Chapter 4, Math Tools: Multivariate Calculus Video Solutions, Molecular Driving Forces

Problem 1

Which of the following are exact differentials?
(a) $6 x^{5} d x+d y$
(b) $x^{2} y^{2} d x+3 x^{2} y^{3} d y$
(c) $(1 / y) d x-\left(x / y^{2}\right) d y$
(d) $y d x+2 x d y$
(e) $\cos x d x-\sin y d y$
(f) $\left(x^{2}+y\right) d x+\left(x+y^{2}\right) d y$
(g) $x d x+\sin y d y$

Linh Vu

Numerade Educator

00:59

Problem 2

Compute the partial derivatives $(\partial f / \partial x)_{y}$ and $(\partial f / \partial y)_{x}$ for the following functions:
(a) $f(x, y)=3 x^{2}+y^{5}$
(b) $f(x, y)=x^{10} y^{1 / 2}$
(c) $f(x, y)=x+y^{2}+3$
(d) $f(x, y)=5 x$

Sriram Soundarrajan

Numerade Educator

01:43

Problem 3

Minimizing a single-variable function subject to a constraint. Given the function $y=(x-2)^{2}$, find $x^{*}$, the value of $x$ that minimizes $y$ subject to the constraint $y=x$.

Sriram Soundarrajan

Numerade Educator

05:07

Problem 4

Maximizing a multivariate function without constraints. Find the maximum $\left(x^{*}, y^{*}, z^{*}\right)$ of the function $f(x, y, z)=d-(x-a)^{2}-(y-b)^{2}-(z-c)^{2} .$

Zachary Watson

Numerade Educator

01:38

Problem 5

Extrema of multivariate functions with constraints.
(a) Find the maximum of the function $f(x, y)=$ $-(x-a)^{2}-(y-b)^{2}$ subject to the constraint $y=k x$.
(b) Find the minimum of the paraboloid $f(x, y)=$ $\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}$ subject to the constraint $y=2 x$.

James Kiss

Numerade Educator

00:49

Problem 6

Composite functions.
(a) Given the functions $f(x, y(u))=x^{2}+3 y^{2}$ and $y(u)=5 u+3$, express $d f$ in terms of changes $d x$ and $d u$.
(b) What is $\left(\frac{\partial f}{\partial u}\right)_{x, u=1} ?$

Zachary Mitchell

Numerade Educator

03:06

Problem 7

Converting to an exact differential. Given the expression $d x+(x / y) d y$, show that dividing by $x$ results in an exact differential. What is the function $f(x, y)$ such that $d f$ is $d x+(x / y) d y$ divided by $x$ ?

Samuel Hannah

Numerade Educator

02:36

Problem 8

Propagation of error. Suppose that you can measure independent variables $x$ and $y$ and that you have a dependent variable $f(x, y)$. The average values are $\bar{x}, \bar{y}$, and $f$. We define the error in $x$ as the deviations $\varepsilon_{x}=x-\hat{x}$, in $y$ as $\varepsilon_{y}=y-\bar{y}$, and in $f$ as $\varepsilon_{f}=f-\bar{f}$.
(a) Use a Taylor series expansion to express the error $\varepsilon_{f}$ in $f$, as a function of the errors $\varepsilon_{x}$ and $\varepsilon_{y}$ in $x$ and $y$.
(b) Compute the mean-squared error $\left\langle\varepsilon_{f}^{2}\right\rangle$ as a function of $\left\langle e_{x}^{2}\right\rangle$ and $\left(\varepsilon_{y}^{2}\right)$.

Matt Just

Numerade Educator

01:31

Problem 9

Small differences in large numbers can lead to nonsense. Using the results from Problem 8, show that the propagated error is larger than the difference itself for $f(x, y)=x-y$, with $x=20 \pm 2$ and $y=19 \pm 2$.

