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.
1/(3³√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 4.7.10b
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-(1/2)x⁻³ᐟ²
Guida verificata passo dopo passo1
Identify the function to find the antiderivative of: \(-\frac{1}{2} x^{-\frac{3}{2}}\).
Recall the power rule for antiderivatives: For \(x^n\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), provided \(n \neq -1\).
Apply the power rule by increasing the exponent by 1: \(-\frac{3}{2} + 1 = -\frac{1}{2}\).
Write the antiderivative as \(-\frac{1}{2} \cdot \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C\).
Simplify the expression and add the constant of integration \(C\) to complete the antiderivative.
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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 using known integral formulas. The result includes a constant of integration since differentiation of a constant is zero.
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Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, for any real number n ≠ -1. This rule is essential for integrating functions with variable exponents, including negative and fractional powers, by increasing the exponent by one and dividing by the new exponent.
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 step confirms the correctness of the antiderivative and helps identify any mistakes in the integration process.
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Percorso guidato
05:53Finding Differentials
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.
1 / 2x³
37
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.
∫3(2x + 1)² dx = (2x + 1)³ + C
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Domanda del libro di testo
[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.
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b. Find the domain of V for the problem situation and graph V over this domain.
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:
b. On what open intervals is f increasing or decreasing?
f′(x) = x(x − 1)
194
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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:
b. On what open intervals is f increasing or decreasing?
f′(x) = 1− 4/x², x ≠ 0
161
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