Answers to Selected Exercises

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May 3, 2023
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- Functions Defined by Improper Integrals (Trench)

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$2\max(a,b)$

$2\min(a,b)$

$4\max(a,b,c)$

$4\min(a,b,c)$

$\infty$ (no);

$-1$ (yes)

$3$ (no);

$-3$ (no)

$\sqrt7$ (yes);

$-\sqrt7$ (yes)\

$2$ (no);

$-3$ (no)

$1$ (no);

$-1$ (no)

$\sqrt7$ (no); $-\sqrt7$ (no)

$2^n/(2n)!$

$2\cdot3^n/(2n+1)!$

$2^{-n}(2n)!/(n!)^2$

$n^n/n!$ \

$\dst{A_n=\frac{x^n}{n!}\left(\ln x-\sum_{j=1}^n\frac{1} {j}\right)}$

$f_n(x_1,x_2, \dots,x_n)=2^{n-1}\max(x_1,x_2, \dots,x_n)$ , $g_n(x_1,x_2, \dots,x_n)=2^{n-1}\min(x_1,x_2, \dots,x_n)$

$[\frac{1}{2},1)$ ;

$(-\infty,\frac{1}{2})\cup[1,\infty)$ ;

$(-\infty,0]\cup(\frac{3}{2},\infty)$ ;

$(0,\frac{3}{2}]$ ;

$(-\infty,0]\cup(\frac{3}{2},\infty)$ ;

$(-\infty,\frac{1}{2}]\cup[1,\infty)$

$(-3,-2) \cup (2,3)$ ;

$(-\infty,-3]\cup[-2,2]\cup[3,\infty)$ ;

$\emptyset$ ;

$(-\infty,\infty)$ ;

$\emptyset$ ;

$(-\infty,-3]\cup[-2,2]\cup[3,\infty)$

$\emptyset$ ; $(-\infty,\infty)$ ; $\emptyset$ ; $(-\infty,\infty)$ ; $\emptyset$ ; $(-\infty,\infty)$

$\emptyset$ ; $(-\infty,\infty)$ ; $[-1,1]$ ; $(-\infty,-1)\cup(1,\infty)$ ; $[-1,1]$ ; $(-\infty,\infty)$

$(0,3]$

$[0,2]$

$(-\infty,1)\cup(2,\infty)$

$(-\infty,0]\cup(3,\infty)$

$\frac{1}{4}$

$\frac{1}{6}$

$6$

$1$

neither;

$(-1,2)\, \cup\, (3,\infty)$ ;

$(-\infty,-1)\,\cup\,(2,3)$ ;

$(-\infty,-1]\,\cup\,(2,3)$ ;

$(-\infty,-1]\,\cup\,[2,3]$ open;

$S$ ;

$(1,2)$ ;

$[1,2]$ closed;

$(-3,-2)\,\cup\,(7,8)$ ;

$(-\infty,-3)\,\cup\,(-2,7)\,\cup(8,\infty)$ ;

$(-\infty-3]\,\cup[-2,7]\,\cup[8,\infty)$

closed; $\emptyset$ ; $\bigcup\set{(n,n+1)}{n=\mbox{integer}}$ ; $(-\infty,\infty)$

$\set{x}{x=1/n,\ n=1,2, \dots}$ ;

$\emptyset$ ,

$S_1=$ rationals,

$S_2=$ irrationals any set whose supremum is an isolated point of the set , the rationals

$S_1=$ rationals, $S_2=$ irrationals

$D_f=[-2,1)\cup[3,\infty)$ , $D_g=(-\infty,-3]\cup[3,7)\cup(7,\infty)$ , $D_{f\pm g}=D_{fg}=[3,7)\cup(7,\infty)$ , $D_{f/g}=(3,4)\cup(4,7)\cup(7,\infty)$

$\set{x}{x\ne(2k+1)\pi/2\mbox{ where }k=\mbox{integer}}$

$\set{x}{x\ne0,1}$

$\set{x}{x\ne0}$

$[1,\infty)$

$4$

$12$

$-1$

$2$

$-2$

$\frac{11}{17}$

$-\frac{2}{3}$

$\frac{1}{3}$

$2$

$0,2$

$0$ , none

$-\frac{1}{3},\frac{1}{3}$

none, $0$

$0$

$0$ none

$0$

$0$ none none none

$0$

$\infty$

$-\infty$

$\infty$

$-\infty$

none

$\infty$

none

$\infty$

$-\infty$ none

$\infty$

$\frac{3}{2}$

$\infty$

$-\infty$

$\infty$

$\frac{1}{2}$

$\lim_{x\to\infty}r(x)=\infty$ if $n>m$ and $a_n/b_m>0$ ; $=-\infty$ if $n>m$ and $a_n/b_m<0$ ; $=a_n/b_m$ if $n=m$ ; $=0$ if $n<m$ . $\lim_{x\to-\infty}r(x)=(-1)^{n-m}\lim_{x\to\infty}r(x)$

$\lim_{x\to x_0}f(x)=\lim_{x\to x_0}g(x)$

$\limsup_{x\to x_0-}(f-g)(x)\le\limsup_{x\to x_0-}f(x)-\liminf_{x\to x_0-}g(x)$ ; $\liminf_{x\to x_0-}(f-g)(x)\ge\liminf_{x\to x_0-}f(x)-\limsup_{x\to x_0-}g(x)$

from the right continuous none continuous none continuous

from the left

$[0,1)$ ,

$(0,1)$ ,

$[1,2)$ ,

$(1,2)$ ,

$(1,2]$ ,

$[1,2]$

$[0,1)$ ,

$(0,1)$ ,

$(1,\infty)$

$\tanh x$ is continuous for all $x$ , $\coth x$ for all $x\ne0$

$[-1,1]$ ,

$[0,\infty)$

$\bigcup_{n=-\infty}^\infty (2n\pi,(2n+1)\pi)$ ,

$(0,\infty)$

$\bigcup_{n=-\infty}^\infty(n\pi,(n+1)\pi)$ ,

$(-\infty,-1)\cup(-1,1)\cup(1,\infty)$

$\bigcup_{n=-\infty}^\infty[n\pi,(n+\frac{1}{2})\pi]$ , $[0,\infty)$

$(-1,1)$

$(-\infty,\infty)$

$x_0\ne(2k+\frac{3}{2}\pi),\ k=$ integer

$x\ne\frac{1}{2}$

$x\ne1$

$x\ne(k+\frac{1}{2}\pi),\ k=$ integer

$x\ne0$

$p(c)=q(c)$ and $p'_{-}(c)=q'_{+}(c)$

$f^{(k)}(x)=n(n-1)\cdots(n-k-1)x^{n-k-1}|x|$ if $1\le k\le n-1$ ; $f^{(n)}(x)=n!$ if $x>0$ ; $f^{(n)}(x)=-n!$ if $x<0$ ; $f^{(k)}(x)=0$ if $k>n$ and $x\ne0$ ; $f^{(k)}(0)$ does not exist if $k\ge n$ .

