Question

Steady states If a function f represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) 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 \(p(t)=\frac{3500 t}{t+1}\)

Short answer

If so, what is the steady-state value? Answer: Yes, a steady state exists. The steady-state value is 3500.

Step-by-step solution

Write down the given function

The given function, which represents the population of a culture of tumor cells, is: \(p(t) = \frac{3500t}{t + 1}\)

Calculate the limit of the function as t approaches infinity

To find the steady-state value, we need to find the limit of the function as \(t\) approaches infinity. In other words, we need to evaluate the following expression: \(\lim_{t \rightarrow \infty} \frac{3500t}{t + 1}\) Using L'Hôpital's Rule, since we have the form \(\frac{\infty}{\infty}\), we take the derivative of the numerator and the derivative of the denominator and then evaluate the limit again: \(\lim_{t \rightarrow \infty} \frac{\frac{d}{dt}(3500t)}{\frac{d}{dt}(t+1)}\) The derivative of the numerator with respect to \(t\) is: \(\frac{d}{dt}(3500t) = 3500\) The derivative of the denominator with respect to \(t\) is: \(\frac{d}{dt}(t + 1) = 1\) So, the limit becomes: \(\lim_{t \rightarrow \infty} \frac{3500}{1}\) Now, we can directly evaluate the limit of this expression as it is a constant value.

Evaluate the limit and determine the steady-state value

Since the transformed limit is a constant value, we don't need to evaluate for a specific value of \(t\). The limit is simply the constant value itself: \(\lim_{t \rightarrow \infty} \frac{3500t}{t + 1} = \frac{3500}{1} = 3500\) The steady-state value exists, and it is equal to 3500.

Conclude the result

In conclusion, a steady state exists for the given system representing the population of a culture of tumor cells. The steady-state value is 3500.

Key concepts

Limit of a Function

Understanding the limit of a function is crucial when analyzing how a function behaves as its input approaches a certain value. In calculus, limits help us grasp the concept of something that approaches a value but doesn't necessarily reach it. For example, in the given exercise, we are interested in what happens to the population of tumor cells, represented by the function \(p(t)=\frac{3500 t}{t+1}\), as time \(t\) goes to infinity.

By examining the limit as \(t\) approaches infinity, we seek to find the population size that the function 'settles into' over a long period. If the limit exists and is finite, it represents a steady state or equilibrium value for the tumor cell population. In this case, the steady state value is 3500, meaning no matter how far into the future we project, the population will approach but not exceed this number.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate the limits of functions that result in an indeterminate form, such as \(\frac{\infty}{\infty}\). When faced with such forms, directly substituting the limit value does not yield a meaningful result. L'Hôpital's Rule states that under certain conditions, if the limit of the ratio of the derivatives of the numerator and denominator exists, it is equal to the limit of the ratio of the original functions.

In our exercise, we applied L'Hôpital's Rule to determine the steady state value of the tumor cell population as \(t\) approaches infinity. By taking the derivatives of the numerator and denominator and then applying the rule, we simplified the calculation, finding a finite limit and thus, the steady state.

System Equilibrium

System equilibrium, or steady state, in the context of dynamic systems such as population growth, is the condition where the system's variables are unchanging over time. In terms of mathematics, it's when the rate of change equals zero or when the function representing that system has reached a constant value.

In the exercise regarding tumor cell population, we discovered that when the limit of the function as \(t\) approaches infinity exists and is a constant value, it represents the system's equilibrium. The steady-state value was determined to be 3500, indicating that the cell population will stabilize at this number, given sufficient time and no external changes to the growth conditions.

Infinite Limits

The concept of infinite limits refers to the value a function approaches as the dependent variable moves towards infinity. If the output of the function grows without bound as the input becomes very large, we say that the function has an infinite limit. However, if the function approaches a specific finite number, as is the case with the tumor cell population, the limit is not infinite, but finite.

The notion of infinite limits is used to describe unbounded growth or decline, but when a limit leads to a specific constant, as seen in the current exercise, it shows that the function will even out at that steady value, which is an indication of equilibrium, not extensive growth or decay.