Final answer:
The slope of the tangent line on a graph represents the rate of change of the function at that point. To determine the points where the slope of the tangent is greatest and least, we need to look for the steepest and flattest parts of the graph.
Step-by-step explanation:
The slope of a tangent line on a graph represents the rate of change of the function at that point. To determine the points where the slope of the tangent is greatest and least, we need to look for the steepest and flattest parts of the graph. The greatest slope occurs at the point where the graph has the steepest upward or downward slope. The least slope occurs at the point where the graph is relatively flat.
For example, in a graph of position versus time, if the graph has a steep upward slope, the tangent at that point will have a steep positive slope. If the graph has a steep downward slope, the tangent at that point will have a steep negative slope. On the other hand, if the graph is relatively flat, the tangent at that point will have a slope close to zero.
Therefore, to find the points with the greatest and least slope of the tangent, we need to visually analyze the graph and identify the moments with the steepest upward or downward slopes (greatest slope) and the moments with relatively flat slopes (least slope).