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 4.7.5a
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 / x²
검증된 단계별 안내1
Recognize that the function given is \(\frac{1}{x^2}\), which can be rewritten as \(x^{-2}\) to make it easier to apply the power rule for antiderivatives.
Recall the power rule for antiderivatives: for any function \(x^n\) where \(n \neq -1\), the antiderivative is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
Apply the power rule to \(x^{-2}\) by increasing the exponent by 1: \(-2 + 1 = -1\), so the antiderivative will be \(\frac{x^{-1}}{-1} + C\).
Simplify the expression to get the antiderivative in a more standard form: \(-x^{-1} + C\), which can also be written as \(-\frac{1}{x} + C\).
Verify your answer by differentiating \(-\frac{1}{x} + C\) and checking that the derivative is \(\frac{1}{x^2}\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Antiderivatives (Indefinite Integrals)
An antiderivative of a function is another function whose derivative equals the original function. It represents the reverse process of differentiation and is expressed as an indefinite integral with a constant of integration, C, 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 for any real number n ≠ -1, the integral of x^n dx is (x^(n+1)) / (n+1) + C. This rule is essential for finding antiderivatives of polynomial and power functions like 1/x², which can be rewritten as x^(-2).
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 identify any mistakes in the integration process.
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 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π
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교과서 질문
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = sin 2x, 0 ≤ x ≤ π
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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.
k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0
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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.
sec²x
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