• Home
  • Textbooks
  • A Course in Calculus and Real Analysis
  • Elementary Transcendental Functions

A Course in Calculus and Real Analysis

Sudhir R. Ghorpade, Balmohan V. Limaye

Chapter 7

Elementary Transcendental Functions - all with Video Answers

Educators

Chapter Questions

01:53

Problem 1

For every $x \in \mathbb{R}$ with $x>1$, show that
$$
\sum_{k=1}^{[x]} \frac{1}{k}-\frac{[x]}{x} \leq \ln x \leq \sum_{k=2}^{[x]-1} \frac{1}{k}+\frac{[x]}{x}
$$
where $[x]$ denotes the integral part of $[x] .$ In particular, show that
$$
\frac{13}{22} \leq \ln 2.2 \leq \frac{11}{10} \quad \text { and } \quad 1 \leq \ln 3.6 \leq \frac{17}{10} \text { . }
$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:41

Problem 2

Consider the sequence $\left(c_{n}\right)$ defined by
$$
c_{n}:=1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln n \quad \text { for } n \in \mathbb{N}
$$
Show that $\left(c_{n}\right)$ is convergent. (Hint: $\left(c_{n}\right)$ is monotonically decreasing and $c_{n} \geq 0$ for all $n \in \mathbb{N}$.)
[Note: The limit of the sequence $\left(c_{n}\right)$ is known as Euler's constant. It is usually denoted by $\gamma$. Approximately, $\gamma=0.5772156649 \ldots$, but it is not known whether $\gamma$ is rational or irrational.]

Adriano Chikande
Adriano Chikande
Numerade Educator
01:08

Problem 3

Let $a>0$ and $r \in \mathbb{Q}$. Show that $\ln a x^{r}=\ln a+r \ln x$ for all $x \in(0, \infty)$, assuming only that $(\ln )^{\prime} x=1 / x$ for all $x \in(0, \infty)$.

Carson Merrill
Carson Merrill
Numerade Educator
01:18

Problem 4

Show that for all $x>0$
$$
x-\frac{x^{2}}{2}<\ln (1+x)<x-\frac{x^{2}}{2}+\frac{x^{3}}{3}
$$

Carson Merrill
Carson Merrill
Numerade Educator
01:12

Problem 5

Let $\alpha \in \mathbb{R}$ and $f:(0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)=\alpha / x$ for all $x \in(0, \infty)$ and $f(1)=0$. Show that $f(x)=\alpha \ln x$ for all $x \in(0, \infty)$. (Compare Exercise 4 of Chapter 4.)

Carson Merrill
Carson Merrill
Numerade Educator
02:22

Problem 6

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be continuous and satisfy
$$
\int_{1}^{x y} f(t) d t=y \int_{1}^{x} f(t) d t+x \int_{1}^{y} f(t) d t \quad \text { for all } x, y \in(0, \infty)
$$
Show that $f(x)=f(1)(1+\ln x)$ for all $x \in(0, \infty)$. (Hint: Consider $F(x):=$ $\left(\int_{1}^{x} f(t) d t\right) / x$ for $x \in(0, \infty)$ and use Exercise 5.)

Pawan Yadav
Pawan Yadav
Numerade Educator
00:58

Problem 7

Show that $2.5<e<3 .$ (Hint: Divide $[1,2.5]$ and $[1,3]$ into subintervals of length $\left.\frac{1}{4} .\right)$

Lucas Finney
Lucas Finney
Numerade Educator
02:37

Problem 8

Show that
(i) $\int_{a}^{b} \ln x d x=b(\ln b-1)-a(\ln a-1)$ for all $a, b \in(0, \infty)$,
(ii) $\int_{a}^{b} \exp x d x=\exp b-\exp a$ for all $a, b \in \mathbb{R}$.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
01:12

Problem 9

Let $\alpha \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}=\alpha f$ and $f(0)=1$. Show that $f(x)=e^{\alpha x}$ for all $x \in \mathbb{R}$. (Compare Exercise 5 of Chapter $4 .$ )

Carson Merrill
Carson Merrill
Numerade Educator
06:12

Problem 10

The hyperbolic sine and hyperbolic cosine functions from $\mathbb{R}$ to $\mathbb{R}$ are defined by
$$
\sinh x:=\frac{e^{x}-e^{-x}}{2} \quad \text { and } \quad \cosh x:=\frac{e^{x}+e^{-x}}{2} \quad \text { for } x \in \mathbb{R} .
$$
Show that for any $t \in \mathbb{R}$, the point $(\cosh t, \sinh t)$ is on the hyperbola $x^{2}-y^{2}=1$. Also, show that
(i) $\sinh 0=0, \cosh 0=1$ and $\cosh ^{2}-\sinh ^{2}=1$ for all $x \in \mathbb{R}$.
(ii) $(\sinh )^{\prime} x=\cosh x$ and $(\cosh )^{\prime} x=\sinh x$ for all $x \in \mathbb{R}$.
(iii) $\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$ and
$\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y$ for all $x, y \in \mathbb{R}$
Sketch the graphs of the functions sinh and cosh.

