Textbook Question
Show that the linearization of f(x) = (1 + x)ᵏ at x = 0 is L(x) = 1 + kx.
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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 3.9.38
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change
Verified step by step guidance1
Identify the formula for the lateral surface area of a right circular cone: \( S = \pi r \sqrt{r^2 + h^2} \).
Recognize that the problem asks for the change in the lateral surface area when the radius changes from \( r_0 \) to \( r_0 + dr \), while the height \( h \) remains constant.
To estimate the change in \( S \), use the concept of differentials. The differential \( dS \) is given by the derivative of \( S \) with respect to \( r \), multiplied by \( dr \).
Calculate the derivative \( \frac{dS}{dr} \) using the product rule and chain rule: \( \frac{dS}{dr} = \pi \left( \sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}} \right) \).
The differential formula that estimates the change in the lateral surface area is \( dS = \pi \left( \sqrt{r^2 + h^2} + \frac{r^2}{\sqrt{r^2 + h^2}} \right) dr \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Calculus
Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this context, it is used to estimate how small changes in the radius of a cone affect its lateral surface area, assuming the height remains constant.
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Lateral Surface Area of a Cone
The lateral surface area of a right circular cone is given by the formula S = πr√(r² + h²), where r is the radius and h is the height. Understanding this formula is crucial for determining how changes in the radius impact the surface area when the height is fixed.
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Differential Formula
A differential formula is used to approximate the change in a function's value due to small changes in its variables. For the cone's surface area, the differential formula involves calculating the derivative of the surface area with respect to the radius, providing an estimate of the change when the radius increases by a small amount dr.
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