Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
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Ch. 3 - 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 3, Problem 3.9.4
Finding Linearizations
In Exercises 1–5, find the linearization L(x) of f(x) at x = a.
f(x) = ∛x, a = −8
Verified step by step guidance1
Identify the function f(x) = ∛x and the point of interest x = a = -8.
Recall the formula for the linearization of a function at a point: L(x) = f(a) + f'(a)(x - a).
Calculate f(a) by substituting a = -8 into the function: f(-8) = ∛(-8).
Find the derivative f'(x) of the function f(x) = ∛x. Use the power rule for derivatives: f(x) = x^(1/3) implies f'(x) = (1/3)x^(-2/3).
Evaluate the derivative at the point a = -8: f'(-8) = (1/3)(-8)^(-2/3). Substitute f(a) and f'(a) into the linearization formula to find L(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linearization
Linearization is the process of approximating a function near a given point using the tangent line at that point. The linearization of a function f(x) at x = a is given by L(x) = f(a) + f'(a)(x - a). This provides a simple linear model that approximates the function's behavior close to x = a.
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Linearization
Derivative
The derivative of a function, denoted as f'(x), represents the rate at which the function's value changes with respect to changes in x. It is the slope of the tangent line to the function at any given point. For linearization, the derivative at x = a, f'(a), is crucial as it determines the slope of the linear approximation.
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Cube Root Function
The cube root function, f(x) = ∛x, is a type of root function where the output is the number that, when cubed, gives x. Understanding its behavior, especially around specific points like x = -8, is essential for calculating derivatives and linearizations. The function is continuous and differentiable for all real numbers, which facilitates finding its linear approximation.
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Related Practice
Textbook Question
In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?
54. x³ + y² = sin²y
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Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x² + x + 8
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Textbook Question
If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.
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Textbook Question
Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form
P = (nRT / (V − nb)) − (an² / V²),
in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)
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184
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Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
xy² − 4x³/² − y = 0
186
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