$c'=ac-bs$ ,

$s'=bc+as$

$c(x)=e^{ax}\cos bx$ , $s(x)=e^{ax}\sin bx$

$f(x)=-1$ if

$x\le0$ ,

$f(x)=1$ if

$x>0$ ; then

$f'(0+)=0$ , but

$f_+'(0)$ does not exist.

continuous from the right

There is no such function (Theorem~2.3.9).

Counterexample: Let $x_0=0$ , $f(x)=|x|^{3/2}\sin(1/x)$ if $x\ne0$ , and $f(0)~=0~$ .

Counterexample: Let $x_0=0$ , $f(x)=x/|x|$ if $x\ne0$ , $f(0)=0$ .

$-\infty$ if $\alpha\le0$ , $0$ if $\alpha>0$

$\infty$ if $\alpha>0$ , $-\infty$ if $\alpha\le0$

$-\infty$ $-\infty$ if $\alpha\le0$ , $0$ if $\alpha>0$

$0$

Suppose that $g'$ is continuous at $x_0$ and $f(x)=g(x)$ if $x\le x_0$ , $f(x)=1+g(x)$ if $x>x_0$ .

$1$

$e$

$1$ $e^L$

$f^{(n+1)}(x_0)/(n+1)!$ .

Counterexample: Let $x_0=0$ and $f(x)=x|x|$ .

Let $g(x)=1+|x-x_0|$ , so $f(x)=(x-x_0)(1+|x-x_0|)$ .

Let $g(x)=1+|x-x_0|$ , so $f(x)=(x-x_0)^2(1+|x-x_0|)$ .

$1$ ,

$2$ ,

$0$

$0$ ,

$-\pi$ ,

$3\pi/2$ ,

$-4\pi+\pi^3/2$ \

$-\pi^2/4$ ,

$-2\pi$ ,

$-6+\pi^2/4$ ,

$4\pi$

$-2$ , $5$ , $-16$ , $65$

$0$ , $-1$ , $0$ , $5$

$0$ ,

$1$ ,

$0$ ,

$5$

$-1$ ,

$0$ ,

$6$ ,

$-24$

$\sqrt2$ ,

$3\sqrt2$ ,

$11\sqrt2$ ,

$57\sqrt2$

$-1$ ,

$3$ ,

$-14$ ,

$88$ min neither min max min neither min

min

$f(x)=e^{-1/x^2}$ if $x\ne0$ , $f(0)=0$ (Exercise~)

None if $b^2-4c<0$ ; local min at $x_1=(-b+\sqrt{b^2-4c})/2$ and local max at $x_1=(-b-\sqrt{b^2-4c})/2$ if $b^2-4c>0$ ; if $b^2=4c$ then $x=-b/2$ is a critical point, but not a local extreme point.

$\dst\frac{1}{6}\left(\frac{\pi}{20}\right)^3$

$\dst\frac{1}{8^3}$

$\dst\frac{\pi^2}{512\sqrt2}$

$\dst\frac{1}{4(63)^4}$

$M_3h/3$ , where

$M_3=\sup_{|x-c|\le h}|f^{(3)}(c)|$ \

$M_4h^2/12$ where $M_4=\sup_{|x-c|\le h}|f^{(4)}(c)|$

$k=-h/2$

monotonic functions

Let $[a,b]=[0,1]$ and $P=\{0,1\}$ . Let $f(0)=f(1)=\frac{1}{2}$ and $f(x)=x$ if $0<x<1$ . Then $s(P)=0$ and $S(P)=1$ , but neither is a Riemann sum of $f$ over $P$ .

$\frac{1}{2}$ ,

$-\frac{1}{2}$

$\frac{1}{2}$ , $1$ $e^b-e^a$ $1-\cos b$ $\sin b$

$f(a)[g_1-g(a)]+f(d)(g_2-g_1)+f(b)[g(b)-g_2]$

$f(a)[g_1-g(a)]+f(b)[g(b)-g_p]+\sum_{m=1}^{p-1}f(a_m)(g_{m+1}-g_m)$

If $g\equiv1$ and $f$ is arbitrary, then $\int_a^b f(x)\,dg(x)=0$ .

$\overline{u}=c=\frac{2}{3}$

$\overline{u}=c=0$

$\overline{u}=(e-2)/(e-1),\ c=\sqrt{\overline{u}}$

$n!$

$\frac{1}{2}$ divergent

$1$

$-1$

$0$

divergent convergent divergent convergent convergent

divergent

$p<2$

$p<1$

$p>-1$

$-1<p<2$ none none

$p<1$

$p-q<1$

$p,q<1$

$-1<p<2q-1$

$q>-1$ ,

$p+q>1$

$q+1<p<3q+1$

$\deg g-\deg f\ge 2$

$2$

$1$

$0$

$1/2 \quad$

$\sqrt A$

$1$

$-\infty$

$0$

If $s_n=1$ and $t_n=-1/n$ , then $(\lim_{n\to\infty}s_n)/(\lim_{n\to\infty}t_n)=1/0=\infty$ , but $\lim_{n\to\infty}s_n/t_n=-\infty$ .

$\infty$ ,

$0$

$\infty$ ,

$-\infty$ if

$|r|>1$ ;

$2$ ,

$-2$ if

$r=-1$ ;

$0$ ,

$0$ if

$r=1$ ;

$1$ ,

$-1$ if

$|r|<1$

$\infty$ ,

$-\infty$ if

$r<-1$ ;

$0$ ,

$0$ if

$|r|<1$ ;

$\frac{1}{2}$ ,

$\frac{1}{2}$ if

$r=1$ ;

$\infty$ ,

$\infty$ if

$r>1$ \

$\infty$ ,

$\infty$

$|t|$ , $-|t|$

$1$ ,

$-1$

$2$ ,

$-2$

$3$ ,

$-1$

$\sqrt{3}/2$ , $-\sqrt{3}/2$

If $\{s_n\}=\{1,0,1,0, \dots\}$ , then $\lim_{n\to\infty}t_n=\frac{1}{2}$

$\lim_{m\to\infty}s_{2m}=\infty$ ,

$\lim_{m\to\infty}s_{2m+1}=-\infty$ \

$\lim_{m\to\infty}s_{4m}=1$ ,

$\lim_{m\to\infty}s_{4m+2}=-1$ ,

$\lim_{m\to\infty}s_{2m+1}=0$ \

$\lim_{m\to\infty}s_{2m}=0$ ,

$\lim_{m\to\infty}s_{4m+1}=1$ ,

$\lim_{m\to\infty}s_{4m+3}=-1$ \

$\lim_{n\to\infty}s_{n}=0$

$\lim_{m\to\infty}s_{2m}=\infty$ ,

$\lim_{m\to\infty}s_{2m+1}=0$ \

$\lim_{m\to\infty}s_{8m}=\lim_{m\to\infty}s_{8m+2}=1$ , $\lim_{m\to\infty}s_{8m+1}=\sqrt2$ ,\ $\lim_{m\to\infty}s_{8m+3}=\lim_{m\to\infty}s_{8m+7}=0$ , $\lim_{m\to\infty}s_{8m+5}=-\sqrt2$ ,\ $\lim_{m\to\infty}s_{8m+4}=\lim_{m\to\infty}s_{8m+6}=-1$

$\{1,2,1,2,3,1,2,3,4,1,2,3,4,5, \dots\}$

Let $\{t_n\}$ be any convergent sequence and $\{s_n\}=\{t_1,1,t_2,2, \dots,t_n,n, \dots\}$ .

