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.
x² − 2x + 1
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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 4.7.15c
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
−π csc (πx/2) cot (πx/2)
Guida verificata passo dopo passo1
Recognize that the given function is \(-\pi \csc\left(\frac{\pi x}{2}\right) \cot\left(\frac{\pi x}{2}\right)\), which involves trigonometric functions cosecant and cotangent with an inner function \(\frac{\pi x}{2}\).
Recall the derivative formula: \(\frac{d}{dx}[\csc(u)] = -\csc(u) \cot(u) \cdot u'\) where \(u\) is a function of \(x\).
Identify the inner function \(u = \frac{\pi x}{2}\), so its derivative is \(u' = \frac{\pi}{2}\).
Notice that the given function resembles the derivative of \(\csc\left(\frac{\pi x}{2}\right)\) multiplied by a constant. Use this to set up the antiderivative as a constant multiple of \(\csc\left(\frac{\pi x}{2}\right)\).
Adjust the constant factor to account for the difference between the given function and the derivative formula, then write the antiderivative as \(F(x) = A \csc\left(\frac{\pi x}{2}\right) + C\), where \(A\) is the constant to be determined and \(C\) is the constant of integration.
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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 an indefinite integral with a constant of integration. This process helps solve problems involving accumulation and area under curves.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Trigonometric Functions and Their Derivatives
Understanding the derivatives of trigonometric functions like sine, cosine, cosecant, and cotangent is essential. For example, the derivative of cotangent is -csc²(x), and the derivative of cosecant is -csc(x)cot(x). Recognizing these relationships aids in identifying antiderivatives involving trigonometric expressions.
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06:35Derivatives of Other Inverse Trigonometric Functions
Chain Rule and Substitution Method
The chain rule helps differentiate composite functions, and its reverse is used in integration by substitution. When the integrand involves a function and its derivative, substitution simplifies finding antiderivatives. This technique is crucial for handling functions like csc(πx/2) cot(πx/2) where the argument is a linear function of x.
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05:02Intro to the Chain Rule
Pratica correlata
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.
x⁻⁴ + 2x + 3
29
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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:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = x(x − 1)
137
views
Domanda del libro di testo
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫tanθ sec²θ dθ = (1/2) sec²θ + C
26
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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:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)²(x + 2)
157
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = 1− 4/x², x ≠ 0
120
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