Domanda del libro di testo
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.3.51a
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
g(x) = (x − 2) / (x²−1), 0 ≤ x < 1
Guida verificata passo dopo passo1
First, find the derivative of the function g(x) = \( \frac{x - 2}{x^2 - 1} \). Use the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \). Here, \( u = x - 2 \) and \( v = x^2 - 1 \).
Calculate the derivatives: \( u' = 1 \) and \( v' = 2x \). Substitute these into the quotient rule formula to find \( g'(x) \).
Set the derivative \( g'(x) \) equal to zero to find critical points. Solve the equation \( \frac{(x^2 - 1) - (x - 2)(2x)}{(x^2 - 1)^2} = 0 \) to find the values of x where the derivative is zero.
Check the critical points and endpoints within the domain \( 0 \leq x < 1 \) to determine if they are local extrema. Evaluate the function g(x) at these points.
Use the first or second derivative test to classify the critical points as local minima, maxima, or neither. The first derivative test involves checking the sign of \( g'(x) \) around the critical points, while the second derivative test involves evaluating \( g''(x) \) at the critical points.
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Local Extrema
Local extrema refer to the points in a function where it reaches a local maximum or minimum within a specific interval. To identify these points, one typically examines the function's derivative to find critical points where the derivative is zero or undefined, and then uses the second derivative test or other methods to determine the nature of these points.
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05:58
Finding Extrema Graphically
Critical Points
Critical points are values of x where the derivative of a function is either zero or undefined. These points are potential candidates for local extrema. To find them, calculate the derivative of the function and solve for x where the derivative equals zero or does not exist. Analyzing these points helps in determining the behavior of the function around them.
Video consigliato:
04:50Critical Points
Domain Restrictions
Domain restrictions define the set of x-values for which a function is considered. In this problem, the domain is 0 ≤ x < 1, meaning the function is only analyzed within this interval. Understanding domain restrictions is crucial as they limit where extrema can occur and affect the behavior of the function, especially near boundaries or points of discontinuity.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
Pratica correlata
Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
csc x cot x
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Domanda del libro di testo
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.
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Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
(3/2)√x
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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:
a. What are the critical points of f?
f′(x) = (x − 1)²(x + 2)
172
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Domanda del libro di testo
Roots (Zeros)
a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.
i. y = x² − 4
231
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