Page 21
•
Perfect Polar curves
1. (a) Use the basic results about parametric curves discussed in class to verify
the validity of the following formula for the slope ofthe polar curve r
=~e) at the point (r, 6):
dy
f(B)cosB + f'(B)sinB
dx - - f(B)sinB + f'(O)cosB
(b)
* 0)
Use the result of (a) to show that if for a specific angle a, tra)=O and
f'(a)
=f-
0
then the slope ofthe tangent line at the pole is tana. [Note that there may
more than one value ofthe angle for which the curve passes through the
pole and hence more than one tangent line to some polar curves at the
pole].
•
•
(if dx / dB
(c)
Find aU the tangent lines at the pole (in polar form) to the curve r =
3sin(26) at the pole.
(d)
Repeat the directions of part (c) for r = 3(I-sinB)
(e)
repeat for r=3cos2B
Page 22
•
2. Convert the following rectangular equations to polar equations and sketch the
graphs - you may find it instructive to use both rectangular and polar or
parametric modes on your calculator to check your answers:
(a) x 2 + y2 == a 2
(h)
y
=4
(c) 3x - Y + 2
=0
(d) y2 == 9x
3. Convert the polar equation to rectangular form and sketch its graph:
(a) r = 3
(b) r = sinO
= 3secO
(c) r
(d) r = 2(hcosO + k sin 0)
•
4. Use your graphing utility to graph the polar curve and find dy/dx at the given
value ofe:
(a) r
(h)
=3(1 -
cos 0), 0 =tr / 2
r =3 sin 0, () -=
tr / 3
5. Find all points of horizontal and vertical tangency to the curves:
(a) r = 1+sin6
(b) r = 2sin6
(c) r = 4sin6cos2
e
6. Sketch the graph ofthe polar equation and find the tangents at the pole:
•
(a)
(b)
(c)
r= 3sin6
r = 2cos36
r = 3sin26
Page 23
•
7. Find the area bounded by the graph of the polar equation by integration and say
why your answer looks reasonable (if it doesn't look reasonable, do it over until it
does!):
(a)
(b)
(e)
(d)
(e)
(f)
r = 8sin6
One petal of r = 2eos36
One petal of r = cos26
interior of r = l-sin6
Inner loop of r = I+cos6
Area between loops of r = I+2cos6
8. Find the points of intersection ofthe graphs below. Use your graphing utility to
help you:
(a)
(b)
•
r=3(1+sin6) and r=3(I-sin6)
r = 1 +cos6 and r = l-cos6
9. Find the area ofthe indicated region:
(a)
(b)
Common interior of r = 4sin26 and r = 2
Common interior of r = 3-2sin6 and r = -3+2sin6
10. Find the length of the graph over the indicated interval:
(a)
(b)
r = l+sin6 over [0, 21t]
r = 5(1+cos6) over [0, 21t]
11. Find the surface area formed by revolving the curve about the given line:
(a)
(b)
•
a
r =2cos6 about the x-axis, in [0, n/2]
r = a(l+oos6), 6 in [0, n], about x-axis
Page 24
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Page 32
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Page 25
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Page 33
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Page 26
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Page 34
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Page 27
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Page 35
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Page 28
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Page 36
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