Domanda del libro di testo
114. Parabolas
b. When is the parabola concave up? Concave down?
383
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 4.3.63b
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.
Guida verificata passo dopo passo1
Step 1: Observe the graph of f'(x), which represents the derivative of f(x). The derivative indicates the slope of the function f(x). When f'(x) > 0, f(x) is increasing, and when f'(x) < 0, f(x) is decreasing.
Step 2: Identify the intervals where f'(x) is positive (above the x-axis). From the graph, f'(x) is positive on the intervals (-2, -1) and (0, 1). This means f(x) is increasing on these intervals.
Step 3: Identify the intervals where f'(x) is negative (below the x-axis). From the graph, f'(x) is negative on the intervals (-1, 0) and (1, 2). This means f(x) is decreasing on these intervals.
Step 4: Note the points where f'(x) crosses the x-axis. These are the critical points where f'(x) = 0, and f(x) may have local extrema. From the graph, these points occur at x = -1, x = 0, and x = 1.
Step 5: Summarize the behavior of f(x) based on the intervals of positivity and negativity of f'(x). f(x) increases on (-2, -1) and (0, 1), decreases on (-1, 0) and (1, 2), and has potential extrema at x = -1, x = 0, and x = 1.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
First Derivative Test
The First Derivative Test is used to determine where a function is increasing or decreasing. If f'(x) > 0, the function f is increasing, and if f'(x) < 0, f is decreasing. By analyzing the sign changes of f'(x), we can identify intervals where f is positive or negative, which helps in finding local extrema.
Video consigliato:
07:09
The First Derivative Test: Finding Local Extrema
Critical Points
Critical points occur where the derivative f'(x) is zero or undefined. These points are potential locations for local maxima or minima. In the graph, critical points are where the curve crosses the x-axis, indicating a change in the direction of f(x), which is crucial for identifying extrema.
Video consigliato:
04:50Critical Points
Domain of the Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this problem, the domain is (-2, 2), meaning we only consider the behavior of f and f' within this interval. Understanding the domain helps focus the analysis on relevant sections of the graph.
Video consigliato:
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.
-sec²(3x/2)
31
views
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?
162
views
Domanda del libro di testo
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).
217
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = (x − 1)(x + 2)(x − 3)
209
views
Domanda del libro di testo
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.
29
views