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.
−3x⁻⁴
25
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 4.1.52a
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
a. Does f'(0) exist?
Guida verificata passo dopo passo1
To determine if f'(0) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = 0. Differentiability requires the function to be continuous and have a defined derivative at that point.
First, check the continuity of f(x) at x = 0. Since f(x) is an absolute value function, it is continuous everywhere, including at x = 0.
Next, consider the definition of the derivative: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit at x = 0.
To evaluate the derivative, consider the piecewise nature of the absolute value function. For x³ - 9x, identify the points where the expression inside the absolute value changes sign, which are the roots of x³ - 9x = 0.
Calculate the left-hand and right-hand derivatives at x = 0 by considering the limits from both sides. If these one-sided derivatives are equal, then f'(0) exists; otherwise, it does not.
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:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Derivative at a Point
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. For f'(0) to exist, the function must be differentiable at x = 0, meaning it must be continuous and have a defined slope at that point. If the function has a sharp corner or cusp at x = 0, the derivative does not exist.
Video consigliato:
04:50
Critical Points
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x. When applied to a function like f(x) = |x³ − 9x|, it can create points where the function is not smooth, such as cusps or corners, which can affect differentiability. Understanding how the absolute value impacts the function's graph is crucial for analyzing its derivative.
Video consigliato:
Percorso guidato
06:37Average Value of a Function
Piecewise Functions
A piecewise function is defined by different expressions over different intervals. The function f(x) = |x³ − 9x| can be expressed as a piecewise function, where the expression inside the absolute value changes sign. Analyzing the behavior of each piece separately helps determine the function's continuity and differentiability at critical points like x = 0.
Video consigliato:
Percorso guidato
05:36Piecewise Functions
Pratica correlata
Domanda del libro di testo
52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.
a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)
174
views
Domanda del libro di testo
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π
180
views
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.
1 / x²
29
views
Domanda del libro di testo
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
f(t) = t³ − 3t², −∞ < t ≤ 3
194
views
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.
sec²x
31
views