Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.8.18b

Chapter 3, Problem 3.8.18b

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
b. How is ds/dt related to dy/dt and dz/dt if x is constant?

Verified step by step guidance

To find how ds/dt is related to dy/dt and dz/dt, we start by differentiating the given formula for the diagonal length s with respect to time t. The formula is s = √(x² + y² + z²).

Since x is constant, its derivative with respect to time, dx/dt, is zero. Therefore, we only need to consider the derivatives of y and z with respect to time.

Apply the chain rule to differentiate s with respect to t: ds/dt = (1/2) * (1/√(x² + y² + z²)) * (2y * dy/dt + 2z * dz/dt).

Simplify the expression: ds/dt = (y * dy/dt + z * dz/dt) / √(x² + y² + z²).

This equation shows how the rate of change of the diagonal length, ds/dt, is related to the rates of change of the edges y and z, dy/dt and dz/dt, respectively, when x is constant.

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Key Concepts

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

Partial Derivatives

Partial derivatives are used to find the rate of change of a function with respect to one variable while keeping other variables constant. In this problem, since x is constant, we focus on how s changes with respect to y and z, requiring the use of partial derivatives to express ds/dt in terms of dy/dt and dz/dt.

Chain Rule

The chain rule in calculus is a formula for computing the derivative of a composite function. It is essential here to relate ds/dt to dy/dt and dz/dt, as it allows us to differentiate s = √(x² + y² + z²) with respect to time t, considering y and z as functions of t.

Pythagorean Theorem in 3D

The formula s = √(x² + y² + z²) is derived from the Pythagorean theorem extended to three dimensions, representing the diagonal of a rectangular box. Understanding this geometric interpretation helps in visualizing how changes in y and z affect the diagonal length s, which is crucial for setting up the relationship between ds/dt, dy/dt, and dz/dt.

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