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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 4.7.37
Capitolo 4, Problema 4.7.37

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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1
Identify the integral to solve: \(\int 7 \sin\left(\frac{\theta}{3}\right) \, d\theta\).
Recognize that the integral involves a sine function with a linear argument \(\frac{\theta}{3}\). Use substitution to simplify the integral. Let \(u = \frac{\theta}{3}\), so that \(\theta = 3u\).
Calculate the differential \(d\theta\) in terms of \(du\): since \(\theta = 3u\), then \(d\theta = 3 \, du\).
Rewrite the integral in terms of \(u\): \(\int 7 \sin(u) \, d\theta = \int 7 \sin(u) \cdot 3 \, du = \int 21 \sin(u) \, du\).
Integrate \(21 \sin(u)\) with respect to \(u\): recall that \(\int \sin(u) \, du = -\cos(u) + C\), so the integral becomes \(-21 \cos(u) + C\). Finally, substitute back \(u = \frac{\theta}{3}\) to express the answer in terms of \(\theta\).

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Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation and is expressed without limits, showing all possible functions whose derivative matches the integrand.
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Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like sine involves using known antiderivatives, such as ∫sin(ax) dx = -cos(ax)/a + C. Recognizing the inner function and applying the chain rule in reverse is essential for correct integration.
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Introduction to Trigonometric Functions

Substitution Method

The substitution method simplifies integration by changing variables to handle composite functions. For example, setting u = θ/3 transforms the integral into a simpler form, making it easier to integrate and then revert to the original variable.
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Euler's Method
Pratica correlata
Domanda del libro di testo

Sketch the graphs of the rational functions in Exercises 53–60.


𝓍²

y = ------------------

𝓍² ― 4

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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.


∫(x + 1) dx

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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) = {sinx / x, −π ≤ x < 0

0, x = 0

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Domanda del libro di testo

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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Domanda del libro di testo

Finding Critical Points


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


g(x) = √(2x − x²)

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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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