Final answer:
To find a vertical tangent line to a curve, calculate the function's derivative, locate points where it is undefined or infinite, and verify the curve's existence at these points. This process helps identify vertical tangents at specific instances, like t = 25 s in the question's context.
Step-by-step explanation:
To find a vertical tangent line to a curve, you need to follow a series of steps involving calculus. Specifically, you want to find where the derivative of the function representing the curve is undefined or infinite, as this indicates a vertical slope, which is characteristic of a vertical tangent line.
- Calculate the derivative of the function that defines the curve, which gives you the slope of the tangent line at any point along the curve.
- Determine the points where the derivative is undefined or approaches infinity. These points are potential candidates for where a vertical tangent line might occur.
- Verify that the function has a point on the curve at the x-values found in the previous step to confirm the existence of the vertical tangent.
The given scenario describes an application where the slope of a curve at a particular time, t = 25 s, is required. By applying this method, you can find the tangent line to the curve at t = 25 s and calculate its slope using points the curve passes through, such as 1,300 m at 19 s and 3,120 m at 32 s.