Question

Find the equations of the tangent line and normal line to the curve at the specified point?

Answer 1

To find the equations of the tangent and normal lines to a curve, determine the endpoints of the tangent line, calculate its slope, and then construct the equations using this slope and the point of tangency.

Step-by-step explanation:

To find the equations of the tangent line and normal line to the curve at a specified point, we can follow these steps:

1. Identify the point on the curve where the tangent line should be drawn. In this case, at time t = 25 s.
2. Determine the endpoints of the tangent line. From the provided information, these endpoints represent positions at 1300 m at 19 s and 3120 m at 32 s.
3. Calculate the slope of the tangent line, v, using the difference in position over the difference in time.
4. Use this slope to write the equation of the tangent line.
5. The slope of the normal line is the negative reciprocal of the tangent line's slope. With this, you can find the equation of the normal line.

The slope, a, can be calculated as follows:

(260 m/s - 210 m/s) / (51 s - 1.0 s) = 1.0 m/s².

The principle that the slope of a curve at a point equals the slope of the tangent line at that point helps us to solve for this slope.