Preview Activity 11.6.1.
Your self-driving car company, Steer Clear, is doing well and almost ready to launch its first car. Some of your engineers have reported that the car has problems when the air intake encounters a large quantity of large particulate matter (sand, dust, large pollen, smog, etc.) in the air. To fix this issue, you have created a new type of filter that uses a sophisticated mesh and gravity to filter out large particulate matter. However, this new type of filter gets clogged if exposed to too much large particulate matter too quickly. To determine the viability of this filter, you must measure the rate at which your car’s intake will be exposed to large particulate matter per unit time.
You consult a friend Alex who does atmospheric modeling of large particulate matter in your area. Alex created a function that describes the amount of large particulate matter in the air in terms of location relative to her lab. This function is expressed algebraically as \(P(x,y)=10-\frac{1}{2} x^2-\frac{1}{5}y^2\text{.}\) Here \(x\) is the distance east/west from Alex’s lab and \(y\) is the distance north/south from her lab. Both distances are measured in kilometers and \(P\) is measured in parts per million.
Using the self-driving feature of your car, you specify that your car will move along a test course that can be described as
\begin{equation*}
\vr(t) = \langle x(t), y(t) \rangle = \langle 2-t^2, t^3 + 1\rangle\text{,}
\end{equation*}
where the \(x\) and \(y\) coordinates are the same as measured by Alex’s function and \(t\) is measured in minutes.
(a)
Substitute the path component functions \(x(t)\) and \(y(t)\) into the expression for \(P(x,y)\) but do not simplify the resulting expression for \(P(t)\text{.}\) This gives a one-variable function that describes the amount of large particulate matter encountered at each location on your test course as a function of time.
(b)
Use derivative rules from single-variable calculus to find \(P'(t)=\frac{dP}{dt}\text{.}\) Do not simplify this expression. Note that this is not a partial derivative because \(P(t)\) only has one dependent variable.
(c)
Alex has told you that the amount of large particulate matter in the air changes in the \(y\)-direction (north/south) because of the proximity to industrial pollution from a factory complex. Also, the amount of large particulate matter in the air changes in the \(x\)-direction (east/west) because of tree pollen from a species of tree in a nearby forest. You would like to understand how the rate of large particulate matter encountered on the car’s drive is split into these two factors.
Alex suggests that you measure \(\frac{dP}{dt}\) at a location \((x,y)=(a,b)\) by taking the derivative of the linearization for \(P\) at \((a,b)\text{.}\) The linearization of \(P\) at \((a,b)\) is given by
\begin{equation*}
L(x,y)=P(a,b)+P_x(a,b)(x-a)+P_y(a,b)(y-b)
\end{equation*}
Remember that \(x=x(t)\) and \(y=y(t)\) are being considered as functions of time in this scenario.
Write a couple of sentences to explain why
\begin{equation}
\frac{dP}{dt}\approx\frac{dL}{dt} = 0 + P_x(a,b) \frac{dx}{dt}+P_y(a,b) \frac{dy}{dt}\tag{11.6.1}
\end{equation}
and how \(P_x(a,b) \frac{dx}{dt}\) and \(P_y(a,b) \frac{dy}{dt}\) measure the rates of change in pollution coming from tree pollen and industrial pollution, respectively.
(d)
Use the formulas provided at the beginning of the Preview Activity for \(P(x,y)\text{,}\) \(x(t)\text{,}\) and \(y(t)\) to compute each of the following. Then substitute your expressions into Alex’s approximation given in equation (11.6.1). Do not simplify this expression.
\begin{equation*}
P_x(a,b)\qquad P_x(a,b)\qquad \frac{dx}{dt}\qquad \frac{dy}{dt}\qquad
\end{equation*}
(e)
Compare your results for part b and part d and write a couple of sentences that describe how each part of your answer for part d corresponds to different parts of part b. Remember your work for part b is completely in terms of \(t\) so you will need to translate some terms from \(t\) into its meaning in other quantities.
