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

Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill. 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-to-distance graph for a sports car accelerating from a standstill.
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 $t=20 \mathrm{sec}$.

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Key Concepts

Units in Rate Problems

In contexts involving distance and time, the units of the rate (or slope) are determined by the units of the distance and time measurements. For instance, if distance is measured in meters and time in seconds, the speed will be expressed in meters per second, which quantitatively conveys how much distance is covered per unit of time.

Average Rate of Change

The average rate of change is calculated by finding the ratio of the change in the dependent variable (distance) to the change in the independent variable (time) over a specific interval. This is represented graphically by the slope of a secant line connecting two points on the distance-time curve, and it provides an overall measure of how quickly distance is being covered on average between those points.

Instantaneous Rate of Change

The instantaneous rate of change is the limit of the average rate of change as the time interval approaches zero. In practical terms, it represents the car’s speed at an exact moment in time and corresponds to the slope of the tangent line to the distance-time graph at that point. This concept is the foundation of derivatives in calculus.

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