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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.39
Capitolo 4, Problema 4.7.39

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.


∫(−3csc²x)dx

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Recognize that the integral is of the form \(\int -3 \csc^{2}x \, dx\), where \(-3\) is a constant multiplier and \(\csc^{2}x\) is a standard trigonometric function whose integral is known.
Recall the basic integral formula: \(\int \csc^{2}x \, dx = -\cot x + C\), where \(C\) is the constant of integration.
Use the constant multiple rule for integrals, which allows you to factor out constants: \(\int -3 \csc^{2}x \, dx = -3 \int \csc^{2}x \, dx\).
Substitute the known integral result into the expression: \(-3 \int \csc^{2}x \, dx = -3 (-\cot x + C) = 3 \cot x + C'\), where \(C'\) is a new constant of integration.
Verify your result by differentiating \(3 \cot x + C'\) and confirming that it equals the original integrand \(-3 \csc^{2}x\).

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Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and includes all possible functions whose derivative matches the integrand.
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Integration of Trigonometric Functions

Integrating trigonometric functions like csc²x involves knowing standard integral formulas, such as ∫csc²x dx = -cot x + C. Recognizing these forms simplifies finding antiderivatives.
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Verification by Differentiation

After finding an indefinite integral, differentiating the result should return the original integrand. This step confirms the correctness of the antiderivative and helps identify any errors.
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Theory and Examples


Maximum height of a vertically moving body The height of a body moving vertically is given by s = −12gt² + υ₀t + s₀,  g > 0, with s in meters and t in seconds. Find the body’s maximum height.

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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.


∫(cscθ cotθ) / 2 dθ

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[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

26. Factoring a quartic Find the approximate values of r_1 through r_4 in the factorization

8x^4-14x^3-9x^2+11x-1=8(x-r_1)(x-r_2)(x-r_3)(x-r_4)

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Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x − 3x²ᐟ³

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Business and Economics

62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

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Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5

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