rates over time
- If the car travels for , then it moves , or
- If the car travels for , then it moves
- If the car travels for , then it moves
The way we calculate distance is by mulitplying the rate times the length of time.
We can model this graphically.
Area = Accumulated Distance
A Constant Rate
Our car’s rate is a constant function: .
Since time is measured by the horizontal axis. our calculation can be represented visually as the rectangular area under the rate curve.
The area of the rectangle represents the distance travelled.
The height of the rectangle is the height of the graph, which is the value of the rate, .
The width of the rectangle is a time measurement. In the graph above, this is .
The area is height times width, which gives
Let the accumulated miles travelled after .
Graphically, this is the area of the rectangle below the graph from to .
is the accumulated distance travelled in , which makes its derivative.
A Linear Rate
This time our rate function is a linear function: .
Our car goes faster as it goes farther.
is still represented visually as the area under the rate curve.
The area is a triangle this time.
Over the interval , the area is
is the accumulated distance travelled in , which makes its derivative.
Accumulation
Below is the graph of , approximated with
is a piecewise linear function, which means we can calculate area using rectangles and triangles.
The shaded regions are made of triangles, trapezoids, and rectangles. And, we know their formulas from Geometry class.
area = triangle + trapezoid + rectangle + trapezoid + triangle
This is the area under , so it is the approximate area under from to .
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more examples can be found by following this link
More Examples of Approximate Behavior