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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.7.47
4장, 문제 4.7.47

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(1 + cos 4t)/2 dt

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Start by rewriting the integral to separate the terms inside the integral: \(\int \frac{1 + \cos 4t}{2} \, dt = \int \frac{1}{2} \, dt + \int \frac{\cos 4t}{2} \, dt\).
Integrate the first term \(\int \frac{1}{2} \, dt\) which is a constant multiple of \(dt\). Recall that \(\int a \, dt = at\) for constant \(a\).
For the second term \(\int \frac{\cos 4t}{2} \, dt\), factor out the constant \(\frac{1}{2}\) to get \(\frac{1}{2} \int \cos 4t \, dt\).
Use the substitution rule or recall the integral formula \(\int \cos (kt) \, dt = \frac{1}{k} \sin (kt) + C\). Apply this with \(k=4\) to find the antiderivative of \(\cos 4t\).
Combine the results from both integrals and add the constant of integration \(C\) to write the most general antiderivative.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation, providing a family of functions whose derivative equals the integrand.
추천 영상:
가이드 코스
05:04
Introduction to Indefinite Integrals

Integration of Trigonometric Functions

Integrating trigonometric functions like cosine involves using known integral formulas, such as ∫cos(ax) dx = (1/a) sin(ax) + C. Recognizing and applying these formulas simplifies finding antiderivatives.
추천 영상:
가이드 코스
6:04
Introduction to Trigonometric Functions

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the integral and helps adjust any initial guesses.
추천 영상:
가이드 코스
05:53
Finding Differentials
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Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x¹ᐟ³(x + 8)

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In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

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y = √ 3 + 2𝓍 ―𝓍²

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Initial Value Problems


Solve the initial value problems in Exercises 71–90.


d²y/dx² = 2 − 6x; y′(0) = 4, y(0) = 1

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37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?

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Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

80. y' = 1 - cot²θ, for 0 < θ < π

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