Textbook Question
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
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 3.17
Find the derivatives of the functions in Exercises 1–42.
______
𝓻 = √2θ sinθ
Verified step by step guidance1
Identify the function given: \( r = \sqrt{2\theta} \sin\theta \). This is a product of two functions: \( \sqrt{2\theta} \) and \( \sin\theta \).
Apply the product rule for differentiation, which states that if you have a function \( u(\theta) \cdot v(\theta) \), its derivative is \( u'(\theta) \cdot v(\theta) + u(\theta) \cdot v'(\theta) \).
Differentiate \( u(\theta) = \sqrt{2\theta} \). Use the chain rule: \( u'(\theta) = \frac{d}{d\theta}(2\theta)^{1/2} = \frac{1}{2}(2\theta)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2\theta}} \).
Differentiate \( v(\theta) = \sin\theta \). The derivative is \( v'(\theta) = \cos\theta \).
Substitute the derivatives back into the product rule: \( r'(\theta) = \left(\frac{1}{\sqrt{2\theta}}\right) \sin\theta + \sqrt{2\theta} \cos\theta \). Simplify if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule allows us to differentiate it by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions expressed in terms of other variables, such as polar coordinates.
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Polar Coordinates
Polar coordinates represent points in a plane using a distance from a reference point and an angle from a reference direction. In this context, the function 𝓻 = √2θ sinθ is expressed in polar form, where 𝓻 is the radius and θ is the angle. Understanding how to differentiate functions in polar coordinates is essential for finding derivatives in this format.
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Related Practice
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𝔂 = x² sin² (2x²)
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