Final answer:
To determine the equation of the tangent line to a curve at a certain point, calculate the slope at that point and use it with the point of tangency to write the equation. This involves identifying two points on the tangent line and using them to calculate the slope, if that information is provided, as in the motion example at t = 25 s.
Step-by-step explanation:
To find the equation of the line tangent to a given curve at a specific point, you need to determine the slope of the curve at that point. The slope of the curve at a point is equal to the slope of the tangent line. You would then use this slope, along with the coordinates of the point of tangency, to write the equation in point-slope form or slope-intercept form.
For example, at t = 25 s, if the curve corresponds to motion and the position is given at specific times, you can determine the slope by taking two points on the tangent line. Let's say the endpoints of the tangent are at a position of 1300 m at time 19 s and a position of 3120 m at time 32 s. Plug these endpoints into the slope formula to get the slope (v). From this, you can find the equation of the tangent line.