Ryan Mcalister
Ryan Mcalister
Numerade Educator
02:00

Problem 11

Let $a, b \in(0, \infty)$.
(i) Consider the functions $f, g:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x):=\log _{a} x$ and $g(x):=\log _{b} x .$ Show that $f$ and $g$ have the same rate as $x \rightarrow \infty$.
(ii) Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x):=a^{x}$ and $g(x):=b^{x} .$ Show that the growth rate of $f$ is less than that of $g$ as $x \rightarrow \infty$ if and only if $a<b$.

Harshita Goel
Harshita Goel
Numerade Educator
05:35

Problem 12

For $b \in \mathbb{R}$, consider the function $g_{b}:(0, \infty) \rightarrow(0, \infty)$ defined by $g_{b}(x)=$ $x^{b} .$ Show that $g_{b_{1}} \circ g_{b_{2}}=g_{b_{1} b_{2}}=g_{b_{2}} \circ g_{b_{1}}$ for all $b_{1}, b_{2} \in \mathbb{R}$.

Regina Hays
Regina Hays
Numerade Educator
01:12

Problem 13

Let $f:(0, \infty) \rightarrow \mathbb{R}$ satisfy $f(x y)=f(x) f(y)$ for all $x, y \in(0, \infty) .$ If $f$ is continuous at 1, show that either $f(x)=0$ for all $x \in(0, \infty)$, or there is $r \in \mathbb{R}$ such that $f(x)=x^{r}$ for all $x \in(0, \infty)$. (Hint: If $f(1) \neq 0$, then $f(x)>0$ for all $x \in(0, \infty)$, and so we can consider $g=\ln \circ f \circ \exp : \mathbb{R} \rightarrow \mathbb{R}$ and use Exercise 4 of Chapter 3.) (Compare Exercise 19 (ii) of Chapter 1 and Exercise 6 of Chapter 3.)

Carson Merrill
Carson Merrill
Numerade Educator
02:35

Problem 14

Let $r \in \mathbb{R}$ be positive and consider the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=x^{r}$. Show that the growth rate of $\ln x$ is less than that of $f$, while the growth rate of $\exp x$ is more than that of $f$ as $x \rightarrow \infty$.

Mike Gaerlan
Mike Gaerlan
Numerade Educator
01:32

Problem 15

Show that
$$
\frac{x}{1+x^{2}}<\arctan x<x \quad \text { for all } x \in(0,1]
$$
and
$$
1-\frac{1}{2 x}<\arctan x<2-\frac{1}{x} \quad \text { for all } x \in(1, \infty) .
$$

Nick Johnson
Nick Johnson
Numerade Educator
04:51

Problem 16

Prove that
$$
\lim _{x \rightarrow \infty} \int_{1}^{x} \frac{1}{1+t^{2}} d t=\int_{0}^{1} \frac{1}{1+t^{2}} d t=\frac{\pi}{4}
$$
that is, $\lim _{x \rightarrow \infty} \arctan x=\arctan 1=\pi / 4$. Deduce that $2.88<\pi<3.39$.
(Hint: Substitute $t=1 / s$ and use Proposition 6.20. Divide $[0,1]$ into subintervals of length $\frac{1}{4}$.)

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:06

Problem 17

Let $D$ and $E$ be the unions of open intervals defined as follows.
$D=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+1) \pi}{2}\right)$ and $E=\bigcup_{k \in \mathbb{Z}}\left(\frac{(4 k-3) \pi}{2}, \frac{(4 k-1) \pi}{2}\right)$
Show that
$\sin x=\left\{\begin{array}{rl}\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in D, \\ -\frac{\tan x}{\sqrt{1+\tan ^{2} x}} & \text { if } x \in E,\end{array} \quad \cos x=\left\{\begin{array}{c}\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in D, \\ -\frac{1}{\sqrt{1+\tan ^{2} x}} \text { if } x \in E .\end{array}\right.\right.$

Adrian Co
Adrian Co
Numerade Educator
01:13

Problem 18

Show from first principles that the function cos is differentiable at $\pi / 2$ and its derivative at $\pi / 2$ is $-1$.

Carson Merrill
Carson Merrill
Numerade Educator
02:02

Problem 19

Show that $0<x \cos x<\sin x$ for all $x \in(0, \pi / 2)$ and $\sin x<x \cos x<0$ for all $x \in(-\pi / 2,0)$. Hence or otherwise prove that $x<\tan x$ for all $x \in(0, \pi / 2)$ and $\tan x<x$ for all $x \in(-\pi / 2,0)$.

