Final answer:
To find the tangent line to a curve at a specific point, such as t = 25 s, one must determine the endpoints of the tangent, calculate the slope using these endpoints, and then understand that if a secant line is parallel to this tangent line, it will have the same slope.
Step-by-step explanation:
To find the tangent line to a curve at a specific point, one must follow a certain set of steps. This process is rooted in calculus and is an application of derivatives. Let's say we need to find the tangent line to the curve at t = 25 seconds (s).
- Determine the endpoints of the tangent. Assume from a given figure that these endpoints correspond to a position of 1300 meters (m) at time 19 seconds (s) and a position of 3120 meters (m) at time 32 seconds (s).
- Calculate the slope of the tangent, v, using the endpoints. The slope is the change in position (also known as displacement) over the change in time, giving us the velocity.
So, v = (3120 m - 1300 m) / (32 s - 19 s). - Substitute the endpoint values into the slope formula to determine the slope. For example: v = (3120 m - 1300 m) / (32 s - 19 s) = 1820 m / 13 s = 140 m/s, which is the slope of the tangent line at t = 25 s.
If a secant line is given to be parallel to the tangent line, it will have the same slope. In physics, the slope of a position versus time graph gives the velocity, thus in this context, the slope we calculated is also the velocity of the object at t = 25 s.