No; consider $\sum 1/n$

convergent convergent divergent divergent \ convergent convergent divergent

convergent

$p>1$

convergent convergent if

$0<r<1$ , divergent if

$r\ge1$ \ divergent convergent divergent

convergent

convergent convergent convergent

convergent

divergent convergent if and only if

$0<r<1$ or

$r=1$ and

$p<-1$ convergent convergent

convergent

divergent convergent convergent

convergent if $\alpha<\beta-1$ , divergent if $\alpha\ge\beta-1$

divergent convergent convergent

convergent

$\sum(-1)^n$

$\sum(-1)^n/n$ ,

$\sum\dst\left[\frac{(-1)^n}{ n}+\frac{1}{ n\log n}\right]$ \

$\sum(-1)^n2^n$

$\sum(-1)^n$

conditionally convergent conditionally convergent absolutely convergent

absolutely convergent

%\begin{exercisepatch1}

Let $k$ and $s$ be the degrees of the numerator and denominator, respectively. If $|r|=1$ , the series converges absolutely if and only if $s\ge k+2$ . The series converges conditionally if $s=k+1$ and $r=-1$ , and diverges in all other cases, where $s\ge k+1$ and $|r|=1$ .

$\sum(-1)^n/\sqrt n$

$0$

$2A-a_0$

$F(x)=0,\ |x|\le1$

$F(x)=0,\ |x|\le1$ \

$F(x)=0,\ -1<x\le1$

$F(x)=\sin x,\ -\infty<x<\infty$ \

$F(x)=1,\ -1<x\le1$ ;

$F(x)=0,\ |x|>1$

$F(x)=x,\ -\infty<x<\infty$ \

$F(x)=x^2/2,\ -\infty<x<\infty$

$F(x)=0,\ -\infty<x<\infty$ \

$F(x)=1,\ -\infty<x<\infty$

$F(x)=0$

$F(x)=1,\ |x|<1$ ;

$F(x)=0,\ |x|>1$ \

$F(x)=\sin x/x$

$F_n(x)=x^n$ ; $S_k=[-k/(k+1),k/(k+1)]$

$[-1,1]$

$[-r,r]\cup\{1\}\cup\{-1\},\ 0<r<1$

$[-r,r]\cup\{1\},\ 0<r<1$ \

$[-r,r],\ r>0$

$(-\infty,-1/r]\cup[-r,r]\cup[1/r,\infty)\cup\{1\},\ 0<r<1$ \

$[-r,r],\ r>0$

$(-\infty,-r]\cup[r,\infty)\cup\{0\},\ r>0$ \

$[-r,r],\ r>0$

Let $S=(0,1]$ , $F_n(x)=\sin(x/n)$ , $G_n(x)=1/x^2$ ; then $F=0$ , $G=1/x^2$ , and the convergence is uniform, but $\|F_nG_n\|_S=\infty$ .

$3$

$1$

$\frac{1}{2}$

$e-1$

compact subsets of

$(-\frac{1}{2},\infty)$

$[-\frac{1}{2},\infty)$ closed subsets of

$\dst\left(\frac{1-\sqrt5}{2},\frac{1+\sqrt5}{2}\right)$

$(-\infty,\infty)$

$[r,\infty),\ r>1$

compact subsets of $(-\infty,0)\cup(0,\infty)$

Let

$S=(-\infty,\infty)$ ,

$f_n=a_n$ (constant), where

$\sum a_n$ converges conditionally, and

$g_n=|a_n|$ .

``absolutely"

means that

$\sum|f_n(x)|$ converges pointwise and

$\sum f_n(x)$ converges uniformly on

$S$ , while

means that $\sum|f_n(x)|$ converges uniformly on $S$ .

$\dst{\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{ n!(2n+1)}}$

$\dst{\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)(2n+1)!}}$

$1/3e$

$1$

$\frac{1}{3}$

$1$

$\infty$

$1$

$\frac{1}{2}$

$\frac{1}{4}$

$4$

$1/e$

$1$

$x(1+x)/(1-x)^3$ $e^{-x^2}$ \ $\dst{\sum_{n=1}^\infty\frac{(-1)^{n-1}}{ n^2} }(x-1)^n;\ R=1$

$\mbox{Tan}^{-1}x=\dst{\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)}};\ f^{(2n)}(0)=0;\ f^{(2n+1)}(0)=(-1)^2(2n)!$ ;

$\dst{\frac{\pi}{6}=\mbox{Tan}^{-1}\frac{1}{\sqrt3}=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)3^{n+1/2}}}$

$\cosh x=\dst{\sum_{n=0}^\infty\frac{x^{2n}}{(2n)!}}$ , $\sinh x=\dst{\sum_{n=0}^\infty\frac{x^{2n+1}}{(2n+1)!}}$

$(1-x)\sum_{n=0}^\infty x^n=1$ converges for all $x$

$\dst{x+x^2+\frac{x^3}{3}-\frac{3x^5}{40}+\cdots}$

$\dst{1-x-\frac{x^2}{2}+\frac{5x^3}{6}+\cdots}$

$\dst{1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{721x^6}{720}+\cdots}$

$\dst{x^2-\frac{x^3}{2}+\frac{x^4}{6}- \frac{x^5}{6}+\cdots}$

$\dst{1+x+\frac{2x^2}{3}+\frac{x^3}{3}+\cdots}$

$\dst{1-x-\frac{x^2}{2}+\frac{3x^3}{2}+\cdots}$

$\dst{1+\frac{x^2}{2}+\frac{5x^4}{24}+\frac{61x^6}{720}+\cdots}$

$\dst{1+\frac{x^2}{6}+\frac{7x^4}{360}+\frac{31x^6}{15120} +\cdots}$

$\dst{2-x^2+\frac{x^4}{12}-\frac{x^6}{360}+\cdots}$

$\dst{F(x)=\frac{5}{(1-3x)(1+2x)}=\frac{3}{1-3x}+\frac{2}{1+2x}= \sum_{n=0}^\infty[3^{n+1}-(-2)^{n+1}]x^n}$ $1$

$(3,0,3,3)$

$(-1,-1,4)$

$(\frac{1}{6},\frac{11}{12},\frac{23}{24},\frac{5}{36})$

$\sqrt{15}$

$\sqrt{65}/12$

$\sqrt{31}$

$\sqrt3$

$\sqrt{89}$

$\sqrt{166}/12$

$3$

$\sqrt{31}$

$12$

$\frac{1}{32}$

$27$

$\mathbf{X}=\mathbf{X}_0+t\mathbf{U}\ (-\infty<t<\infty)$ in all cases.

$\dots\mathbf{U}$ and $\mathbf{X}_1-\mathbf{X}_0$ are scalar multiples of $\mathbf{V}$ .

