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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 4.2.20
Capitolo 4, Problema 4.2.20

Roots (Zeros)


Show that the functions in Exercises 19–26 have exactly one zero in the given interval.


f(x) = x³ + 4x² + 7, (−∞, 0)

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First, understand that finding the zero of a function means finding the value of x for which f(x) = 0. We need to show that the function f(x) = x³ + 4x² + 7 has exactly one zero in the interval (-∞, 0).
To begin, evaluate the function at the endpoints of the interval. Since the interval is (-∞, 0), consider the behavior of the function as x approaches 0 from the left and as x approaches negative infinity.
Next, calculate the derivative of the function, f'(x) = 3x² + 8x, to analyze the behavior of the function. This will help determine if the function is increasing or decreasing in the interval.
Examine the sign of f'(x) in the interval (-∞, 0). If f'(x) is negative throughout the interval, the function is strictly decreasing, which implies that it can have at most one zero.
Finally, use the Intermediate Value Theorem. Since f(x) is continuous and changes sign in the interval (for example, f(x) is positive at x = 0 and negative as x approaches negative infinity), there must be at least one zero in the interval. Combine this with the previous step to conclude that there is exactly one zero.

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Intermediate Value Theorem

The Intermediate Value Theorem states that for any continuous function f(x) on a closed interval [a, b], if f(a) and f(b) have opposite signs, then there exists at least one c in (a, b) such that f(c) = 0. This theorem is crucial for proving the existence of a zero in a given interval.
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Fundamental Theorem of Calculus Part 1

Continuity of Polynomial Functions

Polynomial functions, such as f(x) = x³ + 4x² + 7, are continuous everywhere on the real number line. This property ensures that the function does not have any breaks, jumps, or holes, which is essential when applying the Intermediate Value Theorem to find zeros.
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Introduction to Polynomial Functions

Behavior of Polynomial Functions at Infinity

Understanding the behavior of polynomial functions as x approaches infinity or negative infinity helps determine the number of zeros. For f(x) = x³ + 4x² + 7, as x approaches negative infinity, the dominant term x³ dictates that f(x) approaches negative infinity, indicating a sign change in the interval (−∞, 0).
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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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Checking Antiderivative Formulas


Verify the formulas in Exercises 57–62 by differentiation.


∫(3x + 5)⁻² dx = −(3x + 5)⁻¹/3 + C

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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.


∫csc θ/(csc θ − sin θ) dθ

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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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