Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.7.63b

Capitolo 4, Problema 4.7.63b

Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫xsinx dx = -x cos x + C

Guida verificata passo dopo passo

Recall the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

Identify parts for integration by parts: let $u = x$ (so $du = dx$) and $dv = \sin x \, dx$ (so $v = -\cos x$).

Apply the formula: $\int x \sin x \, dx = -x \cos x - \int (-\cos x) \, dx = -x \cos x + \int \cos x \, dx$.

Integrate $\int \cos x \, dx$ to get $\sin x$, so the full antiderivative is $-x \cos x + \sin x + C$.

Compare this with the given formula $\int x \sin x \, dx = -x \cos x + C$; since the $\sin x$ term is missing, the given formula is incorrect.

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Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). The indefinite integral symbol ∫ represents the family of all antiderivatives, including a constant of integration C, since differentiation of a constant is zero.

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Introduction to Indefinite Integrals

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du, where u and dv are parts of the original integrand chosen to simplify the integral.

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Integration by Parts for Definite Integrals

Verification of Antiderivatives

To verify if a given formula is an antiderivative, differentiate the proposed function and check if the result matches the original integrand. This method confirms correctness by reversing the integration process.

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Antiderivatives

Pratica correlata

Domanda del libro di testo

[Technology Exercise] 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

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Domanda del libro di testo

114. Parabolas

b. When is the parabola concave up? Concave down?

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Domanda del libro di testo

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.