Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.8.11a

Chapter 3, Problem 3.8.11a

If the original 24 m edge length x of a cube decreases at the rate of 5 m/min, when x = 3 m at what rate does the cube’s
a. surface area change?

Verified step by step guidance

Identify the formula for the surface area of a cube, which is given by \( S = 6x^2 \), where \( x \) is the edge length of the cube.

Differentiate the surface area \( S \) with respect to time \( t \) to find the rate of change of the surface area. Use the chain rule: \( \frac{dS}{dt} = \frac{d}{dt}(6x^2) = 12x \frac{dx}{dt} \).

Substitute the given rate of change of the edge length \( \frac{dx}{dt} = -5 \) m/min into the differentiated equation. This negative sign indicates that the edge length is decreasing.

Substitute \( x = 3 \) m into the equation \( \frac{dS}{dt} = 12x \frac{dx}{dt} \) to find the rate of change of the surface area when the edge length is 3 m.

Calculate the expression \( 12 \times 3 \times (-5) \) to determine the rate at which the surface area is changing at the moment when \( x = 3 \) m.

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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 with respect to another. In this problem, we need to determine how the surface area of a cube changes as its edge length changes over time. This requires understanding how to differentiate equations with respect to time, often using the chain rule.

Surface Area of a Cube

The surface area of a cube is calculated as 6x^2, where x is the length of an edge. Understanding this formula is crucial because it allows us to express the surface area in terms of x, which can then be differentiated to find the rate of change of the surface area as x changes.

Chain Rule in Differentiation

The chain rule is a fundamental technique in calculus used to differentiate composite functions. In this context, it helps us find the derivative of the surface area with respect to time by relating it to the derivative of the edge length with respect to time. This is essential for solving related rates problems where variables are interdependent.

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Intro to the Chain Rule

Related Practice

Textbook Question

Find y'' if:

a. y = csc x

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Textbook Question

Interpreting Derivative Values

Effectiveness of a drug On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.

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a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

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Textbook Question

Interpreting Derivative Values

Growth of yeast cells In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure.

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a. Explain what is meant by the derivative P'(5). What are its units?

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Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

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Textbook Question