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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 4.7.109a
Capitolo 4, Problema 4.7.109a

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫f(x) dx

Guida verificata passo dopo passo
1
First, identify the functions f(x) and g(x) given in the problem. We have f(x) = \(\frac{d}{dx}\) (1 - \(\sqrt{x}\)) and g(x) = \(\frac{d}{dx}\) (x + 2).
Next, compute the derivative f(x) by differentiating the function inside the derivative: 1 - \(\sqrt{x}\). Recall that \(\sqrt{x}\) = x^{1/2}, so use the power rule for differentiation.
After finding f(x), set up the integral \(\int\) f(x) \, dx. Since f(x) is the derivative of (1 - \(\sqrt{x}\)), integrating f(x) with respect to x will give you the original function plus a constant of integration.
Write the integral result as (1 - \(\sqrt{x}\)) + C, where C is the constant of integration, because integration is the inverse operation of differentiation.
Finally, verify your result by differentiating your integral expression to ensure it matches f(x). This confirms the correctness of your integration.

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Derivative and Differentiation

The derivative of a function represents the rate at which the function's value changes with respect to its variable. Differentiation rules, such as the power rule, help find derivatives of expressions like √x or x + 2. Understanding how to compute derivatives is essential to identify f(x) and g(x) as given.
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Antiderivative and Indefinite Integral

The antiderivative or indefinite integral of a function reverses differentiation, finding a function whose derivative is the given function. The integral symbol ∫f(x) dx represents all functions whose derivative is f(x), plus a constant of integration. This concept is key to solving ∫f(x) dx.
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Power Rule for Integration

The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is used to integrate functions involving powers of x, such as those derived from derivatives of √x or polynomials. Applying this rule simplifies finding the antiderivative of f(x).
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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.

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Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


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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.

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Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


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In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

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In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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