If you were given a graph of a function instead of its equation, explain how to find the average rate of change between two points. How does this differ from finding the instantaneous rate of change? Please use a diagram to help you explain.
it is the slope of the line joining the points. Recall that on the interval [a,b] that is
((f(b) - f(a)) / (b-a)
I'm sure your calculus text has illustrations and explanations.
I know google does.
To find the average rate of change between two points on a graph of a function, follow these steps:
1. Select two points on the graph between which you want to find the average rate of change. Let's call these points A and B.
2. Determine the coordinates of point A. Read the x-coordinate and the corresponding y-coordinate from the graph.
3. Determine the coordinates of point B in the same manner.
4. Calculate the change in y-values by subtracting the y-coordinate of point A from the y-coordinate of point B. This represents the vertical change or the change in the function's output.
5. Calculate the change in x-values by subtracting the x-coordinate of point A from the x-coordinate of point B. This represents the horizontal change or the change in the function's input.
6. Divide the change in y by the change in x to find the average rate of change. This value represents how the function's output changes on average for every unit change in its input.
The formula for average rate of change between two points A and B is:
Average rate of change = (yB - yA) / (xB - xA)
Here's a visual representation of the steps using a diagram:
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Graph:
B .
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A
1. Select points A and B on the graph.
B .
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A .
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```
2. Determine the coordinates of point A (xA, yA).
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B .
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A(xA, yA) .
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```
3. Determine the coordinates of point B (xB, yB).
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B(xB, yB) .
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A(xA, yA) .
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```
4. Calculate the change in y-values (yB - yA).
```
B(xB, yB) .
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A(xA, yA) .
|______|
Δy
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```
5. Calculate the change in x-values (xB - xA).
```
B(xB, yB) .
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|______.
A(xA, yA) .
|______| Δx
Δy
.
.
```
6. Divide the change in y by the change in x to find the average rate of change.
```
B(xB, yB) .
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A(xA, yA) .
|______| Δx
______
Average rate of change = Δy / Δx
```
Finding the instantaneous rate of change is different from finding the average rate of change because it involves finding the rate of change at a specific point rather than between two points. To find the instantaneous rate of change, you need to calculate the derivative of the function at the specific point of interest. This derivative represents the slope of the tangent line to the function at that point and gives you the rate of change at that exact moment.