Calculus 1 Labs
Lab 7: Linear approximation
The NASA Q36 Robotic Lunar Rover can travel up to 3 hours on a single charge and has a range of 1.6 miles. After leaving its base and traveling for \(t\) hours, the speed of the Q36 is given by the function \(v(t) = \sin \sqrt{9 t^2}\) in miles per hour. One hour into an excursion, the Q36 will have traveled 0.19655 miles. Two hours into a trip, the Q36 will have traveled 0.72421 miles.
Lab Preparation: Answer the following questions individually and bring your writeup to class.

Use your calculator to graph \(v(t)\). Explain in words what the graph says about how the Q36 moves during a 3hour trip starting with a full charge.

Using a full sheet of paper, carefully sketch a graph of the distance \(x(t)\) traveled by the Q36 measured in miles during this trip as a function of time in hours. Explain precisely why you drew the graph as you did.

Draw tangent lines to the graph of \(x(t)\) at times \(t_0 = 0,\ t_0=1,\ t_0 = 2,\) and \(t_0 = 3\). Label each tangent line with its equation. Use the variables \(x\) and \(t\) in these equations.
Lab: Work with your group on the following questions. We encourage you to collaborate both in and out of class, but you must write up your responses individually.
 In this problem, we will focus on the motion of the Q36 near the \(t_0 = 1\) hour mark.

Approximate how far the Q36 traveled in the 10 minutes immediately following the \(t_0 = 1\) hour mark. Is this an underestimate or overestimate? Explain.

Find an error bound for your approximation.
Note: When using a tangent line at time \(t_0\) to approximate the position at some other time, \(x(t)\), the error bound formula translates to
\[ \text{error} \leq \frac{M}{2} t  t_0^2 \]
where \(M = \maxx''\) between \(t_0\) and \(t\).

Draw a large graph of \(x\) vs. \(t\) zoomed in on the 10minute time interval explored in this question. Draw the tangent line used for Part a labeled with its equation. Label the change in time \( \Delta t = t  t_0\), the corresponding unknown distance \(\Delta x\) that you are approximating, your approximation for this distance, and the error.

 Controllers want to turn the Q36 around to head back to its base after traveling 0.75 miles.

Use the linear approximation at \(t_0 = 2\) to determine approximately what time this will happen. Will the actual time be a little earlier or a little later than your estimate? Explain.

Draw a large graph emphasizing the portion of the trip starting at \(t_0 = 2\) hours until the 0.75 mile mark. Include the tangent line used for your linear approximation. Label the change in distance \(\Delta x\) given in this problem, the corresponding unknown change in time \(\Delta t\), your approximation for this time, and the error.
