91Ó°ÊÓ

Here are the curves given in their parametric form: (a) \(x=t, \quad y=t^{2}\) (b) \(x=\sqrt{t}, \quad y=t\) (c) \(x=e^{t}, \quad y=e^{2 t}\)

For each equation in (a), (b), and (c), sketch the graphs. (a) \(x=t, \quad y=t^{2}\) To sketch this curve, simply plot the points \((t, t^2)\) for various values of \(t\). This will result in a parabolic curve with its vertex at the origin. (b) \(x=\sqrt{t}, \quad y=t\) To sketch this curve, plot the points \((\sqrt{t}, t)\) for various values of \(t\). This will result in a curve that starts from the origin and increases in a slower pace. (c) \(x=e^{t}, \quad y=e^{2 t}\) To sketch this curve, plot the points \((e^{t}, e^{2t})\) for various values of \(t\). This curve will start from the point \((1, 1)\) and grow exponentially along the x and y directions.

Now, compare the graphs of the given curves to determine if they differ or not. (a) and (b) differ in their shapes as (a) is a parabolic curve, and (b) is a curve that starts from the origin and increases in a slower pace. Furthermore, their orientations in the Cartesian plane are also different. In (a), the parabolic curve opens upwards, while in (b), the curve is increasing more in the x direction. (a) and (c) also differ, as both their shapes and orientations are distinct. While (a) is a parabolic curve opening upwards, (c) is an exponential curve growing along both the x and y directions. (b) and (c) differ as well, considering that (b) is a curve that starts from the origin and increases in a slower pace, while (c) is an exponential curve. Their growth rates and the orientation of the curves on the Cartesian plane are distinct. In conclusion, all three curves differ from each other in terms of their shape, orientation, and growth rates.