Final answer:
To find the tangent line to a curve, identify the point of tangency, determine two points on the tangent line, and calculate the slope using the change in y over the change in x.
Step-by-step explanation:
To find a tangent line to a curve given by an equation, you typically follow these steps:
- Identify the specific point at which you need to find the tangent. For example, at t = 25 s.
- Find the endpoints of the tangent line segment you're interested in. These are positions on the curve that the line will pass through, such as 1300 m at 19 s and 3120 m at 32 s.
- Use these endpoints to calculate the slope of the tangent line. The formula for the slope (commonly represented as v or a) is the change in y divided by the change in x, or (y2 - y1)/(x2 - x1).
The slope of a curve at a particular point is the same as the slope of the line tangent to the curve at that point. This is the key concept for solving such problems. If you have a graph, you can visually estimate the tangent line at the point of interest and then calculate the slope using the coordinates of two points on that line.