$\mathbf{X}=(1,-3,4,2)+t(1,3,-5,3)$

$\mathbf{X}=(3,1,-2,1,4,)+t(-1,-1,1,3,-7)$

$\mathbf{X}=(1,2,-1)+t(-1,-3,0)$

$5$

$2$

$1/2\sqrt5$

$\set{(x_1,x_2,x_3,x_4)}{|x_i|\le3\ (i=1,2,3) \mbox{ with at least one equality}}$

$\set{(x_1,x_2,x_3,x_4)}{|x_i|\le3\ (i=1,2,3)}$

$S$

$\set{(x_1,x_2,x_3,x_4)}{|x_i|>3 \mbox{ for at least one of }i=1,2,3}$

$S$

$\emptyset$

$\set{(x,y,z)}{z\ne1\mbox{ or }x^2+y^2>1}$

open neither

closed

$(\pi,1,0)$

$(1,0,e)$

$6$

$2\sqrt5$

$2L\sqrt n$

$\infty$

$\set{(x,y)}{x^2+y^2=1}$

$\dots$ if for $A$ there is an integer $R$ such that $|\mathbf{X}_r|>A$ if $r\ge R$ .

$10$

$3$

$1$

$0$

$a/(1+a^2)$

$\infty$

$\infty$ no

$-\infty$

$0$

$0$ none

$0$

none

$D_f$ is unbounded and for each

$M$ there is an

$R$ such that

$f(\mathbf{X})>M$ if

$\mathbf{X}\in D_f$ and

$|\mathbf{X}|>R$ . Replace $>M$'' by

$<M$ ’’ in

$\lim_{\mathbf{X}\to\mathbf{0}}f(\mathbf{X})=0$ if $a_1+a_2+\cdots+a_n>b$ ; no limit if $a_1+a_2+\cdots+a_n\le b$ and $a_1^2+a_2^2+\cdots+a_n^2\ne0$ ; $\lim_{\mathbf{X}\to\mathbf{0}}f(\mathbf{X})=\infty$ if $a_1=a_2=\cdots=a_n=0$ and $b>0$ .

No; for example, $\lim_{x\to\infty}g(x,\sqrt x)=0$ .

$\mathbb R^3$

$\mathbb R^2$

$\mathbb R^3$

$\mathbb R^2$

$\set{(x,y)}{x\ge y}$

$\mathbb R^n$

$\mathbb R^3-\{(0,0,0)\}$

$\mathbb R^2$

$f(x,y)=xy/(x^2+y^2)$ if $(x,y)\ne(0,0)$ and $f(0,0)=0$

$\dst{\frac{2}{\sqrt3}(x+y\cos x-xy\sin x)-2\sqrt\frac{2}{3}(x\cos x)}$

$\dst{\frac{1-2y}{\sqrt3}e^{-x+y^2+2z}}$

$\dst{\frac{2}{\sqrt n}(x_1+x_2+\cdots+x_n)}$

$1/(1+x+y+z)$

$\phi_1^2\phi_2$

$-5\pi/\sqrt6$

$-2e$

$0$

$f_x=f_y=1/(x+y+2z)$ , $f_z=2/(x+y+2z)$

$f_x=2x+3yz+2y$ ,

$f_y=3xz+2x$ ,

$f_z=3xy$

$f_x=e^{yz}$ ,

$f_y=xze^{yz}$ ,

$f_z=xye^{yz}$

$f_x=2xy\cos x^2y$ , $f_y=x^2\cos x^2y$ , $f_z=1$

$f_{xx}=f_{yy}=f_{xy}=f_{yx}=-1/(x+y+2z)^2$ , $f_{xz}=f_{zx}=f_{yz}=f_{zy}=$ $-2/(x+y+2z)^2$ , $f_{zz}=-4/(x+y+2z)^2$

$f_{xx}=2$ ,

$f_{yy}=f_{zz}=0$ ,

$f_{xy}=f_{yx}=3z+2$ ,

$f_{xz}=f_{zx}=3y$ ,

$f_{yz}=f_{zy}=3x$

$f_{xx}=0$ , $f_{yy}=xz^2e^{yz}$ , $f_{zz}=xy^2e^{yz}$ , $f_{xy}=f_{yx}=ze^{yz}$ , $f_{xz}=f_{zx}=ye^{yz}$ , $f_{yz}=f_{zy}=xe^{yz}$

$f_{xx}=2y\cos x^2y-4x^2y^2\sin x^2y$ , $f_{yy}=-x^4\sin x^2y$ , $f_{zz}=0$ , $f_{xy}=f_{yx}=2x\cos x^2y-2x^3y\sin x^2y$ , $f_{xz}=f_{zx}=f_{yz}=f_{zy}=0$

$f_{xx}(0,0)=f_{yy}(0,0)=0$ ,

$f_{xy}(0,0)=-1$ ,

$f_{yx}(0,0)=1$ \

$f_{xx}(0,0)=f_{yy}(0,0)=0$ , $f_{xy}(0,0)=-1$ , $f_{yx}(0,0)=1$

$f(x,y)=g(x,y)+h(y)$ , where $g_{xy}$ exists everywhere and $h$ is nowhere differentiable.

$df=(3x^2+4y^2+2y\sin x+2xy\cos x)\,dx+(8xy+2x\sin x)\, dy$ ,\

$d_{\mathbf{X}_0}f=16\,dx$ ,

$(d_{\mathbf{X}_0}f)(\mathbf{X}-\mathbf{X}_0)=16x$ \

$df=-e^{-x-y-z}\,(dx+dy+dz)$ ,

$d_{\mathbf{X}_0}f=-dx-dy-dz$ , \

$(d_{\mathbf{X}_0}f)(\mathbf{X}- \mathbf{X}_0)=-x-y-z$ \

$df=(1+x_1+2x_2+\cdots+nx_n)^{-1}\sum_{j=1}^nj\,dx_j$ ,

$d_{\mathbf{X}_0}f=\sum_{j=1}^n j\,dx_j$ , \

$(d_{\mathbf{X}_0}f)(\mathbf{X}-\mathbf{X}_0)=\sum_{j=1}^n jx_j$ ,\

$df=2r|\mathbf{X}|^{2r-2}\sum_{j=1}^nx_j\,dx_j$ , $d_{\mathbf{X}_0}f=2rn^{r-1}\sum_{j=1}^n dx_j$ , \ $(d_{\mathbf{X}_0}f)(\mathbf{X}- \mathbf{X}_0)=2rn^{r-1}\sum_{j=1}^n (x_j-1)$ ,

The unit vector in the direction of $(f_{x_1}(\mathbf{X}_0), f_{x_2}(\mathbf{X}_0), \dots,f_{x_n}(\mathbf{X}_0))$ provided that this is not $\mathbf{0}$ ; if it is $\mathbf{0}$ , then $\partial f(\mathbf{X}_0)/\partial\boldsymbol{\Phi}=0$ for every $\boldsymbol{\Phi}$ .

