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.
csc x cot 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 4.7.7a
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
Guida verificata passo dopo passo1
Rewrite the given function in a form that is easier to integrate. Recall that \(\sqrt{x} = x^{1/2}\). So, the function becomes \(\frac{3}{2} x^{1/2}\).
Use the power rule for integration, which states that for any real number \(n \neq -1\), the antiderivative of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to \(\frac{3}{2} x^{1/2}\). Increase the exponent by 1: \(\frac{1}{2} + 1 = \frac{3}{2}\). Then divide by the new exponent: \(\frac{3}{2} \cdot \frac{x^{3/2}}{3/2}\).
Simplify the expression by multiplying \(\frac{3}{2}\) by the reciprocal of \(\frac{3}{2}\), which will simplify the coefficient.
Add the constant of integration \(C\) to the antiderivative. To check your answer, differentiate your result and verify that you get back the original function \(\frac{3}{2} x^{1/2}\).
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Antiderivative (Indefinite Integral)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is often expressed using the integral symbol without limits. Finding antiderivatives helps solve problems involving accumulation and area.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that the integral of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating polynomial and root functions by rewriting roots as fractional exponents and applying the formula.
Video consigliato:
Percorso guidato
04:04Power Rule for Indefinite Integrals
Verification by Differentiation
After finding an antiderivative, differentiating it should return the original function. This verification step ensures the correctness of the antiderivative and reinforces understanding of the relationship between differentiation and integration.
Video consigliato:
Percorso guidato
05:53Finding Differentials
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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.
170
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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) = (x − 1)²(x + 2)
172
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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) = (x − 2) / (x²−1), 0 ≤ x < 1
208
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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
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:
a. What are the critical points of f?
f′(x) = x(x − 1)
149
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