Leon Druch
Leon Druch
Numerade Educator
02:02

Problem 20

Show that for $x \in(0, \pi / 2)$,
$$
\frac{2 x}{\pi}<\sin x<\min \{1, x\} \text { and } 1-\frac{2 x}{\pi}<\cos x<\min \left\{1, \frac{\pi}{2}-x\right\},
$$
whereas for $x \in(-\pi / 2,0)$,
$$
\max \{-1, x\}<\sin x<\frac{2 x}{\pi} \text { and } 1+\frac{2 x}{\pi}<\cos x<\min \left\{1, \frac{\pi}{2}+x\right\} .
$$

Leon Druch
Leon Druch
Numerade Educator
01:55

Problem 21

Prove that $|\sin x-\sin y| \leq|x-y|$ and $|\cos x-\cos y| \leq|x-y|$ for all $x, y \in \mathbb{R}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:03

Problem 22

Show that
$\int_{a}^{b} \sin x d x=\cos a-\cos b$ and $\int_{a}^{b} \cos x d x=\sin b-\sin a$ for all $a, b \in \mathbb{R}$.

Steven Clarke
Steven Clarke
Numerade Educator
08:46

Problem 23

Let $\beta \in \mathbb{R}$. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions such that
$$
f^{\prime}=\beta g, \quad g^{\prime}=-\beta f, \quad f(0)=0, \quad \text { and } \quad g(0)=1
$$
Show that $f(x)=\sin \beta x$ and $g(x)=\cos \beta x$ for all $x \in \mathbb{R}$. (Hint: Consider $h: \mathbb{R} \rightarrow \mathbb{R}$ given by $h(x):=(f(x)-\sin \beta x)^{2}+(g(x)-\cos \beta x)^{2}$. Find $h^{\prime}$.)
(Compare Exercise 7 of Chapter 4.)

Jon Southam
Jon Southam
Numerade Educator
02:38

Problem 24

Let $\alpha, \beta \in \mathbb{R}$. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions such that
$$
f^{\prime}=\alpha f+\beta g, \quad g^{\prime}=\alpha g-\beta f, \quad f(0)=0, \quad \text { and } \quad g(0)=1
$$
Show that $f(x)=e^{\alpha x} \sin \beta x$ and $g(x)=e^{\alpha x} \cos \beta x$ for all $x \in \mathbb{R}$. (Hint:
Consider $h: \mathbb{R} \rightarrow \mathbb{R}$ given by $h(x):=\left(f(x)-e^{\alpha x} \sin \beta x\right)^{2}+(g(x)-$
$\left.e^{\alpha x} \cos \beta x\right)^{2}$. Find $h^{\prime}$.) (Compare Exercise 6 of Chapter 4.)

Lucas Finney
Lucas Finney
Numerade Educator
07:39

Problem 25

Let $\alpha, \beta \in \mathbb{R}$. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions such that
$$
f^{\prime}=\alpha f+\beta g, \quad g^{\prime}=\alpha g+\beta f, \quad f(0)=0, \quad \text { and } \quad g(0)=1
$$
Show that $f(x)=e^{\alpha x} \sinh \beta x$ and $g(x)=e^{\alpha x} \cosh \beta x$ for all $x \in \mathbb{R}$.

Jon Southam
Jon Southam
Numerade Educator
01:52

Problem 26

Show that $\lim _{x \rightarrow 0}(\sin x) /|x|$ does not exist.

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
02:59

Problem 27

Prove the following for all $x \in \mathbb{R}$. $\sin (\pi-x)=\sin x, \quad \sin ((\pi / 2)-x)=\cos x, \quad \sin ((\pi / 2)+x)=\cos x$,
$\cos (\pi-x)=-\cos x, \quad \cos ((\pi / 2)-x)=\sin x, \quad \cos ((\pi / 2)+x)=-\sin x .$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
06:59

Problem 28

Prove the following for all $x_{1}, x_{2} \in \mathbb{R}:$
(i) $\sin x_{1}+\sin x_{2}=2 \sin \left(\left(x_{1}+x_{2}\right) / 2\right) \cos \left(\left(x_{1}-x_{2}\right) / 2\right)$
(ii) $\sin x_{1}-\sin x_{2}=2 \cos \left(\left(x_{1}+x_{2}\right) / 2\right) \sin \left(\left(x_{1}-x_{2}\right) / 2\right)$,
(iii) $\cos x_{1}+\cos x_{2}=2 \cos \left(\left(x_{1}+x_{2}\right) / 2\right) \cos \left(\left(x_{1}=x_{2}\right) / 2\right)$,
(iv) $\cos x_{1}-\cos x_{2}=2 \sin \left(\left(x_{1}+x_{2}\right) / 2\right) \sin \left(\left(x_{2}-x_{1}\right) / 2\right)$.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
02:57

Problem 29

Prove the following for all $x \in \mathbb{R}$ :
(i) $\sin 2 x=2 \sin x \cos x$,
(ii) $\cos 2 x=\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1=1-2 \sin ^{2} x$,
(iii) $\sin 3 x=3 \sin x-4 \sin ^{3} x$
(iv) $\cos 3 x=4 \cos ^{3} x-3 \cos x$.
Deduce that
$\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}, \quad \sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}=\cos \frac{\pi}{6}, \quad \cos \frac{\pi}{3}=\frac{1}{2}=\sin \frac{\pi}{6}$