$z=2x+4y-6$

$z=2x+3y+1$

$z=(\pi x)/2+y-\pi/2$

$z=x+10y+4$

$5\,du+34\,dv$

$0$

$6\,du-18\,dv$

$8\,du$

$h_r=f_x\cos\theta+f_y\sin\theta$ , $h_\theta=r(-f_x\sin\theta+f_y\cos\theta)$ , $h_z=f_z$

$h_r=f_x\sin\phi\cos\theta+f_y\sin\phi\sin\theta+f_z\cos\phi$ , $h_\theta=r\sin\phi(-f_x\sin\theta+f_y\cos\theta)$ , $h_\phi=r(f_x\cos\phi\cos\theta+f_y\cos\phi\sin\theta-f_z\sin\phi)$

$h_y=g_xx_y+g_y+g_ww_y$ , $h_z=g_xx_z+g_z+g_ww_z$

$h_{rr}=f_{xx}\sin^2\phi\cos^2\theta+f_{yy}\sin^2\phi\sin^2\theta+ f_{zz}\cos^2\phi+f_{xy}\sin^2\phi\sin2\theta+f_{yz}\sin2\phi\sin\theta+ f_{xz}\sin2\phi\cos\theta$ ,

$\dst h_{r\theta} =(-f_x\sin\theta+f_y\cos\theta)\sin\phi+\frac{r}{2}(f_{yy}- f_{xx})\sin^2\phi\sin2\theta +rf_{xy}\sin^2\phi\cos2\theta+\frac{r}{2} (f_{zy}\cos\theta-f_{zx}\sin\theta)\sin2\phi$

$\dst{1+x+\frac{x^2}{2}-\frac{y^2}{2}+\frac{x^3}{6}-\frac{xy^2}{2}}$

$\dst{1-x-y+\frac{x^2}{2}+xy+\frac{y^2}{2}-\frac{x^3}{6}-\frac{x^2y}{2} -\frac{xy^2}{2}-\frac{y^3}{6}}$ \

$0$

$xyz$

$(d_{(0,0)}^2p)(x,y)=(d_{(0,0)}^2q)(x,y)=2(x-y)^2$

$\left[\begin{array}{rrr}3&4&6\\2&-4&2\\7&2&3\end{array}\right]$

$\left[\begin{array}{rr}2&4\\3&-2\\7&-4\\6&1\end{array}\right]$

$\left[\begin{array}{rrrr}8&8&16&24\\0&0&4&12\\ 12&16&28&44\end{array}\right]$

$\left[\begin{array}{rrr}-2&-6&0\\0&-2&-4\\-2&2&-6\end{array}\right]$

$\left[\begin{array}{rrr}-2&2&6\\6&7&-3\\0&-2&6\end{array}\right]$

$\left[\begin{array}{rr}-1&7\\3&5\\5&14\end{array}\right]$

$\left[\begin{array}{rr}13&25\\16&31\\16&25\end{array}\right]$

$\left[\begin{array}{r}29\\50\end{array}\right]$

$\mathbf{A}$ and $\mathbf{B}$ are square of the same order.

$\left[\begin{array}{rrr}7&3&3\\4&7&7\\6&-9&1\end{array}\right]$

$\left[\begin{array}{rr}14&10\\6&-2\\14&2\end{array}\right]$

$\left[\begin{array}{rrr}-7&6&4\\-9&7&13\\5&0&-14\end{array}\right]$ , $\left[\begin{array}{rrr}-5&6&0\\4&-12&3\\4&0&3\end{array}\right]$

$\left[\begin{array}{rrr}6xyz&3xz^2&3x^2y\end{array}\right]$ ;
$\left[\begin{array}{rrr}-6&3&-3\end{array}\right]$

$\cos(x+y)\left[\begin{array}{rr}1&1\end{array}\right]$ ; $\left[\begin{array}{rr}0&0\end{array}\right]$

$\left[\begin{array}{rrr}(1-xz)ye^{-xz}&xe^{-xz}&-x^2ye^{-xz} \end{array}\right]$ ; $\left[\begin{array}{rrr}2&1&-2\end{array}\right]$ \

$\sec^2(x+2y+z)\left[\begin{array}{rrr}1&2&1\end{array}\right]$ ; $\left[\begin{array}{rrr}2&4&2\end{array}\right]$

$|\mathbf{X}|^{-1}\left[\begin{array}{rrrr}x_1&x_2&\cdots&x_n\end{array}\right]$ ; $\dst\frac{1}{\sqrt n}\left[\begin{array}{rrrr}1&1&\cdots&1\end{array}\right]$

$(2,3,-2)$

$(2,3,0)$

$(-2,0,-1)$

$(3,1,3,2)$

$\dst\frac{1}{10}\left[\begin{array}{rr}4&2\\-3&1\end{array}\right]$

$\dst\frac{1}{2}\left[\begin{array}{rrr}-1&1&2\\3&1&-4\\-1&-1&2 \end{array}\right]$

$\dst\frac{1}{25}\left[\begin{array}{rrr}4&3&-5\\6&-8&5\\-3&4&10 \end{array}\right]$

$\dst\frac{1}{2}\left[\begin{array}{rrr}1&-1&1\\-1&1&1\\1&1&-1 \end{array}\right]$

$\dst\frac{1}{7}\left[\begin{array}{rrrr}3&-2&0&0\\2&1&0&0\\0&0&2&-3 \\0&0&1&2\end{array}\right]$

$\dst\frac{1}{10}\left[\begin{array}{rrrr}-1&-2&0&5 \\-14&-18&10&20\\21&22&-10&-25\\17&24&-10&-25\end{array}\right]$

$\mathbf{F}'(\mathbf{X})=\left[\begin{array}{ccc} 2x&1&2\\-\sin(x+y+z)&-\sin(x+y+z)&-\sin(x+y+z)\\[2\jot] yze^{xyz}&xze^{xyz}&xye^{xyz}\end{array}\right]$ ;\[2] $J\mathbf{F}(\mathbf{X})=e^{xyz}\sin(x+y+z) [x(1-2x)(y-z)-z(x-y)]$ ;\[2] $\mathbf{G}(\mathbf{X})=\left[\begin{array}{r}0\\1\\1\end{array}\right] +\left[\begin{array}{rrr}2&1&2\\0&0&0\\0&0&-1\end{array}\right] \left[\begin{array}{c}x-1\\y+1\\z\end{array}\right]$

$\mathbf{F}'(\mathbf{X})=\left[\begin{array}{rr}e^x\cos y&-e^x\sin y\\e^x\sin y&e^x\cos y\end{array}\right]$ ; $J\mathbf{F}(\mathbf{X})=e^{2x}$ ;\[2] $\mathbf{G}(\mathbf{X})=\left[\begin{array}{r}0\\1\end{array}\right] +\left[\begin{array}{rr}0&-1\\1&0\end{array}\right] \left[\begin{array}{c}x\\y-\pi/2\end{array}\right]$ \[2]

