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=et+tx = e^t + t

    • y=et−ty = e^t - t

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

  • Point Plotting:

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

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

  • Computations:

    • For t=−2t = -2:

    • x=e−2+(−2)=5.39x = e^{-2} + (-2) = 5.39

    • y=e−2−(−2)=2.14y = e^{-2} - (-2) = 2.14

    • For t=−1t = -1:

    • Calculate: x=e−1−1<br>ightarrowx=1.72x = e^{-1} - 1 <br>ightarrow x = 1.72

    • y=e−1+1<br>ightarrowy=1.37y = e^{-1} + 1 <br>ightarrow y = 1.37

    • For t=0t = 0:

    • x=e0+0=1x = e^{0} + 0 = 1

    • y=e0−0=1y = e^{0} - 0 = 1

    • For t=1t = 1:

    • x=e1+1=1.37x = e^{1} + 1 = 1.37

    • y=e1−1=1.72y = e^{1} - 1 = 1.72

    • For t=2t = 2:

    • x=e2+2=2.14539x = e^{2} + 2 = 2.14539

    • y=e2−2=0.39y = e^{2} - 2 = 0.39

  • Graphical Representation:

    • Plot points on the graph using calculated coordinates:

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

Chapter 2: Get This Point

  • Continuing Graph:

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

    • At t=0t = 0, point is at (1,1).

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

    • At t=2t = 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=2t = 2.

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