Barsha Rana
Barsha Rana
Numerade Educator
02:18

Problem 30

Prove the following for all $x_{1}, x_{2} \in \mathbb{R}$ :
(i) $\sin x_{1}=\sin x_{2} \Longleftrightarrow x_{2}=m \pi+(-1)^{m} x_{1}$, where $m \in \mathbb{Z}$.
(ii) $\cos x_{1}=\cos x_{2} \Longleftrightarrow x_{2}=2 m \pi \pm x_{1}$, where $m \in \mathbb{Z}$.
(iii) $\sin x_{1}=\sin x_{2}$ and $\cos x_{1}=\cos x_{2} \Longleftrightarrow x_{2}=2 m \pi+x_{1}$, where $m \in \mathbb{Z}$.
(Hint: Exercise 28 and solutions of the equations $\sin x=0, \cos x=0 .$ )

Gokul R Nair
Gokul R Nair
Numerade Educator
00:55

Problem 31

If $x \in \mathbb{R}$ with $x \neq(2 k+1) \pi / 2$ for any $k \in \mathbb{Z}$, then show that
$$
1+\tan ^{2} x=\sec ^{2} x, \quad(\tan )^{\prime} x=\sec ^{2} x, \quad \text { and } \quad(\sec )^{\prime} x=\sec x \tan x
$$

AG
Ankit Gupta
Numerade Educator
View

Problem 32

If $x \in \mathbb{R}$ with $x \neq k \pi$ for any $k \in \mathbb{Z}$, then show that
$1+\cot ^{2} x=\csc ^{2} x, \quad(\cot )^{\prime} x=-\csc ^{2} x, \quad$ and $\quad(\csc )^{\prime} x=-\csc x \cot x$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:55

Problem 33

If $x_{1}, x_{2} \in \mathbb{R}$ are such that none of $x_{1}, x_{2}$, and $x_{1}+x_{2}$ equals $(2 k+1) \pi / 2$ for any $k \in \mathbb{Z}$, then show that
$$
\tan \left(x_{1}+x_{2}\right)=\frac{\tan x_{1}+\tan x_{2}}{1-\tan \left(x_{1}+x_{2}\right)}.
$$

AG
Ankit Gupta
Numerade Educator
06:03

Problem 34

Prove the following for all $y_{1}, y_{2} \in \mathbb{R}$ :
(i) $\tan ^{-1} y_{1}+\tan ^{-1} y_{2}=\tan ^{-1}\left(\frac{y_{1}+y_{2}}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}<1$,
(ii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\frac{\pi}{2}$ if $y_{1} y_{2}=1$
(iii) $\tan ^{-1}\left|y_{1}\right|+\tan ^{-1}\left|y_{2}\right|=\tan ^{-1}\left(\frac{\left|y_{1}\right|+\left|y_{2}\right|}{1-y_{1} y_{2}}\right)$ if $y_{1} y_{2}>1$.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:35

Problem 35

Prove the following:
(i) $\sin \left(\sin ^{-1} y\right)=y$ for all $y \in[-1,1]$ and
$$
\sin ^{-1}(\sin x)=\left\{\begin{array}{ll}
x & \text { if } x \in[-\pi / 2, \pi / 2] \\
\pi-x & \text { if } x \in(\pi / 2,3 \pi / 2]
\end{array}\right.
$$
(ii) $\cos \left(\cos ^{-1} y\right)=y$ for all $y \in[-1,1]$ and $\cos ^{-1}(\cos x)=|x|$ for all
$x \in[-\pi, \pi]$

Rukhmani Jain
Rukhmani Jain
Numerade Educator
02:04

Problem 36

If $y \in(-1,1)$, then show that
$$
\sin ^{-1} y=\int_{0}^{y} \frac{1}{\sqrt{1-t^{2}}} d t \quad \text { and } \quad \cos ^{-1} y=\frac{\pi}{2}-\int_{0}^{y} \frac{1}{\sqrt{1-t^{2}}} d t
$$
Deduce that
$$
\lim _{y \rightarrow 1^{-}} \int_{0}^{y} \frac{1}{\sqrt{1-t^{2}}} d t=\frac{\pi}{2} \text { . }
$$

Hunza Gilgit
Hunza Gilgit
Numerade Educator
04:51

Problem 37

If $y \in(1, \infty)$, then show that
$$
\sec ^{-1} y=\lim _{a \rightarrow 1^{+}} \int_{a}^{y} \frac{1}{t \sqrt{t^{2}-1}} d t \quad \text { and } \quad \csc ^{-1} y=\frac{\pi}{2}-\lim _{a \rightarrow 1^{+}} \int_{a}^{y} \frac{1}{t \sqrt{t^{2}-1}} d t .
$$

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
05:56

Problem 38

Prove the following:
(i) $\cot ^{-1} y=\frac{\pi}{2}-\tan ^{-1} y$ for all $y \in \mathbb{R}$,
(ii) $\csc ^{-1} y=\sin ^{-1} \frac{1}{y}$ for all $y \in \mathbb{R}$ with $|y| \geq 1$,
(iii) $\sec ^{-1} y=\cos ^{-1} \frac{1}{y}$ for all $y \in \mathbb{R}$ with $|y| \geq 1$.
(Hint: $\tan ^{-1}|y|+\tan ^{-1}|1 / y|=\pi / 2$ for all $y \in \mathbb{R}$ with $y \neq 0$.)

