Problem 71 - Calculus Early Transcendentals

Question

Steady states If a function \(f\) represents a system that varies in time, the existence of \(\lim f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value. The population of a culture of tumor cells is given by \(p(t)=\frac{3500 t}{t+1}\)

Step-by-Step Solution

Answer

Answer: The steady-state value of the population of the tumor cell culture system is 3500.

Step 1: Write down the given function

We are given the function \(p(t)=\frac{3500 t}{t+1}\) representing the population of the tumor cell culture.

Step 2: Find the limit of the function as t approaches infinity

To determine if the system has a steady state, we need to find the limit of the function as \(t\) approaches infinity. \(\lim_{t\to\infty} p(t) = \lim_{t\to\infty} \frac{3500 t}{t+1}\)

Step 3: Apply L'Hôpital's Rule

Since we have an indeterminate form \(\frac{\infty}{\infty}\), we can apply L'Hôpital's Rule. Take the derivatives of the numerator and the denominator with respect to \(t\): \(\frac{d}{dt}(3500t) = 3500\) \(\frac{d}{dt}(t + 1) = 1\) Now, find the limit of the new fraction: \(\lim_{t\to\infty} \frac{3500}{1} = 3500\)

Step 4: Interpret the result

The limit of the function exists, and the value is 3500. This means that the system reaches a steady state as time goes to infinity. The steady-state value of the population of the tumor cell culture is 3500.

Key Concepts

L'Hôpital's Rule

When dealing with limits that result in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), L'Hôpital's Rule is a valuable tool. This rule allows us to differentiate the numerator and the denominator independently, and then reevaluate the limit.

In the context of our problem where we have the function \(p(t) = \frac{3500 t}{t+1}\), applying L'Hôpital's Rule helps us to resolve the indeterminate form \(\frac{\infty}{\infty}\).

• First, we find the derivative of the numerator, \(3500t\), which is \(3500\).
• Then, we find the derivative of the denominator, \(t+1\), which is \(1\).
• By applying L'Hôpital's Rule, the limit becomes \(\lim_{t\to\infty} \frac{3500}{1} = 3500\).

This method helps simplify complex expressions and makes finding limits far more straightforward.

Limit of a Function

The concept of the limit of a function is a foundational idea in calculus that helps in understanding the behavior of functions as the input variable approaches a particular value.

In this exercise, we are interested in the limit of \(p(t) = \frac{3500t}{t+1}\) as \(t\) approaches infinity.

• Finding the limit helps us see if the function reaches a consistent value as \(t\) increases.
• For \(p(t)\), we use the limit to determine if there is a steady state for the tumor cell population.
• By calculating \(\lim_{t\to\infty} \frac{3500t}{t+1}\), we find a constant value of 3500.

This means the population grows and stabilizes at 3500 as time goes on.

Equilibrium in Calculus

Equilibrium, or steady state, is a condition where a system becomes stable over time, and the variables stop changing, reaching a consistent value.

In calculus, finding the limit of a function as the input variable approaches a specific point tells us if the system reaches an equilibrium.

• For the given function \(p(t) = \frac{3500 t}{t+1}\), a steady state implies the population stops growing and maintains a stable number.
• When calculating \(\lim_{t\to\infty} p(t)\), the result, 3500, is the equilibrium value.
• This equilibrium represents a balance point in the system, where external changes no longer affect the outcome significantly.

Understanding equilibrium in calculus is crucial for predicting long-term behavior in dynamic systems. It provides insights into how systems respond to large time scales.