Question

Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?

Short answer

#Answer#: a. For both θ=0 and θ=2π, the corresponding points on the curve are (r, θ) = (cos(1) - 1.5, 0) and (r, θ) = (cos(1) - 1.5, 2π). b. The curve is not the same for the intervals [-π, π] and [0, 2π]. c. For the generalized function f(θ) = cos a^θ - b, with a = (1 + 2kπ)^(1/(2π)), we have that f(0) = f(2π), meaning the curve closes on itself. d. The number of "fingers" depends on k value and can be determined empirically through plotting or analyzing the function's behavior.

Step-by-step solution

a. Checking \(f(0)\) and \(f(2\pi)\), and finding their corresponding points on the curve.

To show that \(f(0) = f(2\pi)\), simply plug in the values of \(\theta\) into the given function: $$f(0) = \cos a^{0} - 1.5 = \cos(1) - 1.5$$ $$f(2\pi) = \cos a^{2\pi} - 1.5 = \cos(1 + 12\pi) - 1.5$$ Since \(\cos(2n\pi) = 1\) for any integer \(n\), we have: $$f(2\pi) = \cos(1 + 2\cdot 6 \pi) - 1.5 = \cos(1) - 1.5$$ Therefore, \(f(0) = f(2\pi)\). Now, let's find the corresponding points for these angles: For \(\theta = 0\): \(r = \cos(1) - 1.5\), and the point is \((r, \theta) = (\cos(1) - 1.5, 0)\). For \(\theta = 2\pi\): \(r = \cos(1) - 1.5\), and the point is \((r, \theta) = (\cos(1) - 1.5, 2\pi)\).

b. Checking the curve for the intervals \([-\pi, \pi]\) and \([0, 2\pi]\).

To check if the curve is the same for both intervals, we can evaluate the function for an angle \(\theta + \pi\) and compare it to the function evaluated at the original angle, \(\theta\). We have: $$f(\theta + \pi) = \cos a^{\theta + \pi} - 1.5 = \cos((1+12\pi)^{\frac{1}{2\pi}})^\theta (1+12\pi)^{\frac{1}{2}} - 1.5$$ Which does not simplify further. Therefore, the curve is not the same for both intervals, as it depends on the angle through the exponential function.

c. Generalizing \(f(\theta)\) and checking for curve closure.

Let \(f(\theta) = \cos a^{\theta} - b\), with \(a = (1 + 2k\pi)^{\frac{1}{2\pi}}\). To show that \(f(0) = f(2\pi)\), plug in the values of \(\theta\) into the function: $$f(0) = \cos a^{0} - b = \cos(1) - b$$ $$f(2\pi) = \cos a^{2\pi} - b = \cos(1 + 2k\pi) - b$$ Since \(\cos(2n\pi) = 1\) for any integer \(n\), we have: $$f(2\pi) = \cos(1 + 2k\pi) - b = \cos(1) - b$$ Therefore, \(f(0) = f(2\pi)\), which means the curve closes on itself.

d. Plotting the curve with various values of \(k\) and counting the number of fingers.

By plotting the curve using various values of \(k\), we can observe how the number of "fingers" created by the curve changes. A systematic approach is to start with \(k = 0\) and increment \(k\) until it is clear that no new fingers will be created. The number of fingers are found from the plots. While there is no mathematical formula for the number of fingers, the empirical results can be found through plotting or by analyzing the function's behavior.

Key concepts

Parametric Equations

A polar curve can be described using parametric equations, where the radius \(r\) depends on an angle \(\theta\). For the curve \(r = f(\theta) = \cos(a^{\theta}) - 1.5\), the relationship between \(r\) and \(\theta\) defines how the curve spirals or forms specific shapes on the graph.

Parametric equations often involve parameters like \(a\), which affect how the curve behaves. In our function, we have \(a = (1 + 12\pi)^{1/(2\pi)}\). This parameter influences the rate of growth or reduction in the radius as the angle \(\theta\) changes.

With polar curves, it is crucial to understand that these equations do not distinctly define \(x\) and \(y\) as in Cartesian coordinates. Instead, they define a curve based on a distance from origin and an angle. This allows exploration of more complex curves that are not easily represented in a standard Cartesian equation form.

Function Analysis

Function analysis involves examining the behavior and properties of functions to understand their characteristics. For the given exercise, analyzing \(f(\theta) = \cos(a^{\theta}) - 1.5\) helps to determine points of interest such as continuity and symmetry.

One key aspect is verifying function values at different points, such as when \(\theta = 0\) and \(\theta = 2\pi\). By analyzing these points, we can confirm certain properties, like closure of the curve, meaning it repeats its pattern over specific intervals.

In polar coordinates, this is particularly interesting because it indicates that the curve starts and ends at the same point, showing periodicity. Analyzing intervals such as \([0, 2\pi]\) and \([−\pi, \pi]\) reveals whether the curve remains unchanged or transforms across these intervals. This in-depth check helps to define the nature and periodicity of the curve.

Trigonometric Functions

Trigonometric functions like cosine play a significant role in defining polar curves. In our exercise, \(\cos(a^{\theta}) - 1.5\) is the controlling function defining radius based on angle \(\theta\).

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. The usage of \(\cos\), which repeats every \(2\pi\), ensures that certain radii repeat, producing symmetrical and possibly closed curves, like circles or spirals.

Furthermore, trigonometric properties like \(\cos(2n\pi) = 1\) for any integer \(n\), simplify solving problems like proving \(f(0) = f(2\pi)\). Recognizing these properties makes understanding and predicting the behavior of curves easier.

• This periodic nature can create repetitive patterns in plots, known as 'fingers', which can be empirically determined by varying the parameter \(k\).
• The manipulation and adjustment of trigonometric functions within parametric equations form the basis of many complex and beautiful curve designs in mathematics.