Heena Mahenoor
Heena Mahenoor
Numerade Educator
01:06

Problem 39

For all $y \in[-1,1]$, show that
$$
\sin ^{-1} y+\sin ^{-1}(-y)=0, \cos ^{-1} y+\cos ^{-1}(-y)=\pi, \sin ^{-1} y+\cos ^{-1}(y)=\frac{\pi}{2}
$$
and for all $y \in \mathbb{R}$ with $|y| \geq 1$, show that
$$
\csc ^{-1} y+\sec ^{-1} y=\frac{\pi}{2}.
$$

Rukhmani Jain
Rukhmani Jain
Numerade Educator
00:49

Problem 40

Prove the following:
(i) $\left(\cot ^{-1}\right)^{\prime} y=-\frac{1}{1+y^{2}}$ for all $y \in \mathbb{R}$,
(ii) $\left(\mathrm{csc}^{-1}\right)^{\prime} y=-\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$,
(iii) $\left(\mathrm{sec}^{-1}\right)^{\prime} y=\frac{1}{|y| \sqrt{y^{2}-1}}$ for all $y \in \mathbb{R}$ with $|y|>1$.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
03:27

Problem 41

Let $r_{0} \in \mathbb{R}$, and consider the function $f_{0}: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f_{0}(x)=\left\{\begin{array}{ll}
\sin (1 / x) & \text { if } x \neq 0 \\
r_{0} & \text { if } x=0
\end{array}\right.
$$
(i) Show that $f_{0}$ is not continuous at 0 . Conclude that the function $x \mapsto$ $\sin (1 / x)$ for $x \in \mathbb{R} \backslash\{0\}$ cannot be extended to $\mathbb{R}$ as a continuous function.
(ii) (ii) If $I$ is an interval and $I \subset \mathbb{R} \backslash\{0\}$, then show that $f_{0}$ has the IVP on $I$. If $I$ an interval such that $0 \in I$, then show that $f_{0}$ has the IVP on $I$ if and only if $\left|r_{0}\right| \leq 1$.

Bobby Barnes
Bobby Barnes
University of North Texas
01:05

Problem 42

Consider the function $h: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
h(x):=\left\{\begin{array}{ll}
|x|+|x \sin (1 / x)| & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Show that $h$ has a strict absolute minimum at 0, but for any $\delta>0, h$ is neither decreasing on $(-\delta, 0)$ nor increasing on $(0, \delta)$.

Carson Merrill
Carson Merrill
Numerade Educator
02:50

Problem 43

Consider the function $g: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by $g(x):=\cos (1 / x) .$ Prove the following:
(i) $g$ is an even function.
(ii) $\lim _{x \rightarrow 0} g(x)$ does not exist, but $\lim _{x \rightarrow 0^{+}}[g(x)-g(-x)]$ exists. Also, $g$
cannot be extended to $\mathbb{R}$ as a continuous function.
(iii) For any $\delta>0, g$ is not uniformly continuous on $(0, \delta)$ as well as on $(-\delta, 0)$, but it is uniformly continuous on $(\infty,-\delta] \cup[\delta, \infty)$.
(iv) For any $\delta>0, g$ is not monotonic, not convex, and not concave on $(0, \delta)$ as well as on $(-\delta, 0)$.

Dishary Hossain
Dishary Hossain
Numerade Educator
01:05

Problem 44

Let $r_{0} \in \mathbb{R}$ and consider the function $g_{0}: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by
$$
g_{0}(x):=\left\{\begin{array}{ll}
\cos (1 / x) & \text { if } x \neq 0 \\
r_{0} & \text { if } x=0
\end{array}\right.
$$
Show that $g_{0}$ is not continuous at $0 .$ Define $G_{0}: \mathbb{R} \rightarrow \mathbb{R}$ by $G_{0}(x):=$ $\int_{0}^{x} \cos (1 / t) d t$. Show that $G_{0}$ is differentiable at 0 and $G_{0}^{\prime}(0)=0$, that is,
$$
\lim _{x \rightarrow 0} \frac{1}{x} \int_{0}^{x} \cos \frac{1}{t} d t=0.
$$

Carson Merrill
Carson Merrill
Numerade Educator
00:54

Problem 45

Consider the functions $g_{1}, g_{2}: \mathbb{R} \rightarrow \mathbb{R}$ defined by $g_{1}(x):=\left\{\begin{array}{ll}x \cos (1 / x) & \text { if } x \neq 0, \\ 0 & \text { if } x=0,\end{array} \quad\right.$ and $\quad g_{2}(x):=\left\{\begin{array}{ll}x^{2} \cos (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$
Establish properties of $g_{1}$ and $g_{2}$ similar to those of the functions $f_{1}$ and $f_{2}$ given in Example $7.18$ and Example $7.19$, respectively.

Lucas Finney
Lucas Finney
Numerade Educator
01:09

Problem 46

Consider the function $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by $f(x)=(\sin (1 / x)) / x$. Show that the amplitude of the oscillation of the function $f$ increases without any bound as $x$ tends to 0 .

