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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 4.7.11a
Capitolo 4, Problema 4.7.11a

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

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Identify the given function: \(-\pi \sin \pi x\).
Recall that the antiderivative (indefinite integral) of \(\sin(ax)\) is \(-\frac{1}{a} \cos(ax) + C\), where \(a\) is a constant and \(C\) is the constant of integration.
Apply this rule to the function \(-\pi \sin \pi x\): factor out the constant \(-\pi\) and integrate \(\sin \pi x\).
The integral becomes \(-\pi \int \sin \pi x \, dx = -\pi \left(-\frac{1}{\pi} \cos \pi x \right) + C\).
Simplify the expression by canceling \(\pi\) terms and write the antiderivative in its simplest form.

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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 represented as indefinite integrals with a constant of integration. For example, the antiderivative of cos(x) is sin(x) + C.
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Introduction to Indefinite Integrals

Differentiation of Trigonometric Functions

Understanding how to differentiate sine and cosine functions is essential to verify antiderivatives. For instance, the derivative of sin(πx) is πcos(πx), and the derivative of cos(πx) is -πsin(πx). This knowledge helps confirm if the found antiderivative is correct.
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Introduction to Trigonometric Functions

Constant Multiples in Integration

When integrating functions multiplied by constants, the constant can be factored out of the integral. For example, ∫ -πsin(πx) dx = -π ∫ sin(πx) dx. This simplifies the integration process and helps in finding the antiderivative efficiently.
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Additional Rules for Indefinite Integrals
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Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

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Checking Antiderivative Formulas


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


∫tanθ sec²θ dθ = sec³θ / 3 + C

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Identifying Extrema


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


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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

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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.

(3/2)√x

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Analyzing Functions from Derivatives


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


a. What are the critical points of f?


f′(x) = 1− 4/x², x ≠ 0

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Analyzing Functions from Derivatives


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


a. What are the critical points of f?


f′(x) = x(x − 1)

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