$\mathbf{F}'(\mathbf{X})= \left[\begin{array}{rrr}2x&-2y&0\\0&2y&-2z\\-2x&0&2z\end{array}\right]$ ; $J\mathbf{F}=0$ ;\[2] $\mathbf{G}(\mathbf{X})= \left[\begin{array}{rrr}2&-2&0\\0&2&-2\\-2&0&2\end{array}\right] \left[\begin{array}{c}x-1\\y-1\\z-1\end{array}\right]$

$\mathbf{F}'(\mathbf{X})= \left[\begin{array}{ccc} (x+y+z+1)e^x&e^x&e^x\\(2x-x^2-y^2)e^{-x} &2ye^{-x}&0\end{array}\right]$

$\mathbf{F}'(\mathbf{X})=\left[\begin{array}{c}g_1'(x)\\g_2'(x)\\\vdots\\ g_n'(x)\end{array}\right]$ \

$\mathbf{F}'(r,\theta)= \left[\begin{array}{ccc}e^x\sin yz&ze^x\cos yz&ye^x\cos yz\\ze^y\cos xz&e^y\sin xz&xe^y\cos xz\\ye^z\cos xy&xe^z\cos xy&e^z\sin xy\end{array}\right]$

$\mathbf{F}'(r,\theta)=\left[\begin{array}{rr}\cos\theta&-r\sin\theta \\\sin\theta&r\cos\theta\end{array}\right]$ ; $J\mathbf{F}(r,\theta)=r$ \[2]

$\mathbf{F}'(r,\theta,\phi)=\left[\begin{array}{ccc}\cos\theta\cos\phi& -r\sin\theta\cos\phi&-r\cos\theta\sin\phi\\\sin\theta\cos\phi& \phantom{-}r\cos\theta\cos\phi&-r\sin\theta\sin\phi\\\sin\phi&0&r\cos\phi \end{array}\right]$ ;\ $J\mathbf{F}(r,\theta,\phi)=r^2\cos\phi$ \[2]

$\mathbf{F}'(r,\theta,z)=\left[\begin{array}{ccc}\cos\theta&-r\sin\theta&0 \\\sin\theta&\phantom{-}r\cos\theta&0\\0&0&1\end{array}\right]$ ; $J\mathbf{F}(r,\theta,z)=r$

$\left[\begin{array}{rrr}0&0&4\\0&-\frac{1}{2}&0\end{array}\right]$

$\left[\begin{array}{rr}-18&0\\2&0\end{array}\right]$

$\left[\begin{array}{rr}9&-3\\3&-8\\1&0\end{array}\right]$

$\left[\begin{array}{rrr}4&-3&1\\0&1&1\end{array}\right]$

$\left[\begin{array}{rr}2&0\\2&0\end{array}\right]$

$\left[\begin{array}{rr}5&10\\9&18\\-4&-8\end{array}\right]$

$[1,\pi/2]$

$[1,2\pi]$

$[1,\pi]$

$[2\sqrt2,9\pi/4]$

$[\sqrt2,3\pi/4]$

$[1,-3\pi/2]$

$[1,-2\pi]$

$[1,-\pi]$

$[2\sqrt2,-7\pi/4]$

$[\sqrt2,-5\pi/4]$

Let $f(x)=x\ (0\le x\le\frac{1}{2})$ , $f(x)=x-\frac{1}{2}\ (\frac{1}{2}<x\le1)$ ; then $f$ is locally invertible but not invertible on $[0,1]$ .

$\mathbf{F}(S)=\set{(u,v)}{-\pi+2\phi<\arg(u,v)<\pi+2\phi}$ , where $\phi$ is an argument of $(a,b)$ ;

$\mathbf{F}_S^{-1}(u,v)=(u^2+v^2)^{1/4} \left[\begin{array}{r}\cos(\arg(u,v)/2) \\[2\jot]\sin(\arg(u,v)/2)\end{array}\right],\ 2\phi-\pi<\arg(u,v)<2\phi+\pi$

$\left[\begin{array}{c}x\\y\end{array}\right]= \dst\frac{1}{10}\left[\begin{array}{c}\phantom{3}u-2v\\3u+4v\end{array}\right]$ ; $(\mathbf{F}^{-1})'=\dst\frac{1}{10} \left[\begin{array}{rr}1&-2\\3&4\end{array}\right]$

$\left[\begin{array}{c}x\\y\\z\end{array}\right]= \dst\frac{1}{2}\left[\begin{array}{c}u+2v+3w\\u-w\\u+v+2w\end{array}\right]$ ; $(\mathbf{F}^{-1})'=\dst\frac{1}{2} \left[\begin{array}{rrr}1&2&3\\1&0&-1\\1&1&2\end{array}\right]$

$\mathbf{G}_1(u,v)=\dst\frac{1}{\sqrt2} \left[\begin{array}{r}\sqrt{u+v}\\ \sqrt{u-v}\end{array}\right]$ , $\mathbf{G}_1'(u,v)=\dst\frac{1}{2\sqrt2} \left[\begin{array}{rr} 1/\sqrt{u+v}&1/\sqrt{u+v}\\ 1/\sqrt{u-v}&-1/\sqrt{u-v}\\ \end{array}\right]$

$\mathbf{G}_2(u,v)=\dst\frac{1}{\sqrt2} \left[\begin{array}{r}-\sqrt{u+v}\\ \sqrt{u-v}\end{array}\right]$ , $\mathbf{G}_2'(u,v)=\dst\frac{1}{2\sqrt2} \left[\begin{array}{rr} -1/\sqrt{u+v}&-1/\sqrt{u+v}\\ 1/\sqrt{u-v}&-1/\sqrt{u-v}\\ \end{array}\right]$

$\mathbf{G}_3(u,v)=\dst\frac{1}{\sqrt2} \left[\begin{array}{r}\sqrt{u+v}\\ -\sqrt{u-v}\end{array}\right]$ , $\mathbf{G}_3'(u,v)=\dst\frac{1}{2\sqrt2} \left[\begin{array}{rr} 1/\sqrt{u+v}&1/\sqrt{u+v}\\ -1/\sqrt{u-v}&1/\sqrt{u-v}\\ \end{array}\right]$

$\mathbf{G}_4(u,v)=\dst\frac{1}{\sqrt2} \left[\begin{array}{r}-\sqrt{u+v}\\ -\sqrt{u-v}\end{array}\right]$ , $\mathbf{G}_4'(u,v)=\dst\frac{1}{2\sqrt2} \left[\begin{array}{rr} -1/\sqrt{u+v}&-1/\sqrt{u+v}\\ -1/\sqrt{u-v}&1/\sqrt{u-v}\\ \end{array}\right]$

From solving

$x=r\cos\theta$ ,

$y=r\sin\theta$ for

$\theta=\arg(x,y)$ . Each equation is satisfied by angles that are not arguments of

$(x,y)$ , since none of the formulas identifies the quadrant of

$(x,y)$ uniquely. Moreover,

does not hold if $x=0$ .

