Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problema 9.4.21

Capitolo 9, Problema 9.4.21

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

Guida verificata passo dopo passo

Recall the general form of a logistic function: \[y = \frac{L}{1 + e^{-k(x - x_0)}}\] where \(L\) is the maximum value (carrying capacity), \(k\) is the growth rate, and \(x_0\) is the x-value of the sigmoid's midpoint.

Since the function has values between 0 and 1, set \(L = 1\). The function simplifies to: \[y = \frac{1}{1 + e^{-k(x - x_0)}}\]

The function crosses \(y = \frac{1}{2}\) at \(x = 0\), which means the midpoint \(x_0 = 0\). So the function becomes: \[y = \frac{1}{1 + e^{-kx}}\]

To find the slope at \(x = 0\), compute the derivative of \(y\) with respect to \(x\): \[y' = \frac{d}{dx} \left( \frac{1}{1 + e^{-kx}} \right)\] Use the chain rule to differentiate.

Evaluate the derivative at \(x = 0\) and set it equal to 5 (the given slope). Solve for \(k\) to find the growth rate parameter that satisfies the slope condition.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Logistic Function Formula

A logistic function is an S-shaped curve commonly written as y = L / (1 + e^{-k(x - x_0)}), where L is the maximum value, k controls the steepness, and x_0 is the midpoint. For values between 0 and 1, L = 1, simplifying the formula to y = 1 / (1 + e^{-k(x - x_0)}).

Video consigliato:

5:59

Recursive Formulas

Midpoint and Symmetry of Logistic Function

The logistic function crosses its midpoint y = L/2 at x = x_0. This point is the inflection point where the curve changes concavity. Setting x_0 = 0 ensures the function passes through y = 1/2 at x = 0, aligning with the problem's requirement.

Video consigliato:

Percorso guidato

06:21

Properties of Functions

Slope of Logistic Function at Midpoint

The slope of the logistic function at the midpoint x = x_0 is given by (kL)/4. Since L = 1 and the slope is specified as 5 at x = 0, we solve 5 = k/4 to find k = 20. This parameter controls how steeply the function rises at the midpoint.

Video consigliato:

05:45

Understanding Slope Fields

Pratica correlata

Domanda del libro di testo

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

views

Domanda del libro di testo

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = x(1-y), y(1) = 0, dx = 0.2

views

Domanda del libro di testo

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

y' = y/x + cos ((y-x)/x)

views

Domanda del libro di testo

Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?

views

Domanda del libro di testo

In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.

y' = 2y²(x-1), y(2) = -1/2, dx = 0.1, x* = 3

views

Domanda del libro di testo

Show that (0, 0) and (c/d, a/b) are equilibrium points. Explain the meaning of each of these points.

views