Question

Use the limit definition of the derivative to find the slope of the tangent line to the curve at ?

Answer 1

The slope of the tangent line to the curve at t = 25 s is calculated by determining the endpoints of the tangent, which are the positions at 19 s and 32 s, and then using these to calculate the rate of change of position over time, yielding a slope of 140 m/s.

Step-by-step explanation:

Finding the Slope of the Tangent Line to a Curve

To find the slope of the tangent line to the curve at a specific point, in this case at t = 25 s, we need to follow these steps:

1. Locate the point on the curve at which we desire the tangent line.
2. Determine the endpoints of the tangent line. For our scenario, the endpoints are given as a position of 1,300 m at 19 s and a position of 3,120 m at 32 s.
3. Calculate the slope 'v' by using the coordinates of these endpoints in the slope formula: (change in position)/(change in time). Therefore, the slope v is calculated as:
   ((3,120 m - 1,300 m) / (32 s - 19 s)) which simplifies to (1,820 m / 13 s), resulting in a slope of 140 m/s.

This slope corresponds to the velocity at t = 25 s since velocity is the derivative of position with respect to time.