Question

Properties of the Gompertz solution Verify that the function 1\(M(t)=K\left(\frac{M_{0}}{K}\right)^{\operatorname{csp}(-n)}\) satisfies the properties \(M(0)=M_{0}\) and \(\lim _{t \rightarrow \infty} M(t)=K\)

Short answer

Question: Verify the given properties of the function \(M(t) = K(\frac{M_0}{K})^{\operatorname{csp}(-nt)}\), specifically that \(M(0) = M_0\) and \(\lim_{t \rightarrow \infty} M(t) = K\). Solution: We have successfully verified the two required properties of the given function. By substituting \(t = 0\) into the function, we found that \(M(0) = M_0\). Furthermore, by analyzing the behavior of the function as \(t\) tends to infinity, we determined that \(\lim_{t \rightarrow \infty} M(t) = K\). Therefore, both properties are satisfied.

Step-by-step solution

Substitute t = 0 into the function

First, let's write down the function we are given: $$M(t) = K\left(\frac{M_0}{K}\right)^{\operatorname{csp}(-nt)}$$ Now, we need to find the value of \(M(0)\). To do this, we substitute \(t = 0\) and simplify the expression: $$M(0) = K\left(\frac{M_0}{K}\right)^{\operatorname{csp}(-n\cdot 0)}$$

Simplify the expression for M(0)

Let's simplify the expression we obtained in the previous step: $$M(0) = K\left(\frac{M_0}{K}\right)^{\operatorname{csp}(0)}$$ Knowing that \(\operatorname{csp}(0)=1\), the expression becomes: $$M(0) = K\left(\frac{M_0}{K}\right)^1$$ Further simplification results in: $$M(0) = K\frac{M_0}{K}$$ Which cancels out the \(K\) values, yielding: $$M(0) = M_0$$ This verifies the first property, \(M(0) = M_0\).

Find the limit of M(t) as t approaches infinity

Next, let's find the limit as t approaches infinity: $$\lim_{t \rightarrow \infty} M(t) = \lim_{t \rightarrow \infty} K\left(\frac{M_0}{K}\right)^{\operatorname{csp}(-nt)}$$ To determine the result, we have to examine the behavior of the function when \(t\) tends to infinity.

Analyze the behavior of the function as t tends to infinity

Since \(n\) is not specified, we can consider it as a positive constant. Therefore, as \(t\) tends to infinity, \(-nt\) tends towards \(-\infty\). Thus, the argument of csp function becomes a very large negative number: $$\operatorname{csp}(-nt) \rightarrow \operatorname{csp}(-\infty)$$ The csp function can be approximated as \(e^{-x}\) for large \(x\). As such, when x becomes \(-\infty\), it can be expressed as: $$\operatorname{csp}(-\infty) \approx e^{-(-\infty)} = e^{\infty}$$

Calculate the limit of the function

Now that we have the behavior of the csp function as \(t\) tends to infinity, we can put it into our equation to get the limit: $$\lim_{t \rightarrow \infty} M(t) = \lim_{t \rightarrow \infty} K\left(\frac{M_0}{K}\right)^{e^{\infty}}$$ As \(e^{\infty}\) will tend towards infinity, the expression \((\frac{M_0}{K})^{e^{\infty}}\) will tend towards 0, since \(\frac{M_0}{K}<1\). This is because raising a fraction (between 0 and 1) to an infinitely large power will drive it towards 0. Therefore, we get the limit of the function as follows: $$\lim_{t \rightarrow \infty} M(t) = K\cdot 0$$ Which simplifies to: $$\lim_{t \rightarrow \infty} M(t) = K$$ This verifies the second property, \(\lim_{t \rightarrow \infty} M(t) = K\).

Key concepts

Calculus

Calculus is a significant branch of mathematics focused on the study of change and motion. Through its two major processes, differentiation and integration, it provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. In the context of the Gompertz solution, calculus is used to analyze the behavior of the function over time, especially as time approaches infinity. This is done through the evaluation of limits, which is a fundamental aspect of calculus. It allows us to understand the long-term behavior of growth processes described by the Gompertz function, which is often used in biological and demographic studies to model the growth of populations or tumors.

Limits

Limits in calculus are essential tools used to describe the behavior of functions as they approach a specific point or extend towards infinity. When evaluating limits, we investigate what value the function gravitates towards as the input gets arbitrarily close to a particular value. In the case of the Gompertz function, the limit process is utilized to verify mathematical properties by seeing what happens to the function as time grows without bound. Understanding limits is pivotal to confirming that as time goes to infinity \( t \rightarrow \infty \), the Gompertz function approaches the carrying capacity \( K \), denoting a plateau in growth in accordance with its designed behavior.

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent and typically have the form \( f(x) = a^{bx} \). These functions are known for their rapid growth or decay properties, and they commonly appear in various scientific disciplines, from compound interest calculations in finance to population studies in biology. The Gompertz solution incorporates an exponential function in the form of \( e^{-x} \), where \( x \), in this context, represents the passing of time multiplied by the growth rate negated. Exponential functions are crucial for understanding the Gompertz model as they describe how the population diminishes to a certain level over time, capturing the deceleration in growth as the population approaches the carrying capacity.

Mathematical Properties

Mathematical properties pertain to the characteristics and behaviors that certain equations or functions inherently possess. These properties can often be confirmed or derived using calculus, as seen with the Gompertz function. In our Gompertz solution example, verifying that the function satisfies the initial condition \( M(0) = M_0 \) and the long-term limiting behavior \( \lim_{t \rightarrow \infty} M(t) = K \) is a demonstration of understanding these mathematical properties. By confirming these, we assert that the model accurately depicts an initial population \( M_0 \) and tends toward a maximum capacity \( K \) over time, showing its applicability and utility in modeling scenarios with a saturating growth pattern.