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Chapter 8: Problem 34

Verify that the function $$M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ satisfies the properties \(M(0)=M_{0}\) and \(\lim _{t \rightarrow \infty} M(t)=K\).

Short Answer

Expert verified
Answer: Yes, the function satisfies both properties.

Step by step solution

01

Verify M(0) = M_{0}

Start by plugging t=0 into the function and simplifying. $$M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ $$M(0)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r\cdot0)}$$ Since any number raised to the power of 0 is equal to 1: $$M(0)=K\left(\frac{M_{0}}{K}\right)^{1}$$ $$M(0)=KM_{0}/K$$ It is clear that: $$M(0)=M_{0}$$
02

Verify lim_(t->鈭�) M(t) = K

Now, we look at the limit as t approaches infinity. $$\lim _{t \rightarrow \infty} M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$$ As t approaches infinity, the expression inside the exponential function (-rt) approaches negative infinity, provided that r>0. $$\exp (-\infty)=0$$ Therefore: $$\lim _{t \rightarrow \infty} M(t)=K\left(\frac{M_{0}}{K}\right)^{0}$$ Since any number raised to the power of 0 is equal to 1: $$\lim _{t \rightarrow \infty} M(t)=K$$ Thus, we have shown that the given function satisfies both properties, M(0)=M_{0} and \(\lim _{t \rightarrow \infty} M(t)=K\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Function
The limit of a function explores how a function behaves as the input approaches a particular value, often infinity. For the function \(M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}\), we are interested in the behavior of \(M(t)\) as \(t\) approaches infinity. This means we're determining what value \(M(t)\) tends towards as \(t\) grows larger.
  • When we considered the limit \(\lim_{t \rightarrow \infty} M(t)\), it's essential to note that \(\exp(-rt)\) decreases towards 0 as \(t\) increases if \(r > 0\).
  • Why? Because \(-rt\) becomes very negative, and the exponential of a large negative number is close to zero. So the term turns into \(\left(\frac{M_{0}}{K}\right)^{0} = 1\).
Finally, we find that \(\lim_{t \rightarrow \infty} M(t) = K\), indicating that the function levels off at \(K\) as time goes on. This behavior is typical in models where growth naturally caps at a maximum level, representing a stable state.
Initial Conditions
Initial conditions are critical in solving differential equations or verifying properties of a function. They provide values at specific points which help to determine parameters.
  • In this problem, we verify the initial condition by substituting \(t=0\) into \(M(t)\) to see if itequals \(M_0\).
  • Setting \(t=0\) gives us \(M(0)=K\left(\frac{M_{0}}{K}\right)^{\exp(-r\cdot0)}\).
Since \(\exp(0) = 1\), we simplify this to \(M(0) = K \cdot \frac{M_{0}}{K} = M_{0}\). Thus, the function satisfies the initial condition \(M(0) = M_{0}\). Initial conditions like these allow us to check the validity of model functions in representing real world scenarios.
Power Properties
Power properties are essential tools involving exponents and are widely used in simplifying expressions.
  • The property we use often is that any number raised to the zero power is 1. In algebra, \(x^0 = 1\).
  • This is crucial in the verification where \(\left(\frac{M_{0}}{K}\right)^{0} = 1\), which helps us confirm the limit condition \(\lim_{t \rightarrow \infty} M(t) = K\).
Understanding these power properties allows us to navigate through complex mathematical expressions and analyze their behavior. Having a solid grasp of these concepts reduces errors and enhances comprehension when handling exponential functions. Exponential equations frequently rely on these basic properties to evaluate limits or solve for unknowns.

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Most popular questions from this chapter

Use a calculator or computer program to carry out the following steps. a. Approximate the value of \(y(T)\) using Euler's method with the given time step on the interval \([0, T]\). b. Using the exact solution (also given), find the error in the approximation to \(y(T)\) (only at the right endpoint of the time interval). c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to \(y(T)\). d. Compare the errors in the approximations to \(y(T)\). $$\begin{array}{l}y^{\prime}(t)=6-2 y, y(0)=-1 ; \Delta t=0.2, T=3; \\\y(t)=3-4 e^{-2 t}\end{array}$$

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem $$P^{\prime}(t)=r P\left(1-\frac{P}{K}\right), \quad P(0)=P_{0}$$ $$\text { is } P(t)=\frac{K}{\left(\frac{K}{P_{0}}-1\right) e^{-n}+1}$$

Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let \(r\) be the natural growth rate, \(K\) the carrying capacity, and \(P_{0}\) the initial population. $$r=0.4, K=5500, P_{0}=100$$

A differential equation of the form \(y^{\prime}(t)=f(y)\) is said to be autonomous (the function \(f\) depends only on \(y\) ). The constant function \(y=y_{0}\) is an equilibrium solution of the equation provided \(f\left(y_{0}\right)=0\) (because then \(y^{\prime}(t)=0\) and the solution remains constant for all \(t\) ). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for \(t \geq 0\). c. Sketch the solution curve that corresponds to the initial condition \(y(0)=1\). $$y^{\prime}(t)=\sin y$$

The following models were discussed in Section 1 and reappear in later sections of this chapter. In each case carry out the indicated analysis using direction fields. A model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation \(v^{\prime}(t)=g-b v,\) where \(v(t)\) is the velocity of the object, for \(t \geq 0, g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(b>0\) is a constant that involves the mass of the object and the air resistance. Let \(b=0.1 \mathrm{s}^{-1}\). a. Draw the direction field for \(0 \leq t \leq 60,0 \leq y \leq 150\). b. For what initial values \(v(0)=A\) are solutions increasing? Decreasing? c. What is the equilibrium solution?
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