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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.17
Capitolo 4, Problema 4.7.17

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.


∫(x + 1) dx

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Identify the integral to solve: \(\int (x + 1) \, dx\).
Recall the basic rule for integrating sums: \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\).
Split the integral into two separate integrals: \(\int x \, dx + \int 1 \, dx\).
Use the power rule for integration on the first term: \(\int x \, dx = \frac{x^{2}}{2} + C_1\), where \(C_1\) is a constant of integration.
Integrate the constant term: \(\int 1 \, dx = x + C_2\), where \(C_2\) is another constant of integration. Combine constants into a single constant \(C\) to write the 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, meaning if F'(x) = f(x), then ∫f(x) dx = F(x) + C.
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Introduction to Indefinite Integrals

Basic Integration Rules

Integration rules such as the power rule and linearity simplify finding antiderivatives. For example, ∫x dx = x²/2 + C and ∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx, allowing term-by-term integration.
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Basic Rules for Definite Integrals

Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and ensures no errors in the integration process.
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Finding Differentials
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Domanda del libro di testo

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.


∫7sin(θ/3) dθ

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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) = {sinx / x, −π ≤ x < 0

0, x = 0

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Finding Extrema from Graphs


In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.


f(x) = |x|, −1 < x < 2

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Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


g(x) = √(2x − x²)

197
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Domanda del libro di testo

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.


∫(2x³ − 5x + 7) dx

16
views
Domanda del libro di testo

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) = x²ᐟ³, [−1, 8]

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