Problem 14 - Calculus

Question

Give the units and sign of the derivative. \(T^{\prime}(v),\) where \(T(v)\) is the time, in minutes, that it takes to drive from Tucson to Phoenix at a constant speed of \(v\) miles per hour.

Step-by-Step Solution

Answer

The units of \(T'(v)\) are minutes per mile per hour, and its sign is negative.

Step 1: Identify the function

The function given is \(T(v)\), which represents the time, in minutes, to drive from Tucson to Phoenix at a constant speed \(v\) in miles per hour.

Step 2: Understand the relationship

The relationship between time and speed is inverse; higher speeds mean less time. This implies \(T(v)\) is a decreasing function.

Step 3: Derive the unit of the derivative

Since \(T(v)\) is in minutes and \(v\) is in miles per hour, the derivative \(T'(v)\) will have units of \(\frac{\text{minutes}}{\text{miles per hour}}\).

Step 4: Determine the sign of the derivative

Because \(T(v)\) is a decreasing function (as speed increases, time decreases), its derivative \(T'(v)\) will be negative, indicating a negative rate of change.

Key Concepts

Inverse Relationship

In mathematics, an inverse relationship means that as one variable increases, the other decreases. This is crucial when considering real-world scenarios such as travel time and speed. For the function \(T(v)\), higher speeds lead to lower travel times. Thus, time is inversely related to speed.

This concept fundamentally contrasts with a direct relationship, where an increase in one variable leads to an increase in the other. Understanding inverse relationships helps interpret how different variables affect one another.

In practical terms:

• If you drive faster (increase your speed \(v\)), the time \(T(v)\) taken to reach your destination reduces.
• This negative correlation is a hallmark of an inverse relationship.

Such relationships are common in physics and economics, where variables frequently interact inversely.

Decreasing Function

A decreasing function is one in which, as the input value increases, the output value decreases. In the context of \(T(v)\), this means that as speed \(v\) increases, the time \(T(v)\) decreases. Such functions are represented graphically by a downward slope.

When analyzing a function:

• Look for a negative derivative, as this indicates a decreasing trend.
• In the real world, this shows efficiency: getting somewhere faster by driving at a higher speed.

Decreasing functions highlight the importance of optimization, particularly when time constraints are involved. In this case, by effectively understanding how speed impacts time, you can plan your travel more efficiently.

Units of Derivative

The units of a derivative convey not just the rate of change but how two different quantities are related. In the problem, \(T'(v)\) measures how quickly time changes with respect to speed. The derivative \(T'(v)\) has units of minutes per mile per hour (\(\frac{\text{minutes}}{\text{miles per hour}}\)).

Why is understanding units important?

• It gives a concrete measure of change, helping grasp the pace of variation.
• It aids in comparisons, such as comparing time reductions at different speeds.

Correct unit interpretation is essential for real-world applications, allowing precise assessments of functions and their derivatives. Thus, unpacking units helps bridge the gap between mathematical concepts and practical usage.