Preview Activity 10.2.1.
As the only employee of Steer Clear, you have decided that you need to understand how the timing of position measurements will change different properties related to your self-driving car. You decide to drive in a figure eight path described by \(\vr_8(t)=\langle \cos(t),\sin(2t) \rangle \) for \(0 \leq t \leq 2 \pi\text{.}\) A plot of this path is given in Figure 10.2.1.
A curve in the plane. The curve intersects itself at the origin and is symmetric about both axes. On each side of the vertical axis, the curve forms a sort of squashed loop with a point at the origin.
(a)
In order to understand how often your software should collect location data, you decide to look at your position for a few different times. Calculate the following, rounding the component values to three decimal places. Draw the output vectors of \(\vr_8\) on Figure 10.2.1 in standard position.
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\(\displaystyle \vr_8(3)\)
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\(\displaystyle \vr_8(3.1)\)
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\(\displaystyle \vr_8(3.14)\)
The tips of these vectors correspond to the locations that would be sampled if you wanted to know the location of your car at \(\vr_8(\pi)\) but collected data every second, every tenth of a second, and every hundredth of a second, respectively.
(b)
Write a couple of sentences to describe both geometrically and algebraically what happens to the output of \(\vr_8(t)\) as \(t\rightarrow\pi\text{.}\)
(c)
Calculate \(\vr_8(\pi)\) and \(\vr_8(\pi)-\vr_8(3)\text{.}\) Sketch \(\vr_8(\pi)\) and \(\vr_8(3)\) in standard position in Figure 10.2.1. Also plot \(\vr_8(\pi)-\vr_8(3)\text{,}\) positioned in such a way to illustrate that it is difference of these two position vectors.
Hint.
Remember that \(\vr_8(\pi)\text{,}\) \(\vr_8(3)\text{,}\) and \(\vr_8(\pi)-\vr_8(3)\) will form a triangle as in Figure 9.2.6.
(d)
Compute \(\frac{\vr_8(\pi)-\vr_8(3)}{\pi-3}\) and explain how this calculation is different than the result of the previous step.
(e)
How would you expect \(\frac{\vr_8(\pi)-\vr_8(3.1)}{\pi-3.1}\) to be different than \(\frac{\vr_8(\pi)-\vr_8(3)}{\pi-3}\text{?}\) Use this idea write about what is measured by \(\frac{\vr_8(\pi)-\vr_8(\pi-h)}{h}\) if we look at smaller and smaller values of \(h\text{.}\) Remember to be specific about what aspects of our curve or drive are being measured.
