Solution Found!
Finger curves Consider the curve r = f() = cos (a) - 1.5,
Chapter 8, Problem 100E(choose chapter or problem)
Consider the curve \(r=f(\theta)=\cos(a^{\theta})-1.5,\ \text{where}\ a=(1+12\pi)^{1/2\pi}\ \approx\ 1.78933\) (see figure).
a. Show that \(f(0)=f(2\pi)\) and find the points on the curve that correspond to \(\theta=0\ \text{and}\ \theta=2\pi\).
b. Is the same curve produced over the intervals \([-\pi,\ \pi]\ \text{and}\ [0,\ 2\pi]\)?
c. Let \(f(\theta)=\cos (a^{\theta})-b,\ \text{where}\ a=(1+2k\pi)^{1/2\pi}\), k is an integer, and b is a real number. Show that \(f(0)=f(2\pi)\) and the curve closes on itself.
d. Plot the curve with various values of k. How many fingers can you produce?
Questions & Answers
QUESTION:
Consider the curve \(r=f(\theta)=\cos(a^{\theta})-1.5,\ \text{where}\ a=(1+12\pi)^{1/2\pi}\ \approx\ 1.78933\) (see figure).
a. Show that \(f(0)=f(2\pi)\) and find the points on the curve that correspond to \(\theta=0\ \text{and}\ \theta=2\pi\).
b. Is the same curve produced over the intervals \([-\pi,\ \pi]\ \text{and}\ [0,\ 2\pi]\)?
c. Let \(f(\theta)=\cos (a^{\theta})-b,\ \text{where}\ a=(1+2k\pi)^{1/2\pi}\), k is an integer, and b is a real number. Show that \(f(0)=f(2\pi)\) and the curve closes on itself.
d. Plot the curve with various values of k. How many fingers can you produce?
ANSWER:
Solution 100EGiven r = f() = cos (a) - 1.5, where a = (1 + 12)1/2 1.78933 Step 1:(a). Show that f(0) = f(2) and find the points on the curve that correspond to = 0 and = 2.We can find the values of and Therefore f(0) = f(2)The points on the curve will be written in the form Thus the point on the curve that correspond to = 0 is And the point on the curve that correspond to is .