Curve Attachment and Parametric Equations

Chapter 1: Attach The Curve

• Objective: To plot a curve using parametric equations, indicating the direction of tracing as time (t) increases.

• Parametric Equations:

  • x = e^t + t

  • y = e^t - t

• Interval: The curve is traced between t = -2 and t = 2.

• Point Plotting:

  • Create a chart with columns for t, x, and y.

  • Integer values of t to consider: -2, -1, 0, 1, 2.

• Computations:

  • For t = -2:

  • x = e^{-2} + (-2) = 5.39

  • y = e^{-2} - (-2) = 2.14

  • For t = -1:

  • Calculate: x = e^{-1} - 1
    ightarrow x = 1.72

  • y = e^{-1} + 1
    ightarrow y = 1.37

  • For t = 0:

  • x = e^{0} + 0 = 1

  • y = e^{0} - 0 = 1

  • For t = 1:

  • x = e^{1} + 1 = 1.37

  • y = e^{1} - 1 = 1.72

  • For t = 2:

  • x = e^{2} + 2 = 2.14539

  • y = e^{2} - 2 = 0.39

• Graphical Representation:

  • Plot points on the graph using calculated coordinates:

  • For example: at t = -2, the curve starts at (5.39, 2.14).

Chapter 2: Get This Point

• Continuing Graph:

  • At t = -1, point calculated at (1.72, 1.37).

  • At t = 0, point is at (1,1).

  • At t = 1, point is approximately at (1.37, 1.72).

  • At t = 2, final point plotted at (2.14539, 0.39).

• Direction of Trace:

  • Curve traces from left to right and upwards ultimately finishing at (2.14539, 0.39) when t = 2.

• The traced curve provides a visual understanding of how the parametric equations describe the movement in the given interval.