Problem 39 - Calculus with Concepts in Calculus

Question

For each of the given values of \(a\), calculate the iterates $$ a, \cos a, \cos (\cos a), \ldots $$ until the first three displayed digits do not change. From your results, make a conjecture about the behavior of the iterates for any real number \(a\). a. \(a=1\) b. \(a=12\) c. \(a=100\) d. \(a=-5\)

Step-by-Step Solution

Answer

For any real number \( a \), the iterates converge to approximately 0.739.

Step 1: Calculate Iterates for a = 1

Start with the initial value of \( a = 1 \). Calculate the first iterate: \( \cos(1) \). Using a calculator, \( \cos(1) \approx 0.5403 \). Calculate the next iterate: \( \cos(0.5403) \approx 0.8576 \). Continue calculating until the iterates stabilize to a value with the first three digits not changing: \( 0.8576 \rightarrow 0.6543 \rightarrow 0.7937 \rightarrow 0.7074 \rightarrow 0.7645 \rightarrow 0.7221 \rightarrow 0.7504 \rightarrow 0.7318 \rightarrow 0.7440 \rightarrow 0.7356 \rightarrow 0.7417 \rightarrow 0.7375 \rightarrow 0.7403 \rightarrow 0.7385 \rightarrow 0.7395 \rightarrow 0.7389 \rightarrow 0.7393 \approx 0.739 \). This approximate stabilization occurs around \( 0.739 \).

Step 2: Calculate Iterates for a = 12

Start with the initial value of \( a = 12 \). Calculate the first iterate: \( \cos(12) \). Using a calculator, \( \cos(12) \approx 0.8439 \). Calculate the next iterate: \( \cos(0.8439) \approx 0.6702 \). Continue calculating until stabilization: \( 0.6702 \rightarrow 0.7821 \rightarrow 0.7107 \rightarrow 0.7647 \rightarrow 0.7221 \rightarrow 0.7503 \rightarrow 0.7320 \rightarrow 0.7438 \rightarrow 0.7357 \rightarrow 0.7416 \rightarrow 0.7375 \rightarrow 0.7403 \rightarrow 0.7386 \rightarrow 0.7394 \rightarrow 0.7389 \rightarrow 0.7393 \approx 0.739 \). Stabilization occurs around \( 0.739 \).

Step 3: Calculate Iterates for a = 100

Start with the initial value of \( a = 100 \). Calculate the first iterate: \( \cos(100) \). Using a calculator, \( \cos(100) \approx -0.8623 \). Calculate the next iterate: \( \cos(-0.8623) \approx 0.6526 \). Continue calculating: \( 0.6526 \rightarrow 0.7936 \rightarrow 0.7074 \rightarrow 0.7645 \rightarrow 0.7221 \rightarrow 0.7504 \rightarrow 0.7318 \rightarrow 0.7439 \rightarrow 0.7356 \rightarrow 0.7418 \rightarrow 0.7375 \rightarrow 0.7403 \rightarrow 0.7385 \rightarrow 0.7395 \rightarrow 0.7389 \rightarrow 0.7393 \approx 0.739 \). Stabilization occurs around \( 0.739 \).

Step 4: Calculate Iterates for a = -5

Start with the initial value of \( a = -5 \). Calculate the first iterate: \( \cos(-5) \). Using a calculator, \( \cos(-5) \approx 0.2837 \). Calculate the next iterate: \( \cos(0.2837) \approx 0.9609 \). Continue calculating: \( 0.9609 \rightarrow 0.5728 \rightarrow 0.8403 \rightarrow 0.6702 \rightarrow 0.7803 \rightarrow 0.7106 \rightarrow 0.7648 \rightarrow 0.7220 \rightarrow 0.7503 \rightarrow 0.7320 \rightarrow 0.7438 \rightarrow 0.7357 \rightarrow 0.7416 \rightarrow 0.7375 \rightarrow 0.7403 \rightarrow 0.7385 \rightarrow 0.7395 \rightarrow 0.7389 \rightarrow 0.7393 \approx 0.739 \). Stabilization occurs around \( 0.739 \).

Step 5: Conjecture: Behavior of Iterates

In each of the cases (regardless of whether \(a\) was positive, negative, or large), the iterates converged to approximately \( 0.739 \). This suggests that for any real number \( a \), the iterates of \( \cos \) seem to converge to a fixed point that is approximately equal to \( 0.739 \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often denoted as \( \cos(x) \). It relates the angle \( x \) in a right-angled triangle to the length of the adjacent side over the hypotenuse. This function is not only limited to triangles but extends its application in analyzing periodic phenomena like waves.
The cosine function is periodic with a cycle of \( 2\pi \), meaning \( \cos(x) = \cos(x + 2\pi) \). This property is crucial when exploring iterative methods since the convergence might depend on these repeating values.
In this exercise, the cosine function is used iteratively, which means we apply it repeatedly. This iterative approach allows us to explore how applying cosine several times can lead the results towards a steady value. Interestingly, this behavior is the foundation of fixed point iteration methods.

Convergence

Convergence in mathematical terms refers to the idea of approaching a particular point as a sequence progresses. In simpler words, as you calculate more steps or terms of a sequence, those values will get closer and closer to a specific limit or number. This limit is what we call the point of convergence.
In the context of the exercise, the iterates of the cosine function seem to stabilize at a particular value regardless of the starting point '\( a \)'. This consistent result, approximately \( 0.739 \), suggests that we are dealing with a convergent sequence. Understanding convergence is crucial as it indicates stability and predictability in mathematical operations and computations, such as those seen in iterative methods.

Fixed Point Iteration

Fixed point iteration is an iterative method used to find a stable point \( x \) such that \( f(x) = x \). This involves repeated application of a function to potentially arrive at a value that does not change upon further applications of the function.
In our case, the function is \( \cos(x) \). The iteration process starts with a chosen value of \( a \), and through the sequence \( a, \cos(a), \cos(\cos(a)), \ldots \), we iteratively compute the cosine until the results do not change beyond three decimal places.
The convergence to about \( 0.739 \) suggests there is a fixed point for the cosine function near this value. This is a fascinating outcome because it implies a simple trigonometric function, through repeated application, naturally seeks out a steady solution.

Mathematical Conjecture

A mathematical conjecture is an educated guess or a hypothesis that is based on observations. In mathematics, conjectures often state a general pattern or relationship that seems true but has not yet been formally proven.
From the exercise, the consistent convergence to the value \( 0.739 \) across various initial values of \( a \) leads to a conjecture: regardless of the real number initial value, iterates of the cosine function converge to a fixed number, approximately \( 0.739 \).
This observation begs for further exploration and formal proof, as it fuels curiosity about the properties of trigonometric functions and their behavior under iteration. Conjectures can evolve into theorems once they are proven, making them a vital part of mathematical discovery and understanding.