Calculus: Analytic Geometry and Calculus, with Vectors

544 Polar, cylindrical, and spherical coordinates

Problems 10.29

1 Obtain the standard polar equation of the conic K and use it to sketch

the major and minor (or conjugate) axis of K when the eccentricity e and dis-

tance p from the focus to the directrix are

(a) a=-g,p=6 (b) e=2,p=3

2 Make a hasty sketch of the cardioid having the polar equation

p=1+cos4>.

Show that tan -(I + cos 4>)/sin 4> = - cot -4>. Calculate tan P when

0 = 7r/2 and when 0 = it and make any repairs in your figure that this informa-

tion may require.

3 Find the length of the cardioid of the preceding problem. Ans.: 8.

4 There is something unique about angles at which radial lines from the

origin intersect the exponential spiral having the polar equation p = e°o. What

is it? Ant.: The angles are all equal.

5 As ¢ increases from 0, the point Pr(p,4>) on the polar graph of p = e-0

spirals around the origin. Since it is about 3 and e3 is about 20, the point P

spirals toward the origin so rapidly that the length of the whole path may be not

much greater than the distance from the starting point to the origin. What are

the facts? Ant.: To try to preserve good ideas and perhaps create more, put the

matter this way: If 0 starts at time t = 0 and increases at a constant rate, the

point P must keep moving forever, but its speed decreases so rapidly that the

total distance traveled is always less than and only approaches as t -> oo

and ¢ -> oo. It makes sense to say that the total length of the path is 1/2.

6 Let C be the polar graph of p = f((P), where f(0) = 1, f(27r) = 2, and f is

continuous and monotone increasing over the interval 0 < 4 5 2ir. Try to

decide whether it is easy or difficult or impossible to prove that C must have

finite length.

7 The curve C of the preceding problem lies between the polar graphs of the

equations p = 1 and p = 2. Try to decide whether it is easy or difficult or

impossible to prove that the length of C lies between the lengths of the inner and

outer circles.

8 Let the displacement vector of a particle P at time t be

(1) r(t) = p(t)[cos4>(t)i + sin4>(t)j],

where it is supposed that p and 0 have two derivatives. Forgetting formulas

which we have derived but remembering rules for differentiating products, derive

the velocity and acceleration formulas

(2) v(t) = p'(t)[cos 4>(t)i + sin 0(t)j] + p(t)4>'(t)[- sin ¢(t)i + cos .0(t)j]

(3) a(t) _ [p"(t) - p(t)(4>'(t))2][cos 4>(t)i + sin 4>(t)j]

+ [p(t)4>"(t) + 2p'(t)4>'(t)][- sin 4>(t)i + cos 4,(t)jl.

Observe anew that the two vectors

(4) [cos 4>(t)i + sin 4>(t)j], [- sin 0(t)i + cos 0(t)j]