Final answer:
To find the equation of a tangent line, determine the point of tangency, calculate the slope using the positions at specific times, and then use the point-slope form to write the linear equation of the tangent.
Step-by-step explanation:
To find the equation of a tangent line to a curve using calculus and a calculator, you will follow these steps:
- Identify the specific point (t = 25 s in this case) at which you want to find the tangent line.
- Use the curve's function to determine the endpoint positions at specific times, which are 1300 m at 19 s and 3120 m at 32 s.
- Enter these endpoints into your calculator to find the slope (v). To do this, you would calculate the change in the y-values of the positions divided by the change in the x-values of the times, effectively (3120 m - 1300 m)/(32 s - 19 s).
- The slope found will be the slope of the tangent at t = 25 s. This represents the rate of change of position with respect to time at that specific moment.
- To find the full equation of the tangent line, use the point-slope form of a line: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the curve where the tangent line touches.
- Enter the slope and the original point coordinates into this equation to yield the linear equation of the tangent line.
- For a more refined and accurate calculation, analyze the data rounding to four decimal places as needed.
For assistance with calculations, refer to your calculator's instructions or the TI website. Remember that technology tools like graphing calculators can greatly simplify this process by calculating slopes and equations automatically.