Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 66d
Capitolo 3, Problema 66d

{Use of Tech} Population growth Consider the following population functions.
d. Evaluate and interpret lim t→∞ p(t).
p(t) = 600 (t²+3/t²+9)

Guida verificata passo dopo passo
1
Step 1: Identify the function p(t) = 600 \(\left\)(\(\frac{t^2 + 3}{t^2 + 9}\)\(\right\)) and recognize that you need to evaluate the limit as t approaches infinity.
Step 2: Simplify the expression \(\frac{t^2 + 3}{t^2 + 9}\) by dividing both the numerator and the denominator by t^2, the highest power of t in the expression.
Step 3: After simplification, the expression becomes \(\frac{1 + \frac{3}{t^2}\)}{1 + \(\frac{9}{t^2}\)}.
Step 4: Evaluate the limit of the simplified expression as t approaches infinity. As t becomes very large, the terms \(\frac{3}{t^2}\) and \(\frac{9}{t^2}\) approach 0.
Step 5: Conclude that the limit of the expression \(\frac{1 + \frac{3}{t^2}\)}{1 + \(\frac{9}{t^2}\)} as t approaches infinity is 1, and therefore, lim_{t \(\to\) \(\infty\)} p(t) = 600 \(\times\) 1 = 600. Interpret this as the population stabilizing at 600 as time goes to infinity.

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:
6m

Concetti chiave

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

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. Evaluating limits helps us understand the function's behavior at points where it may not be explicitly defined or at infinity. In this case, we are interested in the limit of the population function p(t) as t approaches infinity, which provides insight into the long-term behavior of the population.
Video consigliato:
05:50
One-Sided Limits

Population Functions

Population functions model the growth of a population over time, often represented as p(t), where t is time. These functions can take various forms, such as exponential or logistic growth models. Understanding the specific form of the population function given, p(t) = 600(t² + 3)/(t² + 9), is crucial for evaluating its behavior as time progresses, particularly as t approaches infinity.
Video consigliato:
Percorso guidato
06:21
Properties of Functions

Asymptotic Behavior

Asymptotic behavior refers to the behavior of a function as its input approaches a limit, often infinity. In the context of population growth, it helps us determine the maximum population size that can be sustained over time. By analyzing the limit of p(t) as t approaches infinity, we can interpret the long-term population trend and understand whether the population stabilizes, grows indefinitely, or declines.
Video consigliato:
03:07
Cases Where Limits Do Not Exist
Pratica correlata
Domanda del libro di testo

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

282
views
Domanda del libro di testo

Population growth Consider the following population functions.

b. What is the instantaneous growth rate at t=5?

p(t) = 600 (t²+3/t²+9)

301
views
Domanda del libro di testo

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

414
views
Domanda del libro di testo

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

298
views
Domanda del libro di testo

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x)=x²+1; Q(3, 6)

258
views
Domanda del libro di testo

Population growth Consider the following population functions.

c. Estimate the time when the instantaneous growth rate is greatest.

p(t) = 600 (t²+3/t²+9)

292
views