Raj Bala
Raj Bala
Numerade Educator
00:56

Problem 47

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f(x):=\left\{\begin{array}{ll}
x^{2} \sin \left(1 / x^{2}\right) & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Show that $f$ is differentiable on $\mathbb{R}$, but for any $\delta>0, f^{\prime}$ is not bounded on $[-\delta, \delta] .$ Thus $f^{\prime}$ has an antiderivative on the interval $[-1,1]$, but it is not Riemann integrable on $[-1,1]$.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:21

Problem 48

Let $n \in \mathbb{N}$ and consider the function $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$$
f_{n}(x):=\left\{\begin{array}{ll}
x^{n} \sin (1 / x) & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Prove the following: (i) If $n$ is odd and $k:=(n-1) / 2$, then $f_{n}^{(k)}$ exists and is continuous on $\mathbb{R}$, but $f_{n}^{(k+1)}$ does not exist at 0 . (ii) If $n$ is even and $k:=n / 2$, then $f_{n}^{(k)}$ exists on $\mathbb{R}$, but it is not continuous at $0 .$ (Compare Exercise 12 of Chapter 4.)

Yujie Wang
Yujie Wang
College of San Mateo
02:19

Problem 49

Find the polar coordinates of the points in $\mathbb{R}^{2}$ whose Cartesian coordinates are as follows:
(i) $(1,1)$,
(ii) $(0,3)$,
(iii) $(2,2 \sqrt{3})$,
(iv) $(2 \sqrt{3}, 2)$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:46

Problem 50

If $x, y \in \mathbb{R}$ are not both zero and $(r, \theta)$ are the polar coordinates of $(x, y)$, then determine the polar coordinates of (i) $(y, x)$, and (ii) $(t x, t y)$, where $t$ is any positive real number.

Aditya Sood
Aditya Sood
Numerade Educator
03:15

Problem 51

Let $r$ be a positive real number and $\theta \in(-\pi, \pi]$ and $\alpha \in \mathbb{R}$ be such that $\theta+\alpha \in(-\pi, \pi] .$ If $P$ and $P_{\alpha}$ denote the points with polar coordinates $(r, \theta)$ and $(r, \theta+\alpha)$, respectively, then find the Cartesian coordinates of $P_{\alpha}$ in terms of the Cartesian coordinates of $P$. [Note: The transformation $P \mapsto P_{\alpha}$ corresponds to a rotation of the plane by the angle $\alpha$.]

Nathan Mankovich
Nathan Mankovich
Numerade Educator
03:15

Problem 52

Find the angle(s) between the curves $x^{2}+y^{2}=16$ and $y^{2}=6 x$ at their point(s) of intersection.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
00:26

Problem 53

Determine whether the following functions are algebraic or transcendental:
(i) $f(x)=\pi x^{11}+\pi^{2} x^{5}+9$ for $x \in \mathbb{R}$
(ii) $f(x)=\frac{e x^{2}+\pi}{\pi x^{2}+e}$ for $x \in \mathbb{R}$,
(iii) $f(x)=\ln _{10} x$ for $x>0$,
(iv) $f(x)=x^{\pi}$ for $x>0$.

Linh Vu
Linh Vu
Numerade Educator
01:53

Problem 54

Is it possible that
$\ln x=\left(\sqrt[3]{e x^{2}+(\pi-2 e) x+e-\pi}+\sqrt{\pi x^{2}+(\sqrt{2}-2 \pi) x+\pi-\sqrt{2}}\right)^{1 / 17}$
for all $x>0$ ? Justify your answer.

