Let a curve be described by r = f(), where f() > 0 on its domain. Referring

Let a curve be described by r = f(θ), where f(θ) > 0 on its domain. Referring to the figure of Exercise 62, a curve is isogonal provided the angle φ is constant for all θ. 

a. Prove that φ is constant for all u provided cot φ = f'(θ)/f(θ) is constant, which implies that d/dθ (ln f(θ)) = k, where k is a constant.

b. Use part (a) to prove that the family of logarithmic spirals r = Ce^kθ consists of isogonal curves, where C and k are constants.

c. Graph the curve r = 2e^2θ and confirm the result of part (b).

Data from Exercise 62

Transcribed Image Text:

yA r= f(0) P(x, y) = P(r, 6)