Taylor Shimono

Numerade Educator

05:51

Problem 10

Finding extrema. Find the point $\left(x^{*}, y^{*}, z^{*}\right)$ that is at the minimum of the function
$$
f(x, y, z)=2 x^{2}+8 y^{2}+z^{2}
$$
subject to the constraint equation
$$
g(x, y, z)=6 x+4 y+4 z-72=0
$$

Ryan Swift

Numerade Educator

03:45

Problem 11

Derivative of a multivariable composite function. For the function $f(x, y(v))=x^{2} y+y^{3}$, where $y=m v^{2}$, compute $d f / d v$ around the point where $m=1, v=2$, and $x=3$.

Elijah Dejonge

Numerade Educator

01:57

Problem 12

Volume of a cylinder. For a cylinder of radius $r$ and height $h$, the volume is $V=\pi r^{2} h$, and the surface area is $A=2 \pi r^{2}+2 \pi r h$.
(a) Derive the height $h\left(r_{0}\right)$ that maximizes the volume of a cylinder with a given area $A=a_{0}$ and given radius $r_{0}$.
(b) Compute the change in volume, $\Delta V_{,}$from $\left(n_{1}, h_{1}\right)=$ $(1,1)$ to $\left(r_{2}, h_{2}\right)=(2,2)$.
(c) Compute the component volume changes $\Delta V_{a}$ and $\Delta V_{b}$ that sum to $\Delta V$, where $\Delta V_{a}$ is the change from $\left(r_{1}, h_{1}\right)=(1,1)$ to $\left(r_{2}, h_{1}\right)=(2,1)$ and $\Delta V_{b}$ is the change from $\left(r_{2}, h_{1}\right)=(2,1)$ to $\left(r_{2}, h_{2}\right)=(2,2)$.
(d) Should (b) equal (c)? Why or why not?

Tanishq Gupta

Numerade Educator

01:18

Problem 13

Equations of state. Which of the following could be the total derivative of an equation of state?
(a)
$$
\frac{2 n R T}{(V-n b)^{2}} d V+\frac{R(V-n b)}{n b^{2}} d T
$$
(b)
$$
-\frac{n R T}{(V-n b)^{2}} d V+\frac{n R}{V-n b} d T
$$

Akshaya Rs

Numerade Educator

01:55

Problem 14

Distance from the circumference of a circle. The circle shown in Figure $4.12$ satisfies the equation $x^{2}+y^{2}=4$. Find the point $\left(x^{*}, y^{*}\right)$ on the circle that is closest to the point $(3,2)$, (That is, minimize the distance $\left.f(x, y)=\Delta r^{2}=(x-3)^{2}+(y-2)^{2} .\right)$

Sriram Soundarrajan

Numerade Educator

01:52

Problem 15

Find $d f$ and $\Delta f$.
(a) If $f(x, y)=x^{2}+3 y$, express $d f$ in terms of $d x$ and $d y$.
(b) For $f(x, y)=x^{2}+3 y$, integrate from $(x, y)=$ $(1,1)$ to $(x, y)=(3,3)$ to obtain $\Delta f$.

Lucas Finney

Numerade Educator

01:43

Problem 16

Derivative of a composite function. For $f(x, y)=$ $x^{2} y+y^{3}$, where $y=m r^{2}$, find $d f / d r$.

Adrian Co

Numerade Educator

01:58

Problem 17

Maximum volume of a solid with constraints. You have a rectangular solid of length $x$, width $y$, and height $z$. Find the values of $x, y$, and $z$ that give the maximum volume, subject to a fixed surface area: $g(x, y, z)=$ $2 x z+2 y z+2 x y-$ constant $=0$.

Lucas Finney

Numerade Educator

00:49

Problem 18

Short-answer questions.
(a) Compute the partial derivatives: $(\partial f / \partial x) y$ and $(\partial f / \partial y)_{x}$ for the following functions:
$$
f(x, y)=\ln (2 x)+5 y^{3},
$$

Zachary Mitchell

Numerade Educator