Lab 8: Quadratic 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.

In Lab 7, you used tangent lines to approximate the distance $x(t)$ traveled by the Q36 Rover. Lines with slope $m$ through the point $(t_0, x_0)$ can be written in point-slope form as $x = x_0 + m(t - t_0)$ . You used the derivative $v(t) = x'(t)$ to find the slope at $t_0$ .

We could improve our approximations by using "best-fit parabolas." For the following problems, note that $x = x_0 + c(t - t_0) + c(t - t_0)^2$ is the equation of a parabola that passes through the point $(t_0, x_0)$ . Changing the parameters $m$ and $c$ will change the shape of the parabola without changing the fact that it passes through that point.

Lab Preparation: Answer the following questions individually and bring your write-up to class.

Sketch a parabola on your large graph of $x(t)$ that you think represents the best-fit parabola at time $t_0 = 1$ . Then determine the equation of this parabola using the form $x = x_0 + m(t - t_0) + c(t - t_0)^2$ and the point $(t_0, x_0) = (1, 0.19655)$ . To do this, find the first and second derivatives of the equation for this parabola, set $x'(1) = v(1)$ and $x''(1) = v'(1)$ , then solve for $m$ and $c$ .
Sketch a parabola on your large graph of $x(t)$ that you think represents the best-fit parabola at time $t_0 = 2$ . Then determine the equation of this parabola using the form $x = x_0 + m(t - t_0) + c(t - t_0)^2$ and the point $(t_0, x_0) = (2, 0.72421)$ . To do this, find the first and second derivatives of the equation for this parabola, set $x'(2) = v(2)$ and $x''(2) = v'(2)$ , then solve for $m$ and $c$ .

Lab: Verify that everyone in your group found the same quadratic approximations at $t_0 = 1$ and $t_0 = 2$ , then answer the following questions. We encourage you to collaborate both in and out of class, but you must write up your responses individually.

Use your quadratic equation at time $t_0 = 1$ to find a more accurate approximation to the distance traveled in the 10 minutes after the $t_0 = 1$ hour mark (i.e., more accurate than your linear approximation in Lab 7). Then find an error bound for this approximation.

The error bound for a quadratic approximation centered at $t_0$ and evaluated at $t$ is

$E(t) \leq \frac{M}{6}| t - t_0|^3 \text{ where } M = \max|x''| \text{ between } t \text{ and } t_0.$

(I recommend graphing $x''$ to determine a value for $M$ .)
Use your quadratic equation at time $t_0 = 2$ to find a more accurate approximation to the time the rover reaches the 0.75 mile mark (i.e. more accurate than your linear approximation in Lab 7).