Alex Roush
Alex Roush
Numerade Educator
16:17

Problem 55

Let $p, q \in(1, \infty)$ be such that $(1 / p)+(1 / q)=1$.
(i) If $f:[0, \infty) \rightarrow \mathbb{R}$ is defined by $f(x):=(1 / q)+(1 / p) x-x^{1 / p}$, then show that $f(x) \geq f(1)$ for all $x \in[0, \infty)$.
(ii) Show that $a b \leq\left(a^{p} / p\right)+\left(b^{q} / q\right)$ for all $a, b \in[0, \infty)$. (Hint: If $b \neq 0$, let $x:=a^{p} / b^{q}$ in (i).)
(iii) (Hölder Inequality for Sums) Given any $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$ in $\mathbb{R}$, prove that
$$
\sum_{i=1}^{n}\left|a_{i} b_{i}\right| \leq\left(\sum_{i=1}^{n}\left|a_{i}\right|^{p}\right)^{1 / p}\left(\sum_{i=1}^{n}\left|b_{i}\right|^{q}\right)^{1 / q}
$$
Deduce the Cauchy-Schwarz inequality as a special case. (iv) (Hölder Inequality for Integrals) Given any continuous functions $f, g:[a, b] \rightarrow \mathbb{R}$, prove that
$$
\int_{a}^{b}|f(x) g(x)| d x \leq\left(\int_{a}^{b}|f(x)|^{p} d x\right)^{1 / p}\left(\int_{a}^{b}|g(x)|^{q} d x\right)^{1 / q}
$$
(v) (Minkowski Inequality for Sums) Given any $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$ in $\mathbb{R}$, prove that
$$
\left(\sum_{i=1}^{n}\left|a_{i}+b_{i}\right|^{p}\right)^{1 / p} \leq\left(\sum_{i=1}^{n}\left|a_{i}\right|^{p}\right)^{1 / p}+\left(\sum_{i=1}^{n}\left|b_{i}\right|^{p}\right)^{1 / p}
$$
(Hint: The $p$ th power of the expression on the left can be written as $\sum_{i=1}^{n}\left|a_{i}\right|\left(\left|a_{i}+b_{i}\right|\right)^{p-1}+\sum_{i=1}^{n}\left|b_{i}\right|\left(\left|a_{i}+b_{i}\right|\right)^{p-1} ;$ now use (iii). $)$
(vi) (Minkowski Inequality for Integrals) Given any continuous functions $f, g:[a, b] \rightarrow \mathbb{R}$, prove that
$$
\left(\int_{a}^{b}|f(x)+g(x)|^{p} d x\right)^{1 / p} \leq\left(\int_{a}^{b}|f(x)|^{p} d x\right)^{1 / p}+\left(\int_{a}^{b}|g(x)|^{p} d x\right)^{1 / p}.
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:31

Problem 56

Let $n \in \mathbb{N}$. By applying L'Hôpital's rule $n$ times, prove the following:
(i) $\lim _{x \rightarrow 0} \frac{\exp x-\sum_{k=0}^{n} x^{k} / k !}{x^{n+1}}=\frac{1}{(n+1) !}$,
(ii) $\lim _{x \rightarrow 1} \frac{\ln x-\sum_{k=1}^{n}(-1)^{k}(x-1)^{k} / k}{(x-1)^{n+1}}=\frac{(-1)^{n}}{(n+1)}$,
(iii) $\lim _{x \rightarrow 0} \frac{\sin x-\sum_{k=0}^{\lceil(n-2) / 2\rceil}(-1)^{k} x^{2 k+1} /(2 k+1) !}{x^{n+1}}=\left\{\begin{array}{c}\frac{(-1)^{n / 2}}{(n+1) !} \text { if } n \text { is even, } \\ 0 \quad \text { if } n \text { is odd }\end{array}\right.$
(iv) $\lim _{x \rightarrow 0} \frac{\cos x-\sum_{k=0}^{\lfloor n / 2\rfloor}(-1)^{k} x^{2 k} /(2 k) !}{x^{n+1}}=\left\{\begin{array}{c}\frac{(-1)^{(n+1) / 2}}{(n+1) !} \text { if } n \text { is odd, } \\ 0 & \text { if } n \text { is even. }\end{array}\right.$

Khushbu Rani
Khushbu Rani
Numerade Educator
10:10

Problem 57

Let $p, q \in \mathbb{N}$. For $n \in \mathbb{N}$, consider the function $f_{n}:[0, p / q] \rightarrow \mathbb{R}$ defined by $f_{n}(x):=x^{n}(p-q x)^{n} / n !$. Prove the following results:
(i) $f_{n}(0)=0=f_{n}(p / q) .$ Also, $f_{n}^{(k)}(0)=-f_{n}^{(k)}(p / q) \in \mathbb{Z}$ for each $k \in \mathbb{N}$
in fact, $f_{n}^{(k)}(0)=0=f_{n}^{(k)}(p / q)$ if $k \leq n$ or $k>2 n$.
(ii) $\max \left\{f_{n}(x): x \in[0, p / q]\right\}=f_{n}(p / 2 q)$, and $f_{n}(p / 2 q) \rightarrow 0$ as $n \rightarrow \infty$.
(iii) Let, if possible, $\pi=p / q$, and consider $a_{n}:=\int_{0}^{\pi} f_{n}(x) \sin x d x .$ Then $a_{n} \in \mathbb{Z}$ for each $n \in \mathbb{N}$ (by repeated use of Integration by Parts), whereas $0<a_{n}<1$ for all large $n \in \mathbb{N}$.
(iv) $\pi$ is irrational.

