Final answer:
To find the equation of a tangent line to a curve, calculate the derivative to find the slope at the specific point, and then use the point-slope form of a line to get the equation.
Step-by-step explanation:
To find the equation of a tangent line to a curve at a given point, one must understand that the slope of a curve at any point is equivalent to the slope of the tangent line at that point. The process typically involves calculating the derivative of the equation representing the curve to find the slope at the specific point. Here's a step-by-step guide:
- First, determine the derivative of the function that represents the curve. This derivative function gives us the slope of the tangent at any point on the curve.
- Next, evaluate this derivative at the given point. If the point is at t = 25 s, we plug in t = 25 into the derivative to get the slope at that point, denoted as v.
- With the slope known, and the coordinates of the given point (t, y), we can use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the point.
For example, if the endpoints of the tangent are given by positions 1300 m at time 19 s and 3120 m at time 32 s, use these to calculate the slope v by the formula for slope: v = (y2 - y1) / (t2 - t1). You can then use one of the endpoints as your point to plug into the point-slope form and get the equation of the tangent line.