Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.9

Chapter 3, Problem 3.6.9

The speed of sound (in m/s) in dry air is approximated the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v' (T) and interpret its meaning.

Verified step by step guidance

Step 1: Identify the given function v(T) = 331 + 0.6T, which represents the speed of sound in dry air as a function of temperature T.

Step 2: Recognize that v'(T) represents the derivative of the function v(T) with respect to T, which gives the rate of change of the speed of sound with respect to temperature.

Step 3: Differentiate the function v(T) = 331 + 0.6T with respect to T. Since 331 is a constant, its derivative is 0. The derivative of 0.6T with respect to T is 0.6.

Step 4: Conclude that v'(T) = 0.6. This means that for each degree Celsius increase in temperature, the speed of sound increases by 0.6 meters per second.

Step 5: Interpret the result: The derivative v'(T) = 0.6 indicates that the speed of sound in dry air increases linearly with temperature at a constant rate of 0.6 m/s per degree Celsius.

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Key Concepts

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. In this context, v'(T) represents the rate of change of the speed of sound with respect to temperature. It provides insight into how sensitive the speed of sound is to variations in temperature.

Function Interpretation

Understanding the function v(T) = 331 + 0.6T involves recognizing that it is a linear function where the speed of sound increases with temperature. The constant 331 m/s represents the speed of sound at 0 degrees Celsius, while the coefficient 0.6 indicates the increase in speed for each degree rise in temperature.

Physical Context of Derivatives

In physics, derivatives often have practical interpretations. For this problem, v'(T) not only quantifies how the speed of sound changes with temperature but also reflects the physical phenomenon that sound travels faster in warmer air. This relationship is crucial for applications in meteorology and acoustics.

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