Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = 3 csc(1 − 2√x)
4. Applications of Derivatives
Differentials
1:17 minutesProblem 3.9.40
Textbook Question
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 = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change
Verified step by step guidance1
Identify the formula for the lateral surface area of a right circular cylinder, which is given by \( S = 2\pi rh \), where \( r \) is the radius and \( h \) is the height.
Recognize that the problem asks for the change in the lateral surface area when the height changes from \( h_0 \) to \( h_0 + dh \), while the radius \( r \) remains constant.
To estimate the change in the lateral surface area, use the concept of differentials. The differential \( dS \) represents the approximate change in \( S \) and is given by the derivative of \( S \) with respect to \( h \), multiplied by \( dh \).
Calculate the derivative of \( S \) with respect to \( h \): \( \frac{dS}{dh} = 2\pi r \). This derivative represents the rate of change of the surface area with respect to the height.
Express the differential formula for the change in surface area: \( dS = 2\pi r \cdot dh \). This formula estimates the change in the lateral surface area when the height changes by a small amount \( dh \).
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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 with respect to a variable. In this context, it helps estimate how a small change in one variable (like height) affects another quantity (like surface area). Understanding differentials is crucial for approximating changes in geometric properties.
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Lateral Surface Area of a Cylinder
The lateral surface area of a right circular cylinder is given by the formula S = 2πrh, where r is the radius and h is the height. This formula calculates the area of the curved surface of the cylinder, excluding the top and bottom bases. Knowing this formula is essential for determining how changes in height affect the surface area.
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Partial Derivatives
Partial derivatives are used to find the rate of change of a multivariable function with respect to one variable while keeping others constant. In this problem, the partial derivative of the lateral surface area with respect to height (h) is needed to estimate the change in surface area when the height changes, while the radius remains constant.
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