WZ
Wen Zheng
Numerade Educator
03:33

Problem 58

(i) Show that for any $n \in \mathbb{N}, \int_{0}^{\pi / 2} \sin ^{n} x d x=\frac{n-1}{n} \int_{0}^{\pi / 2} \sin ^{n-2} x d x$.
(ii) Show that for any $k \in \mathbb{N}$,
$$
\int_{0}^{\pi / 2} \sin ^{2 k} x d x=\frac{(2 k-1)(2 k-3) \cdots 3 \cdot 1}{(2 k)(2 k-2) \cdots 4 \cdot 2} \cdot \frac{\pi}{2}=\frac{(2 k) !}{\left[2^{k} k !\right]^{2}} \cdot \frac{\pi}{2}
$$
and
$$
\int_{0}^{\pi / 2} \sin ^{2 k+1} x d x=\frac{2 k(2 k-2) \cdots 4 \cdot 2}{(2 k+1)(2 k-1) \cdots 3 \cdot 1}=\frac{\left[2^{k} k !\right]^{2}}{(2 k+1) !} .
$$
(iii) For $k \in \mathbb{N}$, let
$$
\mu_{k}:=\frac{\int_{0}^{\pi / 2} \sin ^{2 k} x d x}{\int_{0}^{\pi / 2} \sin ^{2 k+1} x d x}
$$
Show that $1 \leq \mu_{k} \leq(2 k+1) / 2 k$ for each $k \in \mathbb{N}$ and consequently that $\mu_{k} \rightarrow 1$ as $k \rightarrow \infty$. Deduce that
$$
\sqrt{\pi}=\lim _{k \rightarrow \infty} \frac{(k !)^{2} 2^{2 k}}{(2 k) ! \sqrt{k}}
$$
Thus, $\pi \sim(k !)^{4} 2^{4 k} /[(2 k) !]^{2} k .\left(\right.$ Hint: $\sin ^{2 k+1} x \leq \sin ^{2 k} x \leq \sin ^{2 k-1} x$
for all $x \in[0, \pi / 2] .)$
[Note: This result is known as the Wallis formula.]

Joseph Liao
Joseph Liao
Numerade Educator
25:36

Problem 59

(i) Show that for any $n \in \mathbb{N}$,
$$
\frac{1}{n+\frac{1}{2}} \leq \int_{n}^{n+1} \frac{d x}{x} \leq \frac{1}{2}\left(\frac{1}{n}+\frac{1}{n+1}\right)
$$
(ii) Let $\left(a_{n}\right)$ be the sequence defined by $a_{n}:=n ! e^{n} / n^{n} \sqrt{n}$ for $n \in \mathbb{N}$. Show that
$$
\ln \left(\frac{a_{n}}{a_{n+1}}\right)=\left(n+\frac{1}{2}\right) \ln \left(1+\frac{1}{n}\right)-1
$$
and hence
$$
1 \leq \frac{a_{n}}{a_{n+1}} \leq \exp \left(\frac{1}{4}\left[\frac{1}{n}-\frac{1}{n+1}\right]\right) \quad \text { for all } n \in \mathbb{N}
$$
Deduce that $\left(a_{n}\right)$ is a monotonically decreasing sequence of positive real numbers and it is convergent. Let $\alpha:=\lim _{n \rightarrow \infty} a_{n}$.
(iii) Use the inequalities in (ii) to show that
$$
1 \leq \frac{a_{n}}{a_{n+k}} \leq \exp \left(\frac{1}{4}\left[\frac{1}{n}-\frac{1}{n+k}\right]\right) \quad \text { for all } n, k \in \mathbb{N}
$$
Taking the limit as $k \rightarrow \infty$, deduce that $\alpha>0$ and furthermore, $1 \leq\left(a_{n} / \alpha\right) \leq \exp (1 / 4 n)$ for all $n \in \mathbb{N}$
(iv) Show that the Wallis formula given in Exercise 58 can be written as $\sqrt{2 \pi}=\lim _{n \rightarrow \infty} a_{n}^{2} / a_{2 n} .$ Deduce that $\alpha=\sqrt{2 \pi}$
(v) Use (iii) and (iv) to show that for all $n \in \mathbb{N}$,
$$
(\sqrt{2 \pi}) n^{n+\frac{1}{2}} e^{-n} \leq n ! \leq(\sqrt{2 \pi}) n^{n+\frac{1}{2}} e^{-n+(1 / 4 n)}
$$
and conclude that
$$
\lim _{n \rightarrow \infty} \frac{n !}{(\sqrt{2 \pi n}) n^{n} e^{-n}}=1
$$
Thus, $n ! \sim(\sqrt{2 \pi n}) n^{n} e^{-n}$
[Note: This result is known as Stirling's Formula.]

MK
Musashi Koyama
Numerade Educator
01:06

Problem 60

Let $r, s \in \mathbb{R}$ and consider the function $F:[0,1] \rightarrow \mathbb{R}$ defined by
$$
F(x):=\left\{\begin{array}{ll}
x^{r} \sin \left(1 / x^{s}\right) & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.
$$
Prove the following:
(i) $F$ is continuous $\Longleftrightarrow r>0$.
(ii) $F$ is differentiable $\Longleftrightarrow r>1$.
(iii) $F^{\prime}$ is bounded $\Longleftrightarrow r \geq 1+s$.
(iv) $F^{\prime}$ is continuous $\Longleftrightarrow r>1+s$.
(v) $F$ is twice differentiable $\Longleftrightarrow r>2+s$.
(vi) $F^{\prime \prime}$ is bounded $\Longleftrightarrow r \geq 2+2 s$.
(vii) $F^{\prime \prime}$ is continuous $\Longleftrightarrow r>2+2 s$.

Carson Merrill
Carson Merrill
Numerade Educator
02:11

Problem 61

Prove that the secant function, the cosecant function, and the cotangent function are transcendental.

Lauren Shelton
Lauren Shelton
Numerade Educator