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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.109c
Capitolo 4, Problema 4.7.109c

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

Guida verificata passo dopo passo
1
First, identify the functions f(x) and g(x) as derivatives given in the problem: \( f(x) = \frac{d}{dx} (1 - \sqrt{x}) \) and \( g(x) = \frac{d}{dx} (x + 2) \).
Calculate \( f(x) \) by differentiating the function inside: \( 1 - \sqrt{x} = 1 - x^{1/2} \). Use the power rule for differentiation: \( \frac{d}{dx} x^{n} = n x^{n-1} \).
Express \( f(x) \) explicitly as \( f(x) = 0 - \frac{1}{2} x^{-1/2} = -\frac{1}{2 \sqrt{x}} \).
The integral to find is \( \int -f(x) \, dx \). Substitute \( f(x) \) into the integral to get \( \int -\left(-\frac{1}{2 \sqrt{x}}\right) dx = \int \frac{1}{2 \sqrt{x}} \, dx \).
Rewrite the integral in terms of exponents: \( \int \frac{1}{2} x^{-1/2} \, dx \). Use the power rule for integration: \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).

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Derivative and Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change or slope at any point. In this problem, f(x) and g(x) are defined as derivatives of given functions, so understanding how to compute derivatives is essential.
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Integral and Antiderivative

Integration is the reverse process of differentiation, used to find the original function from its derivative. The integral ∫[−f(x)] dx requires finding the antiderivative of the negative of f(x), which involves applying integration rules to the derived function.
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Antiderivatives

Properties of Square Roots and Power Functions

The function inside the derivative includes a square root, which can be rewritten as a power function (x^(1/2)). Understanding how to differentiate and integrate power functions, including fractional exponents, is crucial for correctly solving the problem.
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Properties of Functions
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Domanda del libro di testo

Theory and Examples


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


Let f(x) = |x³ − 9x|.


d. Determine all extrema of f.

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Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[f(x) + g(x)] dx

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Domanda del libro di testo

Theory and Examples


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


Let f(x) = |x³ − 9x|.


b. Does f'(-3) exist?

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Domanda del libro di testo

105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?

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Domanda del libro di testo

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = 1− 4/x², x ≠ 0

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106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?

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