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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.91
Capitolo 4, Problema 4.7.91

Initial Value Problems


Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

Guida verificata passo dopo passo
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Identify the given differential equation from the problem: the slope of the curve at each point is given by \(\frac{dy}{dx} = 3\sqrt{x}\).
Rewrite the slope expression in terms of a power of \(x\) to make integration easier: \(\frac{dy}{dx} = 3x^{\frac{1}{2}}\).
Integrate both sides with respect to \(x\) to find the general form of \(y\): \(y = \int 3x^{\frac{1}{2}} \, dx\).
Perform the integration using the power rule for integrals: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(9) = 4\) to solve for the constant \(C\) by substituting \(x=9\) and \(y=4\) into the integrated function.

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Initial Value Problems

An initial value problem involves finding a function that satisfies a given differential equation and passes through a specific point, called the initial condition. This ensures a unique solution by fixing the constant of integration after solving the differential equation.
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Initial Value Problems

Differential Equations and Slope Fields

A differential equation relates a function to its derivatives. Here, the slope of the curve y = f(x) is given as a function of x, meaning dy/dx = 3√x. Understanding how to interpret and solve such equations is key to finding the original function.
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Understanding Slope Fields

Integration to Find the Original Function

To find y = f(x) from its derivative dy/dx, integrate the given slope function with respect to x. After integration, use the initial condition to solve for the constant of integration, yielding the specific curve passing through the given point.
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Integrals of General Exponential Functions
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Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


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Domanda del libro di testo

Identifying Extrema


In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.


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Domanda del libro di testo

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

______

y = √𝓍² ― 1

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Domanda del libro di testo

Absolute Extrema on Finite Closed Intervals


In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.


f(t) = 2 − |t|, −1 ≤ t ≤ 3

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Domanda del libro di testo

Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


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Domanda del libro di testo

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ

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