Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫√(2x + 1) dx = √(x² + x +C)
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 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?
Verified step by step guidance1
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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05:36Piecewise Functions
Related Practice
Textbook Question
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.)
Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).
Textbook Question
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
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Textbook Question
20.The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. a.What dimensions will give a box with a square end the largest possible volume?
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Textbook Question
Identifying Extrema
In Exercises 41–52:
a. Identify the function’s local extreme values in the given domain, and say where they occur.
f(x) = √(x² − 2x − 3), 3 ≤ x < ∞