$\left[\begin{array}{c}x\\y\end{array}\right]= \mathbf{G}(u,v)=(u^2+v^2)^{1/4} \left[\begin{array}{r}\cos[\frac{1}{2}\arg(u,v)] \\[2\jot] \sin(\arg(u,v)/2)\end{array}\right]$ ,

where $\beta-\pi/2<\arg(u,v)<\beta+\pi/2$ and $\beta$ is an argument of $(a,b)$ ;

$\mathbf{G}'(u,v)=\dst\frac{1}{2(x^2+y^2)} \left[\begin{array}{rr}x&y\\-y&x\end{array}\right]$

If $\mathbf{F}(x_1,x_2, \dots,x_n)=(x_1^3,x_2^3, \dots,x_n^3)$ , then $\mathbf{F}$ is invertible, but\ $J\mathbf{F}(\mathbf{0})~=0$ .

$\mathbf{A}(\mathbf{U})= \left[\begin{array}{r}1\\-1\end{array}\right] -\dst\frac{1}{25}\left[\begin{array}{rr}5&5\\3&8\end{array}\right] \left[\begin{array}{c}u+5\\v-4\end{array}\right]$

$\mathbf{A}(\mathbf{U})= \left[\begin{array}{r}1\\1\end{array}\right] +\dst\frac{1}{6}\left[\begin{array}{rr}4&-2\\-3&3\end{array}\right] \left[\begin{array}{c}u-2\\v-3\end{array}\right]$

$\mathbf{A}(\mathbf{U})= \left[\begin{array}{r}0\\1\\1\end{array}\right]+ \left[\begin{array}{rrr}0&-1&1\\-1&1&0\\1&0&0\end{array}\right] \left[\begin{array}{c}u-1\\v-1\\w-2\end{array}\right]$

$\mathbf{A}(\mathbf{U})= \left[\begin{array}{c}1\\\pi/2\\\pi\end{array}\right]+ \left[\begin{array}{rrr}0&-1&0\\1&0&0\\0&0&-1\end{array}\right] \left[\begin{array}{c}u\\v+1\\w\end{array}\right]$

$\mathbf{G}'(x,y,z)= \left[\begin{array}{ccc} \cos\theta\cos\phi&\sin\theta\cos\phi&\sin\phi\\[2\jot] -\dst\frac{\sin\theta}{ r\cos\phi}&\dst\frac{\cos\theta}{ r\cos\phi}&0\\[2\jot] -\dst\frac{1}{ r}\cos\theta\sin\phi&-\dst\frac{1}{ r}\sin\theta\sin\phi&\dst\frac{1}{ r}\cos\phi\end{array}\right]$

$\mathbf{G}'(x,y,z)= \left[\begin{array}{ccc}\cos\theta&\sin\theta&0\\[2\jot] -\dst\frac{1}{ r}\sin\theta&\dst\frac{1}{ r}\cos\theta&0\\[2\jot] 0&0&1\end{array}\right]$

$\left[\begin{array}{c}u\\v\end{array}\right]= \dst\frac{1}{2}\left[\begin{array}{rr}-3&4\\1&-2\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right]$

$\left[\begin{array}{c}u\\v\\w\end{array}\right]=\dst -\frac{1}{2} \left[\begin{array}{rr} 3&3\\-1&2\\2&3 \end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right]$

$\left[\begin{array}{c}u\\v\end{array}\right]= \dst\frac{1}{5}\left[\begin{array}{rr}2&-1\\-1&3\end{array}\right] \left[\begin{array}{c}-y+\sin x\\-x+\sin y\end{array}\right]$

$u=-x$ , $v=-y$ , $z=-w$

$f_i(\mathbf{X},\mathbf{U})=\dst\left(\sum_{j=1}^n a_{ij}(x_j-x_{j0})\right)^r-(u_i-u_{i0})^s$ ,

$1\le i\le m$ , where

$r$ and

$s$ are positive integers and not all

$a_{ij}=0$ .

$r=s=3$ ;

$r=1$ ,

$s=3$ ;

$r=s=2$

$u_x(1,1)=-\frac{5}{8}$ , $u_y(1,1)=-\frac{1}{2}$

$u_x(1,1,1)=\frac{5}{8}$ , $u_y(1,1,1)=-\frac{9}{8}$ , $u_z(1,1,1)=\frac{1}{2}$

$u(1,2)=0$ , $u_x(1,2)=u_y(1,2)=-4$

$u(-1,-2)=2$ , $u_x(-1,-2)=1$ , $u_y(-1,-2)=-\frac{1}{2}$

$u(\pi/2,\pi/2)=u_x(\pi/2,\pi/2)=u_y(\pi/2,\pi/2)=0$

$u(1,1)=1$ , $u_x(1,1)=u_y(1,1)=-1$

$u_1(1,1)=1$ , $\dst\frac{\partial u_1(1,1)}{\partial x}=5$ , $\dst\frac{\partial u_1(1,1)}{\partial y}=2$

$u_2(1,1)=2$ , $\dst\frac{\partial u_2(1,1)}{\partial x}=-14$ ; $\dst\frac{\partial u_2(1,1)}{\partial y}=-2$

$u_k(0,\pi)=(2k+1)\pi/2$ , $\dst\frac{\partial u_k(0,\pi)}{\partial x}=0$ , $\dst\frac{\partial u_k(0,\pi)}{\partial y}=-1$ , $k=$ integer

$\dst\frac{1}{5}\left[\begin{array}{rrr}-1&-2&1\\-1&-2&1\end{array}\right]$ $u'(0)=3$ , $v'(0)=-1$

$\dst\frac{1}{6}\left[\begin{array}{rr}5&5\\-5&-5\\6&6\end{array}\right]$

$\mathbf{U}_1(1,1)=\left[\begin{array}{r}3\\1\end{array}\right]$ , $\mathbf{U}_1'(1,1)=\left[\begin{array}{rr}1&3\\-1&2\end{array}\right]$ ;

$\mathbf{U}_2(1,1)=-\left[\begin{array}{r}3\\1\end{array}\right]$ , $\mathbf{U}_2'(1,1)=-\left[\begin{array}{rr}1&3\\-1&2\end{array}\right]$

$u_x(0,0,0)=2$ , $v_x(0,0,0)= w_x(0,0,0)=-2$

$y_x=-\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(x,z,u)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$ , $y_v=-\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(v,z,u)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$ , $z_x=-\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(y,x,u)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$ ,

$z_v= -\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(y,v,u)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$ , $u_x=-\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(y,z,x)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$ , $u_v=-\dst\frac{\dst\frac{\partial(f,g,h)}{\partial(y,z,v)}}{ \dst\frac{\partial(f,g,h)}{\partial(y,z,u)}}$

$x=-2y-u$ , $z=-2v$ ; $x=-2y-u$ , $v=-\dst\frac{z}{2}$ ; $y=-\dst\frac{x}{2}-\dst\frac{u}{2}$ , $z=-2v$ ; $y=-\dst\frac{x}{2}-\dst\frac{u}{2}$ , $v=-\dst\frac{z}{2}$ ; $z=-2v$ , $u=-x-2y$ ; $u=-x-2y$ , $v=-\dst\frac{z}{2}$

