교과서 질문
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
4장, 문제 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
검증된 단계별 안내1
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.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
가이드 코스
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.
추천 영상:
가이드 코스
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.
추천 영상:
가이드 코스
05:53Finding Differentials
관련 실천
교과서 질문
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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교과서 질문
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 < ∞
170
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교과서 질문
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
36
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교과서 질문
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
views
교과서 질문
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
31
views