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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 4.7.23
Capitolo 4, Problema 4.7.23

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(1/x² − x² − 1/3) dx

Guida verificata passo dopo passo
1
Rewrite the integral by expressing each term with exponents: \(\int \left( x^{-2} - x^{2} - \frac{1}{3} \right) \, dx\).
Use the linearity of the integral to split it into separate integrals: \(\int x^{-2} \, dx - \int x^{2} \, dx - \int \frac{1}{3} \, dx\).
Apply the power rule for integration to each term: For \(\int x^{n} \, dx\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Integrate each term individually: \(\int x^{-2} \, dx = \frac{x^{-1}}{-1} + C\), \(\int x^{2} \, dx = \frac{x^{3}}{3} + C\), and \(\int \frac{1}{3} \, dx = \frac{1}{3}x + C\).
Combine the results, remembering to include a single constant of integration \(+ C\) at the end, and write the most general antiderivative.

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Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function and is expressed with a constant of integration, C. It reverses differentiation and is written as ∫f(x) dx, yielding a function F(x) such that F'(x) = f(x).
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating polynomial terms by increasing the exponent by one and dividing by the new exponent.
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Power Rule for Indefinite Integrals

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral solution and helps identify any errors in the integration process.
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Finding Differentials
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Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


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Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

85. y' = x^(-2/3) (x - 1)

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35. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

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41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?

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Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = x⁴ᐟ⁵, [0, 1]

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Find values of a and b such that the function


ƒ(𝓍) = (a𝓍 + b) / 𝓍² ―1)


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