$y_x(1,-1,-2)=-\frac{1}{2}$ , $v_u(1,-1,-2)=1$

$u_w(0,-1)=\frac{5}{6}$ , $u_y(0,-1)=0$ , $v_w(0,-1)=-\frac{5}{6}$ , $v_y(0,-1)=0$ ,\ $x_w(0,-1)=1$ , $x_y(0,-1)=-1$

$u_x(1,1)=0$ , $u_y(1,1)=0$ , $v_x(1,1)=-1$ , $v_y(1,1)=-1$ , $u_{xx}(1,1)=2$ ,\ $u_{xy}(1,1)=1$ , $u_{yy}(1,1)=2$ , $v_{xx}(1,1)=-2$ , $v_{xy}(1,1)=-1$ , $v_{yy}(1,1)=-2$

$u_x(1,-1)=0$ , $u_y(1,-1)=\dst\frac{1}{2}$ , $v_x(1,-1)=-\dst\frac{1}{2}$ , $v_y(1,-1)=0$ ,\ $u_{xx}(1,-1)=-\dst\frac{1}{8}$ , $u_{xy}(1,-1)=\dst\frac{1}{8}$ , $u_{yy}(1,-1)=\dst\frac{1}{8}$ , $v_{xx}(1,-1)=-\dst\frac{1}{8}$ ,\ $v_{xy}(1,-1)=-\dst\frac{1}{8}$ , $v_{yy}(1,-1)=\dst\frac{1}{8}$

$28$

$\frac{1}{4}$ $3(b-a)(d-c)$ , $0$ $\set{(m,n)}{m,n=\mbox{integers}}$

$12$

$\frac{79}{20}$

$-1$

$(1-\log2)/2$

$\frac{7}{4}$

$17$

$\frac{2}{3}(\sqrt2-1)$

$1/4\pi$

$\frac{3}{8}$ ,

$\frac{5}{8}$

$\frac{3}{8}$ ,

$\frac{5}{8}$

$\frac{3}{4}$ ,

$\frac{5}{4}$

$\frac{3}{4}\left(z+\frac{1}{2}\right)$ ,

$\frac{5}{4}\left(z+\frac{1}{2}\right)$

$z+\frac{1}{2}$ , $1$

$-285$

$0$

$\frac{1}{4}(e-\frac{5}{2})$

$324$

$\frac{1}{6}$

$1$ $\frac{52}{15}$

$36$

$1$

$\frac{64}{3}$

$(e^6+17)/2$

$\frac{2}{27}$

$\frac{1}{2}(e-\frac{5}{2})$

$\frac{1}{24}$

$\frac{1}{36}$

$16\pi$

$\frac{1}{6}$

$\frac{128}{21}$

$\frac{\pi}{2}$

$\frac{1}{2}(b_1-a_1)\cdots(b_n-a_n)\sum_{j=1}^n(a_j+b_j)$

$\frac{1}{3}(b_1-a_1)\cdots(b_n-a_n)\sum_{j=1}^n(a_j^2+a_jb_j+b_j^2)$

$2^{-n}(b_1^2-a_1^2)\cdots(b_n^2-a_n^2)$

$\int_{-\sqrt3/2}^{\sqrt3/2}dx\int_{1/2}^{\sqrt{1-x^2}}f(x,y)\,dy$ $\frac{1}{2}$

Let $S_1$ and $S_2$ be dense subsets of $\mathbb R$ such that $S_1\cup S_2=\mathbb R$ .

$-1$ ; $c$ (constant); $1$ $(u_2-u_1)(v_2-v_1)/|ad-bc|$

$\frac{5}{6}$

$\frac{4}{9}$

$\log\frac{5}{2}$ $3$ $\frac{1}{2}$ $\frac{5}{4}e(e-1)$

$\frac{4}{3}\pi abc$ $2\pi(e^{25}-e^9)$ $16\pi/3$ $21/64$

$(\pi/8)\log5$

$(\pi/4)(e^4-1)$

$2\pi/15$

$\pi^2a^4/2$

$(\beta_1-\alpha_1)\cdots(\beta_n-\alpha_n)/|\det(\mathbf{A})|$ $|a_1a_2\cdots a_n|V_n$

\end{document}

{}

Suppose $f\in C[a,b]$ . Acording to the , if $\epsilon>0$ , there is a polynonmial $p$ such that $|f(x)-p(x)|<\epsilon$ , $a\le x\le b$ . Now suppose that

$a\le x_{1n} < x_{2n}<\cdots<x_{nn}\le b,\quad a\le y_{1n} < y_{2n}<\cdots<y_{nn}\le b,\quad n\ge 1. \nonumber$ and

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}|x_{in}-y_{in}|=0. \nonumber$ Show that if

$f\in C[a,b]$ , then

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}|f(x_{in})-f(y_{in})|=0. \nonumber$

By the triangle inequality,

$|f(x_{in})-f(y_{in})|\le |f(x_{in})-p(x_{in})| +|p(x_{in})-p(y_{in})|+|p(y_{in})-f(y_{in})|, \nonumber$ so

$\begin{equation} \tag{A} |f(x_{in})-f(y_{in})|<|p(x_{in})-p(y_{in})|+2\epsilon. \end{equation} \nonumber$ Let

$M=\max_{a\le x\le b}|p'(x)|$ . By the mean value theorem,

$|p(x_{in})-p(y_{in})|\le M |x_{in}-y_{in}|. \nonumber$ This and (A) imply that

$\frac{1}{n}\sum_{i=1}^n|f(x_{in})-f(y_{in})|< 2\epsilon+\frac{M}{n}\sum_{i=1}^n|x_{in}-y_{in}|. \nonumber$ From this and (A),

$\limsup_{n\to\infty} \frac{1}{n}\sum_{i=1}^n|f(x_{in})-f(y_{in})|\le 2\epsilon. \nonumber$ Since

$\epsilon$ is arbitrary, this implies the conclusion.

{} In the setting of Exercise~1, let

$y_{in}=a+\frac{i}{n}\sum_{i=1}^{n}(b-a), \quad 1\le i\le n,\quad n\ge 1. \nonumber$ Show that

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n} f(x_{in})=\frac{1}{b-a}\int_{a}^{b}f(x)\,dx. \nonumber$

{Solution 2.} Since $f\in C[a,b]$ , $\int_{a}^{b}f(x)\,dx$ exists (Theorem~3.2.8, p.~133).

Therefore, from Definition~3.1.1 (p._114),

$\lim_{n\to\infty}\sum_{i=1}^{n} f(y_{in})=\frac{1}{b-a}\int_{a}^{b}f(x)\,dx. \nonumber$ Since

$\frac{1}{n}\left| \sum_{i=1}^{n}f(x_{in})-\frac{1}{b-a}\int_{a}^{b}f(x)\,dx\right|\le \frac{1}{n}\sum_{i=1}^{n}|f(x_{in})-f(y_{in})|+ \left|\frac{1}{n} \sum_{i=1}^{n}f(y_{in})-\frac{1}{b-a}\int_{a}^{b}f(x)\,dx\right|, \nonumber$ Exercise1 implies the conclusion.

Suppose $g$ is continuous and nondecreasing on $[c,d]$ . For

\end{document}

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