Domanda del libro di testo
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = −x² − 6x − 9,−4 ≤ x < ∞
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.7.64a
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
Guida verificata passo dopo passo1
Identify the integral to be checked: \(\int \tan\theta \sec^{2}\theta \, d\theta\).
Recall the derivatives of common trigonometric functions: for example, \(\frac{d}{d\theta}(\sec\theta) = \sec\theta \tan\theta\) and \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{2}\theta \cdot \sec\theta \tan\theta = 3 \sec^{3}\theta \tan\theta\).
Compare the integrand \(\tan\theta \sec^{2}\theta\) with the derivative of \(\sec^{3}\theta\): since \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{3}\theta \tan\theta\), the integrand is \(\frac{1}{3}\) of this derivative.
Therefore, the integral \(\int \tan\theta \sec^{2}\theta \, d\theta\) should be \(\frac{1}{3} \sec^{3}\theta + C\), matching the given formula \(\frac{\sec^{3}\theta}{3} + C\).
Conclude that the formula is correct because the derivative of the proposed antiderivative matches the integrand.
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Antiderivatives and Indefinite Integrals
An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C. Understanding how to find antiderivatives is essential for verifying integral formulas.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Integration by Substitution
Integration by substitution simplifies integrals by changing variables to make the integral easier to solve. It often involves identifying a part of the integrand as a derivative of another function. Recognizing substitution opportunities helps verify or find antiderivatives involving composite functions.
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04:27Substitution With an Extra Variable
Derivatives of Trigonometric Functions
Knowing the derivatives of trigonometric functions like tan(θ) and sec(θ) is crucial for checking antiderivatives. For example, the derivative of sec(θ) is sec(θ)tan(θ), and the derivative of sec³(θ) involves the chain rule. This knowledge allows one to differentiate the proposed antiderivative to confirm correctness.
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06:35Derivatives of Other Inverse Trigonometric Functions
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
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.
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Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
πcos πx
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Domanda del libro di testo
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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Domanda del libro di testo
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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