area of the region below the parametric curve by the following set of
parametric equations:
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11.2 Arc Length with Parametrics (Day 4)
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Example #1: Determine the length of the parametric curve given by the following parametric equations.
=
x 3sin=
t
y 3cos t
0 ≤ t ≤ 2π
Example #2: Find the arc length for the following parametric equations.
x = 3t − t 3
y = 3t 2
− 3≤t ≤ 3
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11.3 Polar Coordinates (Day 1)
20
tan θ =
y
x
To form the polar coordinate system in the plane, fix the point O, called the pole (or origin), and
construct from O an initial ray called the polar axis (positive x-axis). Then each point P in the plane can
be assigned polar coordinates
A)
(r ,θ ) as follows:
r = directed distance from O to P
θ = directed angle, counterclockwise from polar axis to segment OP if positive and clockwise
B)
from polar axis to segment OP if negative.
Practice #1:
π
5π
Graph (2,
) and (3, − )
3
6
below.
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In rectangular coordinates, each point (x, y) has a unique representation. This is not true for polar
( r ,θ )
(r ,θ + 2π )
coordinates. For instance, the coordinates
and
represent the same point.
Another way to obtain multiple representations of a point is to use negative values for r. Because r is a
directed distance, the coordinates
(r ,θ ) and (− r ,θ + π ) represent the same point.
In
(r ,θ ) can be represented as :
(r ,θ ± 2nπ ) add/subtract 2π to θ
general, the point
or
(− r ,θ ± (2n + 1)π )
add/subtract
π to θ
Practice #2: Plot the point (3, − 3π ) below. Find two additional polar representations with r = 3 and
4
two additional polar representations with r = -3 so that −3π ≤ θ ≤ 3π .
(3 ,
( -3,
Practice #3: Graph
) and
) and
(3 ,
( -3,
)
)
5π
5π
π
13π ,
(
1,
) , and (−3, − π ) below.
−
−
(2, ) , (3, − ) , (4, −
)
3
6
3
3
4
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Examples 2 – 4 Sketch the curve with the given polar equation
3π
2. r =
3. r 2 =
sin 2θ
4. θ =
−3cos θ
−
4
Examples 5 – 7 Sketch the curve and find the Cartesian equation for the curve.
5. r
=
θ
3=
6. r sec
7. θ
π
3
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11.3 Polar Coordinates (Day 2)
Problems 1 – 3
1. y
3
Find a polar equation for the curve represented by the given Cartesian equation.
=
2. x 2 + y 2 16
=
3. x + y 9
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Problem #4: a. Find the slope of the tangent line to the given polar curve at the point specified by the
value of θ .
b. Find the point(s) on the curve where the tangent line is horizontal or vertical.
c. Sketch the graph.
π
r=
1 + sin θ , θ =
3
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Problem #5: a. Find the slope of the tangent line to the given polar curve at the point specified by the
value of θ .
b. Find the point(s) on the curve where the tangent line is horizontal.
c. Sketch the graph.
π
r=
2 − sin θ , θ =
3
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11.4 Area Polar Coordinates (Day 1)
Example #1: Find the area enclosed by one loop of the four-leaved rose r = cos 2θ . Sketch the curve.
27
Area enclosed by two curves:
Example #2: Find the area of the region that lies inside the circle r = 3sin θ and outside the cardioid
r = 1 + sin θ .
28
11.4 Arc Length in Polar Coordinates (Day 2)
The derivation is shown on p. 718 in your textbook.
Problems 1 – 4 Set up the integral and use your calculator to find the length of each curve.
Sketch each graph to see if the length you get from the calculator seems reasonable.
1. r = 1 + sin θ .
2.
3.
4.
(no calc)
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