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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.9.37
Chapter 3, Problem 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

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1
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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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 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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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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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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Related Practice
Textbook Question

In Exercises 41–58, find dy/dt.


y = (t⁻³/⁴ sin(t))⁴/³

Textbook Question

Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

Textbook Question

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.


(y - x)² = 2x + 4, (6, 2)

Textbook Question

Find the derivatives of the functions in Exercises 19–40.


s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)

Textbook Question

In Exercises 41–58, find dy/dt.


y = tan²(sin³(t))

Textbook Question

Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?


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