Domanda del libro di testo
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) exist?
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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 4.7.3a
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
−3x⁻⁴
Guida verificata passo dopo passo1
Identify the function to find the antiderivative of: \(-3x^{-4}\).
Recall the power rule for antiderivatives: For \(f(x) = x^n\), an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
Apply the power rule to \(-3x^{-4}\) by increasing the exponent by 1: \(-4 + 1 = -3\).
Divide the coefficient by the new exponent: \(\frac{-3}{-3}\), and write the antiderivative as \(\frac{-3}{-3} x^{-3} + C\).
Simplify the expression and add the constant of integration \(C\) to represent the family of antiderivatives.
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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 expressed with a constant of integration since differentiation loses constant terms.
Video consigliato:
Percorso guidato
05:04
Introduction to Indefinite Integrals
Power Rule for Integration
The power rule for integration states that the antiderivative of x^n (where n ≠ -1) is (x^(n+1)) / (n+1) plus a constant. This rule is essential for integrating polynomial and power functions like x⁻⁴.
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 avoid mistakes in integration.
Video consigliato:
Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
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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.
f(t) = 12t − t³, −3 ≤ t < ∞
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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.
2x
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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.
f(t) = t³ − 3t², −∞ < t ≤ 3
194
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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.
sec²x
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