Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.95c

Chapter 3, Problem 3.95c

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.

c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Verified step by step guidance

Start by identifying the given equation for the total surface area of the cylinder: \( S = 2\pi r^2 + 2\pi rh \).

Recognize that \( S \), \( r \), and \( h \) are all functions of time \( t \), so we need to use implicit differentiation with respect to \( t \).

Differentiate both sides of the equation with respect to \( t \). For the first term \( 2\pi r^2 \), use the chain rule: \( \frac{d}{dt}(2\pi r^2) = 4\pi r \frac{dr}{dt} \).

For the second term \( 2\pi rh \), apply the product rule: \( \frac{d}{dt}(2\pi rh) = 2\pi \left( r \frac{dh}{dt} + h \frac{dr}{dt} \right) \).

Combine the differentiated terms to express \( \frac{dS}{dt} \) in terms of \( \frac{dr}{dt} \) and \( \frac{dh}{dt} \): \( \frac{dS}{dt} = 4\pi r \frac{dr}{dt} + 2\pi \left( r \frac{dh}{dt} + h \frac{dr}{dt} \right) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this context, we are interested in how the surface area of a cylinder changes with respect to time as both the radius and height change. This concept is fundamental in calculus, as it allows us to connect different rates of change through derivatives.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. In the context of the given problem, it allows us to express the derivative of the surface area with respect to time as a function of the derivatives of the radius and height. This is crucial for relating the rates of change of different variables.

Surface Area of a Cylinder

The surface area of a right circular cylinder is given by the formula S = 2πr² + 2πrh, which includes contributions from both the circular bases and the lateral surface. Understanding this formula is essential for applying related rates, as it provides the relationship between the radius, height, and surface area that we need to differentiate with respect to time.

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c. When, if ever, during the interval does the body change direction?

s = 25/(t + 5), −4 ≤ t ≤ 0

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[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:

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s = t² - 3t + 2, 0 ≤ t ≤ 5

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Textbook Question

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In Exercises 51–54,

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