Speed of a car The accompanying figure shows the...

Speed of a car The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill. GRAPH CANT COPY a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure 2.6 What are the appropriate units for these slopes? b. Then estimate the car's speed at time $t=20 \mathrm{sec}$

Speed of a car The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill.
GRAPH CANT COPY
a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure 2.6 What are the appropriate units for these slopes?
b. Then estimate the car's speed at time $t=20 \mathrm{sec}$

Key Concepts

Units Analysis in Slope

When interpreting the slope from a distance-time graph, the units are determined by the units of distance and time. Typically, if distance is measured in meters and time in seconds, then the slope (or speed) will be in meters per second. This analysis is essential for ensuring consistency and making sense of the computed rates.

Graphical Interpretation of Slope

Understanding how to read and interpret slopes on a graph is crucial. In a distance-time graph, the slope at any point indicates the speed, and by estimating the slopes of secants and tangents, one can derive both average and instantaneous speeds.

Average Rate of Change

This concept involves computing the change in the function's output over the change in the input over a specified interval. It is represented by the slope of a secant line between two points on the graph and tells you how fast the quantity (in this case, distance) is changing on average over that interval.

Secant Lines

A secant line connects two points on a graph and is used to estimate the average rate of change of the function between those points. The slope of the secant line approximates how steeply the function is rising or falling over that interval.

Instantaneous Rate of Change

This is the rate at which a function is changing at a specific point, which corresponds to the slope of the tangent line at that point. In the context of the distance-time graph, it represents the car's speed at a particular moment.

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