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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 3.9.40
Chapter 3, Problem 3.9.40

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

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1
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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Fundamental Theorem of Calculus Part 1

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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Example 1: Minimizing Surface Area

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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Derivatives
Related Practice
Textbook Question

For what value or values of the constant m, if any, is


ƒ(x) = { sin 2x, x ≤ 0

{ mx, x > 0


a. continuous at x = 0?

b. differentiable at x = 0?


Give reasons for your answers.

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Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = x√(1 − x²)

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Textbook Question

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.


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Textbook Question

Finding Derivative Values


In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.


f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

291
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Textbook Question

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


3 + sin y = y – x³

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Textbook Question

Derivatives


In Exercises 23–26, find dr/dθ.


r = (1 + sec θ) sin θ

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