Show that the curve r sin tan (called a cissoid of Diocles) has the line x 1 as a

12 Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r . 0 and one with r , 0.

12 Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r . 0 and one with r , 0.

34 Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.

34 Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.

56 The Cartesian coordinates of a point are given. (i) Find polar coordinates sr, d of the point, where r . 0 and 0 <  , 2. (ii) Find polar coordinates sr, d of the point, where r , 0 and 0 <  , 2.

56 The Cartesian coordinates of a point are given. (i) Find polar coordinates sr, d of the point, where r . 0 and 0 <  , 2. (ii) Find polar coordinates sr, d of the point, where r , 0 and 0 <  , 2.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

712 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

Find the distance between the points with polar coordinates s4, 4y3d and s6, 5y3d.

Find a formula for the distance between the points with polar coordinates sr1, 1d and sr2, 2 d.

1520 Identify the curve by finding a Cartesian equation for the curve.

1520 Identify the curve by finding a Cartesian equation for the curve.

1520 Identify the curve by finding a Cartesian equation for the curve.

1520 Identify the curve by finding a Cartesian equation for the curve.

1520 Identify the curve by finding a Cartesian equation for the curve.

1520 Identify the curve by finding a Cartesian equation for the curve.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2126 Find a polar equation for the curve represented by the given Cartesian equation.

2728 For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve

2728 For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

2946 Sketch the curve with the given polar equation by first sketching the graph of r as a function of  in Cartesian coordinates.

4748 The figure shows a graph of r as a function of  in Cartesian coordinates. Use it to sketch the corresponding polar curve

4748 The figure shows a graph of r as a function of  in Cartesian coordinates. Use it to sketch the corresponding polar curve

4748 The figure shows a graph of r as a function of  in Cartesian coordinates. Use it to sketch the corresponding polar curve

4748 The figure shows a graph of r as a function of  in Cartesian coordinates. Use it to sketch the corresponding polar curve

Show that the curve r sin  tan  (called a cissoid of Diocles) has the line x 1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0 < x , 1. Use these facts to help sketch the cissoid.

Sketch the curve sx 2 1 y 2 d 3 4x 2 y 2

(a) In Example 11 the graphs suggest that the limaon r 1 1 c sin  has an inner loop when | c | . 1. Prove that this is true, and find the values of  that correspond to the inner loop. (b) From Figure 19 it appears that the limaon loses its dimple when c 1 2. Prove this.

(a) In Example 11 the graphs suggest that the limaon r 1 1 c sin  has an inner loop when | c | . 1. Prove that this is true, and find the values of  that correspond to the inner loop. (b) From Figure 19 it appears that the limaon loses its dimple when c 1 2. Prove this.

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of .

6164 Find the points on the given curve where the tangent line is horizontal or vertical.

6164 Find the points on the given curve where the tangent line is horizontal or vertical.

6164 Find the points on the given curve where the tangent line is horizontal or vertical.

6164 Find the points on the given curve where the tangent line is horizontal or vertical.

Show that the polar equation r a sin  1 b cos , where ab 0, represents a circle, and find its center and radius.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

Show that the curves r a sin  and r a cos  intersect at right angles.

How are the graphs of r 1 1 sins 2 y6d and r 1 1 sins 2 y3d related to the graph of r 1 1 sin ? In general, how is the graph of r fs 2 d related to the graph of r fsd?

Use a graph to estimate the y-coordinate of the highest points on the curve r sin 2. Then use calculus to find the exact value.

Use a graph to estimate the y-coordinate of the highest points on the curve r sin 2. Then use calculus to find the exact value.

Investigate the family of polar curves r 1 1 cosn

Let P be any point (except the origin) on the curve r fsd. If  is the angle between the tangent line at P and the radial line OP, show that tan  r dryd [Hint: Observe that   2  in the figure.]

a) Use Exercise 77 to show that the angle between the tangent line and the radial line is  y4 at every point on the curve r e . (b) Illustrate part (a) by graphing the curve and the tangent lines at the points where  0 and y2. (c) Prove that any polar curve r fsd with the property that the angle  between the radial line and the tangent line is a constant must be of the form r Cek , where C and k are constants.