Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = (t⁻³/⁴ sin(t))⁴/³
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.9.37
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 surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx
Guida verificata passo dopo passo1
Identify the formula for the surface area of a cube, which is given by \( S = 6x^2 \), where \( x \) is the length of an edge of the cube.
To estimate the change in surface area, we use the concept of differentials. The differential \( dS \) represents the approximate change in the surface area when the edge length changes by a small amount \( dx \).
Calculate the derivative of the surface area with respect to \( x \). This is \( \frac{dS}{dx} = \frac{d}{dx}(6x^2) \).
Compute the derivative: \( \frac{dS}{dx} = 12x \). This derivative tells us how the surface area changes with respect to a small change in \( x \).
The differential formula for the change in surface area is \( dS = 12x \, dx \). This formula estimates the change in surface area \( dS \) when the edge length changes by \( dx \).
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Differential Calculus
Differential calculus focuses on the concept of the derivative, which represents the rate of change of a function. In this context, it helps estimate how a small change in one variable (like the edge length of a cube) affects another variable (such as surface area). Understanding how to compute and apply derivatives is crucial for solving problems involving differential estimates.
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Percorso guidato
06:11
Fundamental Theorem of Calculus Part 1
Surface Area of a Cube
The surface area of a cube is calculated using the formula S = 6x², where x is the length of an edge. This formula is essential for determining how changes in the edge length affect the surface area. Recognizing this relationship allows us to apply differential calculus to estimate changes in surface area when the edge length changes slightly.
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09:07Example 1: Minimizing Surface Area
Differential Formula
A differential formula provides an approximation of how a function changes as its input changes. For a function S = 6x², the differential dS can be expressed as dS = d(6x²) = 12x dx, where dx is a small change in x. This formula is used to estimate the change in surface area when the edge length of the cube changes by a small amount dx.
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Percorso guidato
05:53Finding Differentials
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In Exercises 41–58, find dy/dt.
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