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Understanding the concavity of a function can help you comprehend how the rate of change is behaving over time. In mathematical terms, a function is concave down when its second derivative is negative. For instance, in the function for the average speed of a trip, \( S(t) = \frac{70}{t} \), the second derivative \( S''(t) = \frac{140}{t^3} \) is negative when \( t > 0 \). This implies that as time \( t \) increases, the rate of change of the average speed slows down.

In practical terms, this means that if a trip takes longer, each additional hour doesn't result in as significant a decrease in speed as the previous hour did. The curve of the graph flattens out as time increases, showing that while your speed is still decreasing, it is doing so at a slower rate.

Key Points to Remember:

• The function is concave down if the second derivative is negative.
• Concavity provides insights into the rate of change's acceleration or deceleration.
• In this problem, it reflects the diminishing reduction in speed as time increases.

The relationship between distance and time is fundamental when calculating speed. Speed measures how quickly distance is covered over a particular period, and it's commonly expressed in units like miles per hour (mph).

For this exercise, the total distance of the commute is 70 miles. If we let \( t \) represent the time in hours, the average speed \( S \) can be described by the function \( S(t) = \frac{70}{t} \). This formula shows that if you know how long the journey takes, you can compute the average speed.

Important Aspects:

• A shorter travel time \( t \) increases average speed \( S \).
• A longer travel time \( t \) decreases average speed \( S \).
• Speed is inversely proportional to time when distance is constant.

This reciprocal linkage allows us to draw conclusions about speed from the time taken, providing a clear glance at how variability in trip duration affects overall travel efficiency.

Graphing is a powerful tool when analyzing functions, as it transforms abstract equations into visual data. For the average speed function \( S(t) = \frac{70}{t} \), plotting on a graph helps illustrate how speed changes as time varies.

Steps for Graphing:
1. Choose a range of time \( t \), here from 1 to 3 hours.
2. Calculate \( S(t) \) for various \( t \)-values within the range.
3. Plot these points on a graph where the x-axis is \( t \) (time in hours) and the y-axis is \( S(t) \) (average speed in mph).

The Resulting Graph:

• Shows a hyperbolic curve, starting high at short durations and flattens over longer periods.
• Illustrates the inverse relationship: as time increases, average speed decreases.
• Visually demonstrates the concave down nature by the decreasing slope as \( t \) increases.

This visual representation aids in grasping how speed is sensitive to changes in travel time, facilitating a more intuitive understanding of the relationship between these variables.