Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫xsinx dx = -x cos 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 4.1.52b
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(3) exist?
Verified step by step guidance1
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: f'(x) = lim (h -> 0) [(f(x + h) - f(x)) / h]. We need to evaluate this limit at x = 3.
To evaluate the derivative, consider the piecewise nature of the absolute value function. For x³ - 9x, identify the intervals where the expression inside the absolute value is positive or negative. This will help in determining the left-hand and right-hand derivatives at x = 3.
Calculate the left-hand derivative (approaching from the left of x = 3) and the right-hand derivative (approaching from the right of x = 3). If both derivatives exist and are equal, then f'(3) exists. If they are not equal, f'(3) does not exist.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 f'(x) is found using limits, and it exists at a point if the function is continuous and smooth (differentiable) there.
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Derivatives
Absolute Value Function
The absolute value function, denoted as |x|, outputs the non-negative value of x, regardless of its sign. When dealing with derivatives, the presence of an absolute value can create points where the function is not differentiable, typically at points where the expression inside the absolute value changes sign, leading to a cusp or corner in the graph.
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Differentiability and Continuity
A function is differentiable at a point if it is smooth and has no sharp corners or cusps at that point, which also implies it must be continuous there. For f(x) = |x³ − 9x|, checking differentiability at x = 3 involves ensuring the function is continuous and that the left-hand and right-hand derivatives at x = 3 are equal.
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05:34Intro to Continuity
Related Practice
Textbook Question
[Technology Exercise] 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.
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b. Find the domain of V for the problem situation and graph V over this domain.
Textbook Question
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)
Textbook Question
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) = 1− 4/x², x ≠ 0
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.
x⁻³/2 + x²
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Textbook Question
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)²(x + 2)