The accompanying graph shows the total amount of gasoline A...

The accompanying graph shows the total amount of gasoline $A$ in the gas tank of an automobile after it has been driven for $t$ days. (FIGURE CAN'T COPY) a. Estimate the average rate of gasoline consumption over the time intervals $[0,3],[0,5],$ and [7,10] b. Estimate the instantaneous rate of gasoline consumption at the times $t=1, t=4,$ and $t=8$ c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

The accompanying graph shows the total amount of gasoline $A$ in the gas tank of an automobile after it has been driven for $t$ days. (FIGURE CAN'T COPY)
a. Estimate the average rate of gasoline consumption over the time intervals $[0,3],[0,5],$ and [7,10]
b. Estimate the instantaneous rate of gasoline consumption at the times $t=1, t=4,$ and $t=8$
c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

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Key Concepts

Average Rate of Change

This concept measures how a quantity changes, on average, over a specified interval. It is calculated by finding the difference in the function's values at the endpoints of the interval and dividing by the difference in the input values (?y/?x). This provides a straightforward measure of overall change, often represented graphically by the slope of the secant line connecting two points on a curve.

Instantaneous Rate of Change

This is the rate at which a quantity changes at a specific moment in time. It is determined by taking the derivative of the function with respect to time, which represents the slope of the tangent line to the curve at that point. Understanding the instantaneous rate of change is critical in contexts where precision is needed, such as analyzing consumption rates at a particular moment.

Derivative

The derivative of a function provides a mathematical way to determine the instantaneous rate of change. It represents how a function's output value changes as its input changes, and can be interpreted graphically as the slope of the tangent line at any given point on the function's graph. This concept is fundamental in calculus for analyzing variable rates and is directly applied when estimating rates of consumption or change over time.

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