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Calc 2 9.3

 

<== Calculus 2
Number Question Answer
See the work

1.

 

 

 

 

 

 

Given any Cartesian coordinates, (x,y), there are polar coordinates (r,	heta) with -pi/2 < (theta) < pi/2 .
Find polar coordinates with -pi/2 < (theta) < pi/2 for the following Cartesian coordinates:
(a) If (x,y) = (2, 9) then (r, (theta)) =  ( , ),
(b) If (x,y) = (17, 4) then (r, 	heta) =  ( , ),
(c) If (x,y) = (-9, 5) then (r, 	heta) =  ( , ),
(d) If (x,y) = (13, 10) then (r, 	heta) =  ( , ),
(e) If (x,y) = (-2, 9) then (r, 	heta) =  ( , ),
(f) If (x,y) = (0, -2) then (r, 	heta) =  ( , ).

 

(a) (r, 	heta) =  ( 9.22 , 1.35 )
(b) (r, 	heta) =  ( 17.46 , 0.23 )
(c) (r, 	heta) =  ( see work , -0.51 )
(d) (r, 	heta) =  ( 16.40 , 0.66 )
(e) (r, 	heta) =  ( see work , -1.35 )
(f) (r, 	heta) =  ( -2 , see work )
The work

Point Cost: 2

2.

 

 

 

 

 

A curve in polar coordinates is given by: r = 7 + 2 \cos 	heta.
Point P is at 	heta = rac{16 \pi}{14}.
1) Find polar coordinate r for P, with r > 0 and pi < (theta)< 3pi/2 .
r =
2) Find cartesian coordinates for point P.
x = , y =
3) How may times does the curve pass through the origin when 0 < (theta) < 2\pi ?

 

1) r = 5.1981

2) x = -4.68, y = -2.255

3) 0

 

The work

Point Cost: 2

3.

 

 

 

 

 

A curve with polar equation

r=12/(3 sin(theta)+22cos(theta))
represents a line. This line has a Cartesian equation of the form
y = mx + b ,where m and b are constants.


 

y=4-(22/3)x

The work

Point Cost: 3

4.

 

 

 

 

The equation

r = 4sintheta

represents a circle.
Find the cartesian coordinates of the center.
x= ______
y= ______
r= ______

x= 0
y= 2
r= 2
The work

Point Cost: 2

5.

 

 

 

 

 

The equation

r = 4cos(theta)
represents a circle.
Find the cartesian coordinates of the center.
x= ______
y= ______
r= ______

 

x= 2
y= 0
r= 2

 

The work

Point Cost: 2

6.

 

 

 

 

The equation r = 4 sin (theta) + 4 cos (theta) represents a circle.
The cartesian equation has the form:
(x-A)^2 + (y-B)^2 = R^2
where
A =
B =
R =

A = 2

B = 2

R = 2.82

 

The work

Point Cost: 3

7.

 

 

 

Find a cartesian equation for the polar curve:

a) r = 5 \csc 	heta
Cartesian equation:   y =

b) r =  -3  	an 	heta \sec 	heta
Cartesian equation:   y =

a) y= 5

b) y= see the work

The work

Point Cost: 3

8.

 

 

 

A circle C has center at the origin and radius 9. Another circle K has a diameter with one end at the origin and the other end at the point (0,16). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let (r, (theta) ) be the polar coordinates of P, chosen so that r is positive and 0 < (theta) < 2. Find r and (theta).
r =

	heta =
r = 9

	heta = 0.5974
The work

Point Cost: 4

9.

 

 

 

Find a polar equation for the curve represented by the given Cartesian equation:

a) 6x + y = 4

Write the answer in the form r = f(t) where t stands for (theta).
Polar equation:   r =

b) x^2 - y^2 = 2

Write the answer in the form r^2 = f(t) where t stands for 	heta.
Polar equation:   r^2  =

r =4/(6cos(t)+1sin(t)
r^2  =2sec(2t)

 

The work

Point Cost: 2

10.

 

 

 

Find the slope of the tangent line to the polar curve r = sin (2(theta)) at (theta) = pi/4.

slope =

-1
The work

Point Cost: 3

11.

 

Use calculus to determine the exact y-coordinate of the highest points on the curve r=sin(2(theta)).

y-coordinate of highest point:________

0.7698

The work

Point Cost: 4

12.

 

 

 

 

 

If you have the polar curve r=e^(theta).

(a) List all of the points (r,(theta) where the tangent line is horizontal.
Only find (theta)'s in the range (0<=theta<=2pi)
Find the r per corrisponding (theta) value.

Point 1: (r,	heta) = , )

Point 2: (r,(theta)) = , )


(b) List all of the points (r,theta where the tangent line is vertical.
Only find (theta)'s in the range (0<=theta<=2pi)
Find the r per corrisponding (theta) value.

Point 1: (r,theta) = , )

Point 2: (r,theta) = , )

 

a)

Point 1: (r,theta) = 10.551, 2.356 )
Point 2: (r,theta) = 244.151, 5.498 )


b)

Point 1: (r,theta) = 2.193, 0.785 )
Point 2: (r,theta) = 50.754, 3.927 )

The work

Point Cost: 3