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 4.7.2b

Capitolo 4, Problema 4.7.2b

Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁷

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Recall that an antiderivative of a function is another function whose derivative is the original function. For power functions, the antiderivative rule is: if \(f(x) = x^n\), then an antiderivative is \(F(x) = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration and \(n \neq -1\).

Identify the exponent in the given function \(x^7\). Here, \(n = 7\).

Apply the antiderivative formula by increasing the exponent by 1: \(7 + 1 = 8\).

Divide by the new exponent to get the antiderivative: \(\frac{x^8}{8} + C\).

Verify your answer by differentiating \(\frac{x^8}{8} + C\) and checking that the derivative is \(x^7\).

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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 often expressed with an arbitrary constant, C, since differentiation of a constant is zero.

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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) + C. This rule is fundamental for finding antiderivatives of polynomial functions like x⁷.

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Percorso guidato

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Power 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.

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Finding Differentials

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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

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[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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114. Parabolas

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

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[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a 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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38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

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Theory and Examples

Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.