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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 4.7.8b
Capitolo 4, Problema 4.7.8b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
1/(3³√x)

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1
Rewrite the given function in a form that is easier to integrate. Recall that the cube root of x is \(x^{1/3}\), so the function \(\frac{1}{3\sqrt[3]{x}}\) can be written as \(\frac{1}{3x^{1/3}}\).
Express the function as \(\frac{1}{3} x^{-1/3}\) by bringing the \(x^{1/3}\) in the denominator to the numerator with a negative exponent.
Use the power rule for antiderivatives, which states that for \(f(x) = x^n\), an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule to \(\frac{1}{3} x^{-1/3}\) by increasing the exponent by 1: \(-\frac{1}{3} + 1 = \frac{2}{3}\), and then divide by the new exponent \(\frac{2}{3}\).
Multiply the constant \(\frac{1}{3}\) by the reciprocal of the new exponent \(\frac{3}{2}\) to find the coefficient of the antiderivative, and don't forget to add the constant of integration \(C\).

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Antiderivatives (Indefinite Integrals)

An antiderivative of a function is another function whose derivative equals the original function. Finding antiderivatives involves reversing differentiation, often using basic integration rules. The result includes a constant of integration since derivatives of constants are zero.
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Introduction to Indefinite Integrals

Power Rule for Integration

The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions expressed as powers of x, including fractional and negative exponents.
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Power Rule for Indefinite Integrals

Simplifying Expressions with Radicals and Exponents

Radicals can be rewritten as fractional exponents to simplify integration. For example, the cube root of x is x^(1/3). Converting radicals to exponents allows the use of the power rule directly and makes mental calculation easier.
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Simplifying Exponential Expressions
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Finding Antiderivatives

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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Checking Antiderivative Formulas


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


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Finding Antiderivatives

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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Finding Antiderivatives

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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Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


b. On what open intervals is f increasing or decreasing?


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Finding Antiderivatives

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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