10.3 Polar Coordinates – Math 152, section 4, Spring 2023 Web Assign
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University
Illinois Institute of Technology
Course
Calculus Ii (MATH 152)
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- Homework 1 – Calculus
Preview text
Find two other pairs of polar coordinates of the given polar coordinate, one with and one with Then plot the point.
(a)
(b)
r > 0 r < 0.
( 5 , ÿ/ 3 )
(r, ÷) =
$$5,73π
(r > 0)
(r, ÷) =
$$−5,−5π 3
(r < 0)
( 3 , −5ÿ/ 6 )
1. [5/6 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.501.
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER
(c)
(r, ÷) =
$$3,7π 6
(r > 0)
(r, ÷) =
$$−3,π 6
(r < 0)
(−3, ÿ/ 4 )
(r, ÷) =
$$3,54π
(r > 0)
(r, ÷) =
$$−3,−74π
(r < 0)
The Cartesian coordinates of a point are given.
(a)
(i) Find polar coordinates (r, ÷) of the point, where
(ii) Find polar coordinates (r, ÷) of the point, where
(b)
(i) Find polar coordinates (r, ÷) of the point, where
(ii) Find polar coordinates (r, ÷) of the point, where
( 4 , −4)
r > 0 and 0 ≤ ÷ < 2 ÿ.
(r, ÷) =
$$4√2,74π
r < 0 and 0 ≤ ÷ < 2 ÿ.
(r, ÷) =
$$−4√2,34π
(−1, 3 )
r > 0 and 0 ≤ ÷ < 2 ÿ.
(r, ÷) =
$$2,2π 3
r < 0 and 0 ≤ ÷ < 2 ÿ.
(r, ÷) =
$$−2,5π 3
2. [4/4 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.505.
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Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.
1 < r ≤ 5 , 2 ÿ/ 3 ≤ ÷ ≤ 4 ÿ/ 3
3. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.528.
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Sketch the region in the plane consisting of points whose polar coordinates satisfy the given condition.
5 ≤ r ≤ 8
5. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.533.
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Find a Cartesian equation for the curve and identify it.
$$y=
Find a Cartesian equation for the curve and identify it.
$$y=x 29
r = 8 csc(÷)
hyperbola
circle
line
ellipse
parabola
r = 9 tan(÷) sec(÷)
ellipse
parabola
limaçon
circle
line
6. [2/2 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.521.
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7. [2/2 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.508.
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The gure shows the graph of r as a function of ÷ in Cartesian coordinates. Use it to sketch the corresponding polar curve.
9. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.
MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER
The gure shows the graph of r as a function of ÷ in Cartesian coordinates. Use it to sketch the corresponding polar curve.
10. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.517.
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Sketch the curve with the given polar equation by rst sketching the graph of r as a function of ÷ in Cartesian coordinates.
r = cos( 5 ÷)
12. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.512.
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Sketch the curve with the given polar equation by rst sketching the graph of r as a function of ÷ in Cartesian coordinates.
r = 2 cos( 4 ÷)
13. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.
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Sketch the curve with the given polar equation by rst sketching the graph of r as a function of ÷ in Cartesian coordinates.
r = 5 + 3 sin(÷)
15. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.515.
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Sketch the curve with the given polar equation by rst sketching the graph of r as a function of ÷ in Cartesian coordinates.
r 2 = 6 sin(2÷)
16. [0/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.
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Sketch the curve with the given polar equation.
Find the slope of the tangent line to the given polar curve at the point specied by the value of ÷.
$$√ 3
r = 5 (1 − sin(÷)), ÷ ≥ 0
r = 4 sin(÷), ÷ = ÿ/
18. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.543.
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19. [1/1 Points] DETAILS PREVIOUS ANSWERS SCALC9 10.3.519.
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Video Example
EXAMPLE 4 What curve is represented by the polar equation
SOLUTION The curve consists of all points with Since r represents the distance from
the point to the pole, the curve represents the (No Response) with center O and radius (No
Response). In general, the equation represents a circle with center O and radius (See the
gure.)
Video Example
EXAMPLE 7 Sketch the curve
SOLUTION Instead of plotting points, we rst sketch the graph of in Cartesian
coordinates in the top gure by shifting the sine curve up one unit. This enables us to read at a glance
the values of r that correspond to increasing values of ÷. For instance, we see that as ÷ increases from
0 to ÿ/2, r (the distance from O) increases from (No Response) to (No Response) , so we sketch the
corresponding part of the polar curve in gure (a). As ÷ increases from ÿ/2 to ÿ, the top gure shows
that r decreases from (No Response) to (No Response) , so we sketch the next part of the curve as
in gure (b). As ÷ increases from ÿ to 3 ÿ/2, r decreases from (No Response) to (No Response) as
shown in part (c). Finally, as ÷ increases from 3 ÿ/2 to 2 ÿ, r increases from (No Response) to (No
Response) as shown in part (d). If we let ÷ increase beyond 2 ÿ or decrease beyond 0, we would
simply retrace our path. Putting together the parts of the curve from gures (a)-(d), we sketch the
complete curve in part (e). It is called a cardioid because it’s shaped like a heart.
r = 6?
(r, ÷) r = 6.
r = 6
r = a |a|.
r = 1 + sin(÷).
r = 1 + sin(÷)
20. [–/0 Points] DETAILS SCALC9 10.3.AE. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER
21. [–/0 Points] DETAILS SCALC9 10.3.AE. MY NOTES ASK YOUR TEACHER
10.3 Polar Coordinates – Math 152, section 4, Spring 2023 Web Assign
University: Illinois Institute of Technology
Course: Calculus Ii (MATH 152)
176 Documents
Students shared 176 documents in this course
Find two other pairs of polar coordinates of the given polar coordinate, one with and one with Then plot the point.
(a)
(b)
r
> 0
r
< 0.
(5,
ÿ
/3)
(
r
,
÷
)=
$$5,73
π
(
r
> 0)
(
r
,
÷
)=
$$−5,−5
π
3
(
r
< 0)
(3, −5
ÿ
/6)
1.
[5.25/6 Points] SCALC9 10.3.501.XP.
DETAILS
PREVIOUS ANSWERS
MY NOTES
ASK YOUR TEACHER
PRACTICE ANOTHER
