Slope of secant line between two points, use to estimate instantanous rate of change at a point.
Average Rate of Change
The slope of a secant line between two points on a curve is the average rate of change of the function over that interval. It tells us how much the function changes on average during that interval. To estimate the instantaneous rate of change at a point, we need to make the interval of the secant line smaller and smaller until it approaches zero. When the interval approaches zero, it becomes the tangent line at that point, which is the instantaneous rate of change of the function at that point.
To calculate the slope of a secant line between two points, we can use the formula:
slope = (y2 – y1) / (x2 – x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points on the curve.
For example, let’s say we have the function f(x) = x^2 and we want to estimate the instantaneous rate of change at x = 2. We can choose a point on either side of x = 2, say (1,1) and (3,9), and calculate the slope of the secant line:
slope = (9 – 1) / (3 – 1) = 4
This tells us that on average, the function is increasing at a rate of 4 units per 1 unit increase in x between x = 1 and x = 3. To estimate the instantaneous rate of change at x = 2, we can make the interval smaller, say by choosing a point closer to x = 2 such as (1.5, 2.25) and (2.5, 6.25):
slope = (6.25 – 2.25) / (2.5 – 1.5) = 4
This still gives us the same slope as before, but we can see that as the interval gets smaller, the secant line approaches the tangent line at x = 2, which has a slope of 4. Therefore, we can estimate that the instantaneous rate of change of f(x) at x = 2 is 4.
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