12.6. EXERCISES 269

>>t=[0:.01:7*pi];x=(1+2*sin(5*t)).*cos(t);y=(1+2*sin(5*t)).*sin(t);plot(x,y,’LineWidth’,2,’color’,’green’)axis equalWhen you do this, and press enter, you get

-2 0 2

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12.6 Exercises1. Suppose r = a

1+ε sinθwhere ε ≥ 0. By changing to rectangular coordinates, show that

this is either a parabola, an ellipse or a hyperbola. Determine the values of ε whichcorrespond to the various cases.

2. In Example 12.1.2 suppose you graphed it for θ ∈ [0,kπ] where k is a positive integer.What is the smallest value of k such that the graph will start at (3,0) and end at (3,0)?

3. Suppose you were to graph r = 3+ sin(m

n θ)

where m,n are integers. Can you givesome description of what the graph will look like for θ ∈ [0,kπ] for k a very largepositive integer? How would things change if you did r = 3+ sin(αθ) where α isan irrational number?

4. Graph r = 1+ sinθ for θ ∈ [0,2π].

5. Graph r = 2+ sinθ for θ ∈ [0,2π].

6. Graph r = 1+2sinθ for θ ∈ [0,2π].

7. Graph r = 2+ sin(2θ) for θ ∈ [0,2π].

8. Graph r = 1+ sin(2θ) for θ ∈ [0,2π].

9. Graph r = 1+ sin(3θ) for θ ∈ [0,2π].

10. Graph r = sin(3θ)+2+ cos(3θ) for θ ∈ [0,2π] .

11. Find the area of the bounded region determined by r = 1+ sin(3θ) for θ ∈ [0,2π].

12. Find the area inside r = 1+ sinθ and outside the circle r = 1/2.

13. Find the area inside the circle r = 1/2 and outside the region defined by r = 1+sinθ .