6Week 41

6.1 Readings for this week's lectures

Sections 15.3 and 17.1 in the textbook.

6.2 Readings for this week's exercises

Sections 15.1-15.2 in the textbook.

6.3 Notes

Chain rules

The chain rule for the composite function of two variables $F(x,y)$ , where $x=f(t)$ and $y=g(t)$ is:

$\frac{dF}{dt} = F'_1(x,y)f'(t) + F'_2(x,y)g'(t)$

In Leibniz's notation:

$\frac{dF}{dt} = \frac{\partial F}{\partial x}\frac{dx}{dt} + \frac{\partial F}{\partial y}\frac{dy}{dt}$

This can be extended from one variable ( $t$ ) to two variables ( $t$ and $s$ ) to give the composite function:

$z = F(x,y), \qquad x = f(t,s), \quad y=g(t,s)$

The chain rule for this case is:

$\frac{\partial z}{\partial t} = \frac{\partial F}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial F}{\partial y}\frac{\partial y}{\partial t}$

$\frac{\partial z}{\partial s} = \frac{\partial F}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial F}{\partial y} \frac{\partial y}{\partial s}$

Finally, the general chain rule for a function $F(x_1,x_2,\ldots,x_n)$ of $n$ variables that are in turn functions of $m$ other variables $x_1 = f_1(t_1,t_2,\ldots,t_m), \ldots, x_n = f_n(t_1,t_2,\ldots,t_m)$ is:

$\frac{\partial F}{\partial t_j} = \frac{\partial F}{\partial x_1}\frac{\partial x_1}{\partial t_j} + \frac{\partial F}{\partial x_2}\frac{\partial x_2}{\partial t_j} + \cdots + \frac{\partial F}{\partial x_n}\frac{\partial x_n}{\partial t_j}$

6.4 Problems

A function of two variables is given by the expression

$f(x,y) = \sqrt{9+2\ln(x-2) + 2\ln(y-2)}$ What is the domain?

A function of two variables is given by the expression

$f(x,y) = \frac{1}{e^{2x+2y}-4}$ What is the domain?

Find all the first- and second-order partial derivatives of:

$z = x^4-3x^3y+\ln(x)$

For $f(x,y) = x^4e^{2y}+x^2$ , find the first- and second-order partial derivatives at the point $(x,y)=(1,0)$ .

For the function $z=x + y^2$ , where $x=t^2$ and $y=t^3$ , find $\frac{dz}{dt}$ . Express your answer in terms of $t$ .

For the function $z=10e^{4x}+8\ln(y)$ , where $x=4t^2$ and $y=\frac{5}{t}$ , find $\frac{dz}{dt}$ . Express your answer in terms of $t$ .

Consider a function $z=D(p,q)$ , where $p=400e^{0.2t}$ and $q=50e^{-0.4t}$ . Find an expression for $\frac{dz}{dt}$ .

During summer, the hourly sales of ice cream, $S$ , at a particular beach is given by a function, $S(T, N),$ of the temperature $T$ and the number of people at the beach. Both the temperature and the number of people are functions of the cloud cover, $v$ , which is 0 when there are no clouds and 1 when it is completely overcast.

Construct an expression for $\frac{\mathrm{d} S}{\mathrm{d} v}$ by dragging the correct elements below into the empty boxes.
$\frac{\mathrm{d} S}{\mathrm{d} v}$ =
$\,\cdot\,$
$\,+\,$
$\,\cdot\,$
$\frac{\partial T}{\partial v}$
$\frac{\mathrm{d} T}{\mathrm{d} v}$
$\frac{\mathrm{d} S}{\mathrm{d} N}$
$\frac{\partial S}{\partial N}$
$\frac{\partial N}{\partial v}$
$\frac{\partial S}{\partial v}$
$\frac{\partial S}{\partial T}$
$\frac{\mathrm{d} N}{\mathrm{d} v}$
$\frac{\mathrm{d} S}{\mathrm{d} T}$
Correct! Correct! Wrong. Incorrect. Try again!
Drag the correct elements on to the empty boxes to match the statements.
a. The growth in ice cream sales, as the number of beachgoers increases:
b. The increase in temperature, as the cloud cover increases:
c. The growth in ice cream sales, as the cloud cover increases:
d. The increase in the number of beachgoers, as the cloud cover increases:
e. The growth in ice cream sales, as the temperature increases:

$\frac{\partial T}{\partial v}$
$\frac{\mathrm{d} T}{\mathrm{d} v}$
$\frac{\mathrm{d} S}{\mathrm{d} N}$
$\frac{\partial S}{\partial N}$
$\frac{\partial N}{\partial v}$
$\frac{\partial S}{\partial v}$
$\frac{\partial S}{\partial T}$
$\frac{\mathrm{d} N}{\mathrm{d} v}$
$\frac{\mathrm{d} S}{\mathrm{d} v}$
Correct! Correct! Wrong. Incorrect. Try again!

A utility company wants to make a forecast of the energy consumption, based on historical data. The company assumes that the energy consumption, $E,$ of an average household is a function of the outside temperature, $T,$ and the length of the day $L,$ which are both function of the time of year, $t.$

Introduce and define your own variables and functions, as needed, to answer the questions below. You may also have to make certain assumptions, which you have to argue for.

Write down an expression for $E$ .
You do not know the specific expression for $E$ , but you know that $E$ is a function of $T$ and $L$ , which in turn are functions of $t$ . You can therefore introduce these fuctions and write down a general expression for $E$ .
Write down an expression for $\frac{\mathrm{d} E}{\mathrm{d} t}$ .
Provide an interpretation of all the quantities that are part of the two expressions.
Provide an argument for the sign of each the quantities you found above for:
1. $t$ around 1 May.
2. $t$ around 1 October.
  You may want to state whether you are on the Northern or Southern Hemisphere.

For the function $z=5xy^2-6x^2y$ where $x=3s+t$ and $y=4s-t$ , find $z'_s$ and $z'_t$ .

For the function $z=y^2 e^{2xy}$ , where $x=t-3s$ and $y=ts^2$ , find $z'_s$ . Then find $z'_s$ in the point $(t,s)=(1,1).$

In this exercise we consider Toyota's costs when producing cars.

Toyota produces both electric cars and conventional cars. Let $z$ be the total cost of producing cars, $x$ the number of produced electric cars and $y$ the number of conventional cars.

The total cost is a function, $f$ , og the number of produced cars of each type: $z = f(x, y)$ .

The number of each type of car produced depends on how Toyta sets the price of each type of car. As an example, if they increase the price of the electric cars, more people will probably buy the conventional cars, and fewer electric cars will be produced.

Let $j$ be the price of electric cars and $k$ the price of conventional cars. Then $x=g(j,k)$ and $y=h(j,k)$ , where $g$ and $h$ are two functions.

We are interested in how Toyota's manufacturing costs change when the price of electric cars changes. With the chain rule, we can write this as:

$\frac{\partial z}{\partial j} = f_1'(x,y) \frac{\partial x}{\partial j} + f_2'(x,y) \frac{\partial y}{\partial j}$

Now, match the sentences below to the correct mathematical expressions, by dragging the expressions to the empty boxes.

a) The total cost increases when the number of produced conventional cars increases:

b) The number of produced electric cars falls when the price of electric cars increases:

c) The cost of producing electric cars falls when the price of electric cars increases:

d) The cost of conventional cars increases when the price of electric cars increases:

$f_1'(x,y) \frac{\partial x}{\partial j} > 0$

$f'_1(x,y) > 0$

$f_2'(x,y) > 0$

$\frac{\partial x}{\partial j} < 0$

$f'_2(x,y) \frac{\partial x}{\partial j} > 0$

$f'_1(x,y) \frac{\partial x}{\partial j} < 0$

$\frac{\partial x}{\partial j} > 0$

$\frac{\partial y}{\partial j} > 0$

Consider the function defined by

$w=xy+yz-xz$ where $x=4u+2v$ , $y=4u-5v$ , and $z=uv$ . Find the expression for $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ as functions of $u$ and $v$ . Then, evaluate $\frac{\partial w}{\partial u}$ and $\frac{\partial w}{\partial v}$ at the point $(u,v) = (1,4)$ .