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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 4.2.24
Capitolo 4, Problema 4.2.24

Roots (Zeros)


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


r(θ) = 2θ − cos²θ + √2, (−∞, ∞)

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First, understand that finding the zero of a function means finding the value of θ for which r(θ) = 0. We need to show that there is exactly one such value in the interval (−∞, ∞).
Consider the function r(θ) = 2θ − cos²θ + √2. To find the zeros, set the equation to zero: 2θ − cos²θ + √2 = 0.
To determine if there is exactly one zero, analyze the behavior of the function. Start by finding the derivative r'(θ) to understand the function's monotonicity. The derivative is r'(θ) = 2 + 2cos(θ)sin(θ).
Evaluate the derivative r'(θ). Notice that r'(θ) = 2 + sin(2θ), which is always positive because sin(2θ) ranges from -1 to 1, making 2 + sin(2θ) always greater than 0. This indicates that r(θ) is a strictly increasing function.
Since r(θ) is strictly increasing, it can cross the x-axis at most once. Check the limits as θ approaches -∞ and ∞ to ensure the function crosses the x-axis. As θ approaches -∞, r(θ) approaches -∞, and as θ approaches ∞, r(θ) approaches ∞. Therefore, by the Intermediate Value Theorem, there is exactly one zero in the interval (−∞, ∞).

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Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f(x), takes on different signs at two points, a and b, then it must cross zero at some point between a and b. This theorem is crucial for proving the existence of a root within an interval, as it guarantees that a zero exists if the function changes sign.
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Fundamental Theorem of Calculus Part 1

Continuity of Functions

A function is continuous if there are no breaks, jumps, or holes in its graph. For the function r(θ) = 2θ − cos²θ + √2, continuity is essential to apply the Intermediate Value Theorem. Since polynomials, trigonometric functions, and their combinations are continuous over their domains, r(θ) is continuous over (−∞, ∞).
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Intro to Continuity

Derivative and Monotonicity

The derivative of a function provides information about its monotonicity, indicating whether the function is increasing or decreasing. By analyzing the derivative of r(θ), we can determine if the function is strictly increasing or decreasing, which helps establish that there is exactly one zero in the interval by showing that the function does not change direction.
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Derivatives
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Estimate the open intervals on which the function y = ƒ(𝓍) is


a. increasing.

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Finding Extrema from Graphs


In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.


g(x) = {−x, 0 ≤ x < 1

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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.


∫(4secx tanx − 2 sec²x)dx

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Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.


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109. Suppose the derivative of the function y = f(x) is

y'=(x-1)^2(x-2).

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Theory and Examples


In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.


y = 3x + tan x

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