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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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 4.1.52c
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(-3) exist?
Guida verificata passo dopo passo1
To determine if f'(-3) exists, we need to check if the function f(x) = |x³ − 9x| is differentiable at x = -3. Differentiability requires the function to be continuous and have a defined derivative at that point.
First, check the continuity of f(x) at x = -3. Since f(x) is an absolute value function, it is continuous everywhere, including at x = -3.
Next, consider the definition of the derivative. The derivative f'(x) exists at x = -3 if the limit of the difference quotient exists as x approaches -3 from both sides.
To find the derivative, consider the piecewise nature of the absolute value function. The expression inside the absolute value, x³ - 9x, changes sign at the roots of the equation x³ - 9x = 0. Solve for x to find these critical points.
Evaluate the derivative from the left and right of x = -3 using the piecewise definition of f(x). If the left-hand and right-hand derivatives are equal, then f'(-3) exists. Otherwise, it does not.
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Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is the slope of the tangent line to the function's graph at that point. For a function f(x), the derivative is denoted as f'(x) and is found using differentiation rules.
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Percorso guidato
05:44
Derivatives
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x. It affects the differentiability of functions because it creates sharp corners or cusps in graphs, where the derivative may not exist. For f(x) = |x³ − 9x|, the absolute value impacts the function's smoothness and continuity.
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06:37Average Value of a Function
Differentiability and Continuity
A function is differentiable at a point if it is smooth and has a defined tangent at that point, implying continuity. However, a function can be continuous but not differentiable at points where there are sharp turns or cusps, such as those introduced by absolute values. Checking differentiability involves examining the function's behavior around the point of interest.
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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.
x⁻⁴ + 2x + 3
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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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Domanda del libro di testo
Applications
Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).
Find:
∫[−f(x)] dx
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Domanda del libro di testo
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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