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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.7.11a
Chapter 4, Problem 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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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
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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.
Recommended video:
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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
Related Practice
Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

Textbook Question

Finding displacement from an antiderivative of velocity


a. Suppose that the velocity of a body moving along the s-axis is


ds/dt = v = 9.8t − 3.


i. Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0.

Textbook Question

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)

Textbook Question

Finding Antiderivatives

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

sec²x

2
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Textbook Question

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


f(r) = 3r³ + 16r

Textbook Question

25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper.

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a. Show that L^2=2x^3/(2x-8.5).