Textbook Question
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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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 3.9.37
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
Verified step by step guidance1
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
Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).
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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)
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Textbook Question
Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is
a. perpendicular to the line y = 1 - (x/24).
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b. parallel to the line y = √2 - 12x.
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Textbook Question
In Exercises 41–58, find dy/dt.
y = tan²(sin³(t))
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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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