Question

Estimate the slope of the tangent line at P(15, 1530) by averaging the slopes of two adjacent secant lines

Answer 1

To estimate the slope of the tangent line at a given point, calculate the slopes of secant lines using nearby points on the curve, then average these slopes to approximate the tangent's slope.

Step-by-step explanation:

The procedure to estimate the slope of the tangent line at a given point by averaging slopes of adjacent secant lines involves a few steps:

1. Find two points close to the point of tangency that lie on the curve. In this case, you are given two points representing the endpoints of the tangent line: one at time 19 s with a position of 1,300 m, and another at time 32 s with a position of 3,120 m.
2. Calculate the slope of the secant line through these two points using the formula slope (v) = (change in position) / (change in time).
3. To find the slope at point P(15, 1530), we will assume that the slopes of the secant lines adjacent to P are similar to the slope at P.
4. Finally, plug the corresponding values into the equation to get the average slope, which is an approximation of the slope of the tangent line at P(15, 1530).

For example, if you had slopes from two secants equivalent to (260 m/s - 210 m/s) / (51 s - 1.0 s), resulting in an average slope of 1.0 m/s², a similar process would be applied using the endpoints provided for the specific problem at hand (19 s, 1,300 m) and (32 s, 3,120 m).

Answer 2

Will need at least the function to which the line is tangent.