To calculate the slope of a curve that is not a straight line, the arc method is a useful approach. This method involves finding the slope between two points on the curve, rather than focusing on a single point. The slope of a curve can vary significantly, so selecting different pairs of points will yield different average slopes over the chosen region.
To apply the arc method, begin by identifying two points on the curve. Draw a straight line connecting these points, which represents the average slope over the arc between them. This line approximates the behavior of the curve in that region. The average slope can be calculated using the formula:
\(\text{slope} = \frac{\text{rise}}{\text{run}}\)
Here, the rise is the vertical change between the two points, and the run is the horizontal change. For example, if the first point has a y-coordinate of 4 and the second point has a y-coordinate of 5, the rise is:
\(\text{rise} = 5 - 4 = 1\)
If the x-coordinates of the first and second points are 3 and 6, respectively, the run is:
\(\text{run} = 6 - 3 = 3\)
Substituting these values into the slope formula gives:
\(\text{slope} = \frac{1}{3}\)
This value represents the average slope over the arc defined by the two selected points. It is important to note that if different points are chosen, the average slope may vary, but the method of drawing a line and calculating the slope remains consistent. This approach is particularly useful in many mathematical contexts, allowing for a straightforward estimation of slope over a curve without the need for more complex calculations like drawing tangent lines.