3.7 The Multivariable Chain Rule

The Multivariable Chain Rule

Introduction

The Chain Rule

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 $P(x,y)=10-\frac{1}{2} x^2-\frac{1}{5}y^2$. 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

, where the $x$ and $y$ coordinates are the same as measured by Alex's function and $t$ is measured in minutes.

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 $P(t)$. 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.

Suppose $z(x,y) = x^2+xy^3$. In addition, suppose that $x$ and $y$ are restricted to points that move around the plane by following a circle of radius $2$ centered at the origin that is parameterized by

. - Use the chain rule to find the instantaneous rate of change $\frac{dz}{dt}$. - Substitute $x(t)$ for $x$ and $y(t)$ for $y$ in the rule for $z$ to write $z$ in terms of $t$ and calculate $\frac{dz}{dt}$ using only techniques from single-variable calculus. - Write a couple of sentences comparing your answers above.

Suppose that the temperature on a metal plate is given by the function $T$ with

where the temperature is measured in degrees Fahrenheit and $x$ and $y$ are each measured in feet. - Find $T_x$ and $T_y$. What are the units on these partial derivatives? - Suppose an ant is walking along the $x$-axis at the rate of 2 feet per minute toward the origin. When the ant is at the point $(2,0)$, what is the instantaneous rate of change in the temperature $dT/dt$ that the ant experiences. Include units on your response. - Suppose instead that the ant walks along the ellipse $\vr(t) = \langle 6\cos(t), 3\sin(t)\rangle$, where $t$ is measured in minutes. Use the multivariable chain rule to find $\frac{dT}{dt}$. What does this tell you about the path along which the ant is walking?

Suppose that you are walking along a surface whose elevation is given by a function $f$. Some contours of $f$ are as shown in the image below. Furthermore, suppose that if you consider how your location corresponds to points in the $xy$-plane, you know that when you pass the point $(2,1)$, your velocity vector is $\vv=\langle -1,2\rangle$. Estimate the rate of change $df/dt$ when you pass through $(2,1)$.

Work through this exercise and explain your reasoning step by step.

There are several proposed formulas to approximate the surface area of the human body. One model（DuBois D, DuBois DF. A formula to estimate the approximate surface area if height and weight be known. Arch Int Med 1916;17:863-71.） uses the formula

where $A$ is the surface area in square meters, $h$ is the height in centimeters, and $w$ is the weight in kilograms.

Since a person's height $h$ and weight $w$ change over time $h$ and $w$ are functions of time $t$. Let us think about what is happening to a child whose height is $60$ centimeters and weight is $9$ kilograms. Suppose, furthermore, that $h$ is increasing at an instantaneous rate of 20 centimeters per year and $w$ is increasing at an instantaneous rate of $5$ kg per year.

Determine the instantaneous rate at which the child's surface area is changing at this point in time.