Domanda del libro di testo
33–42. Solving initial value problems Solve the following initial value problems.
p'(x) = 2/(x² + x), p(1) = 0
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 9, Problema 9.5.18
17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.
The population increases from 50 to 60 in the first month and eventually levels off at 150.
Guida verificata passo dopo passo1
Identify the general form of the logistic function: \(f(t) = \frac{L}{1 + Ce^{-kt}}\), where \(L\) is the carrying capacity (the population limit), \(C\) and \(k\) are constants to be determined, and \(t\) represents time in months.
From the problem, note that the population levels off at 150, so set \(L = 150\). Also, the initial population at \(t=0\) is 50, so use this to find \(C\) by substituting \(t=0\) and \(f(0) = 50\) into the logistic function.
Use the population at \(t=1\) month, which is 60, to create a second equation by substituting \(t=1\) and \(f(1) = 60\) into the logistic function. This will involve the unknown constant \(k\).
Solve the system of two equations obtained from steps 2 and 3 to find the values of \(C\) and \(k\). This may require algebraic manipulation and possibly taking natural logarithms.
Write the final logistic function with the determined constants \(L\), \(C\), and \(k\). Then, sketch or graph the function to visualize how the population grows from 50 towards the carrying capacity of 150 over time.
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Logistic Function Model
A logistic function models population growth that starts exponentially but slows as it approaches a maximum limit called the carrying capacity. It is typically expressed as P(t) = L / (1 + Ce^(-kt)), where L is the carrying capacity, and C and k are constants determined by initial conditions.
Video consigliato:
Percorso guidato
06:21
Properties of Functions
Parameter Determination Using Initial Conditions
To find the specific logistic function, you use given data points such as initial population values and growth at certain times. These conditions help solve for constants C and k by substituting values into the logistic equation and solving the resulting system of equations.
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Percorso guidato
05:03Initial Value Problems
Graphing Logistic Functions
Graphing a logistic function involves plotting population over time, showing an S-shaped curve that starts near the initial population, rises rapidly, and then levels off at the carrying capacity. This visual helps interpret growth behavior and verify the model's accuracy.
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Percorso guidato
5:53Graph of Sine and Cosine Function
Pratica correlata
Domanda del libro di testo
9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.
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Domanda del libro di testo
33–42. Solving initial value problems Solve the following initial value problems.
y''(t) = teᵗ, y(0) = 0, y'(0) = 1
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Domanda del libro di testo
45–46. Harvesting problems Consider the harvesting problem in Example 6.
If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?
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Domanda del libro di testo
11–16. Initial value problems Solve the following initial value problems.
y'(t) − 3y = 12, y(1) = 4
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Domanda del libro di testo
27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.
An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C?
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