Problem 10 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Compute solutions until they become negative or imaginary for the systems: a. $$ P_{0}=\pi / 4 \quad \text { b. } \quad P_{0}=5 $$ $$ P_{t+1}=\ln \left(\tan \left(P_{t}\right)\right) \quad P_{t+1}=P_{t}-1 $$ c. \(P_{0}=0.76 \quad\) d. \(\quad P_{0}=2\) $$ P_{t+1}=2 \sqrt{P_{t}}\left(1-P_{t}^{2}\right) \quad P_{t+1}=0.9 P_{t}-0.1 $$

Step-by-Step Solution

Answer

In parts (a), (c) can't be continued, and in (b), (d) results eventually turn negative.

Step 1: Understanding the Problem

We have multiple initial values \( P_0 \) and equations defining \( P_{t+1} \) based on \( P_t \). For each of these, we must compute the sequence until the values become negative or imaginary.

Step 2: Solving Part (a)

Start with \( P_0 = \frac{\pi}{4} \). Using the equation \( P_{t+1} = \ln(\tan(P_t)) \), compute:\- \( P_1 = \ln(\tan(\frac{\pi}{4})) = \ln(1) = 0 \).- \( P_2 = \ln(\tan(0)) = -\infty \). This value is illegal (as \( \ln(0) \) is undefined).

Step 3: Solving Part (b)

Start with \( P_0 = 5 \). Using \( P_{t+1} = \ln(\tan(P_t)) \), compute:\- \( P_1 = \ln(\tan(5)) \) which should be calculated until it becomes negative or imaginary. With \( P_{t+1} = P_t - 1 \),\- \( P_1=5-1=4 \), \( P_2=3 \), continuing until \( P_t < 0 \): \( P_5 = 1 \), \( P_6 = 0 \), \( P_7 = -1 \). Then stop.

Step 4: Solving Part (c)

Begin with \( P_0 = 0.76 \). Using \( P_{t+1} = 2 \sqrt{P_t}(1-P_t^2) \): \- Calculate \( P_1, P_2, \ldots \) until the result becomes negative. Example: Calculate\( P_1 \).

Step 5: Solving Part (d)

Start with \( P_0 = 2 \), using \( P_{t+1} = 0.9 P_t - 0.1 \), compute:\- \( P_1 = 0.9 \times 2 - 0.1 = 1.7 \).- Continue calculating until negative: \( P_2 = 1.43 \), \( P_3 = 1.187, \dots \). Once \( P_t < 0 \), stop.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(x) \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine function to the cosine function, that is, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
The tangent function is periodic with a period of \( \pi \), meaning it repeats every \( \pi \) units.
This function has vertical asymptotes where the cosine function equals zero, so it is undefined at odd multiples of \( \frac{\pi}{2} \).

Its graph extends from negative to positive infinity, flipping direction after each asymptote.
In practical applications, such as solving the given problem, understanding the range and behavior of the tangent function is crucial. For example, the tangent of \( \frac{\pi}{4} \) is 1, rendering \( \ln(1) = 0 \).

• Tangent is undefined for inputs like 0 in computing sequences.
• These peculiarities are why \( \ln(\tan(\text{input})) \) can quickly become problematic.

When using the tangent function in recurrence relations, it's important to note any potential discontinuities, as this may lead to undefined or infinite values.

Logarithm Function

The logarithm function \( \ln(x) \) is the inverse function of the exponential function \( e^x \). It allows one to unwind exponential growth.
When examining sequences, it is frequently used due to its property of transforming multiplication into addition, which simplifies complex multiplier-based problems.

In mathematics, particularly when dealing with sequences and series, logarithms are instrumental for expressing growth rates and scaling factors.

• \( \ln(1) = 0 \)
• \( \ln(x) \) is undefined for \( x \leq 0 \)

When computing sequences that involve logarithms of functions like the tangent, extreme caution must be taken.
Negative or zero values as input lead to undefined results, instantly breaking the sequence.
This nature of the logarithm is crucial in ensuring valid computations, especially in mathematical models or systems.

Sequence Convergence

Sequence convergence refers to the behavior of a sequence over time as it approaches a specific limit.
When we calculate sequences using recurrence relations, we often assess if they converge to a particular value or diverge to infinity. Convergence describes sequences that settle down to a single point.

Some characteristics of convergence include:

• It guarantees that from a certain point onwards, the successive terms of the sequence become arbitrarily close to each other.
• In our context, sequences must be analyzed to determine if and when they hit negative values or become imaginary.

In the problem, convergence and divergence help decide when to terminate sequence computations.
Little changes like adopting different initial conditions or rules can markedly change the convergence behavior, guiding us on where sequences stabilize.

Initial Conditions

Initial conditions are the starting values of a sequence, providing the base case for recursion in recurrence relations.
In sequences like the ones in the exercise, the choice of initial condition \( P_0 \) determines the entire subsequent sequence.

• Determine the possible convergence or divergence based on initial conditions.
• Each choice of \( P_0 \) leads to a different path of computation.

In problems involving recurrence relations, testing various initial conditions might be critical to finding workable and stable solutions.
Not only do they affect convergence, but they also influence values that define the behavior of the sequence over its course.