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8.4 Quiz 2

Please submit solutions to the following via moodle by 23:59 on Wednesday 30th November. Each question is worth [2] marks. See Week07Logic.pdf for information which may help with Q2.3-Q2.5.

Q2.1.Convergence. Which of the following statements about the function $h(x)$ is FALSE, where:

h(x)=2\log(x^{3}+x+2)-3\log(x^{2}+2x+1)\;\;\mbox{?}

(A) $h$ has a stationary point at $x=\frac{5}{3}$ . $\;\;$ (B) $h(R)\rightarrow 0$ as $R\rightarrow\infty$ . $\;\;$ (C) $h(0)=\log 4$ .

(D) $h(x)$ is well-defined for every real number $x$ . $\;\;$ (E) $e^{h(x)}$ is a rational function in $x$ .

Q2.2.Improper integrals. One of the following statements describes the improper integrals

I=\int_{0}^{1}{{dx}\over{\sqrt{x}(1+x)}}\quad{\hbox{and}}\quad J=\int_{0}^{1}{% {dx}\over{(1-x)(1+x^{2})}}.

(A) Both $I$ and $J$ diverge. $\;\;\;\;\;$ (B) The integral $I$ converges to $\pi/2$ , whereas $J$ diverges.

(E) Whereas $I$ diverges, $J$ converges to $\log 2$ .

Q2.3Elements within a range. Let $\color{blue}\colorlet{pgfstrokecolor}{.}{(a_{1},a_{2},\ldots,a_{100})}$ be a sequence with $k$ -th element

\displaystyle a_{k}=\sin\Bigl(\frac{3k\pi}{44}\Bigr).

Find the number of elements of the sequence for which $\color{blue}\colorlet{pgfstrokecolor}{.}{-\frac{3}{4}<a_{k}\leq-\frac{1}{4}}$ .

Q2.4Comparison of sequences. A second sequence $\color{blue}\colorlet{pgfstrokecolor}{.}{(b_{1},b_{2},\ldots,b_{100})}$ has $k$ -th element

\displaystyle b_{k}=\cos\Bigl(\frac{5k\pi}{13}\Bigr).

How many of the 100 pairs of elements satisfy the condition $\color{blue}\colorlet{pgfstrokecolor}{.}{|a_{k}|>\frac{1}{4}\leavevmode% \nobreak\ \leavevmode\nobreak\ \mbox{OR}\leavevmode\nobreak\ \leavevmode% \nobreak\ a_{k}<b_{k}}$ ?

Q2.5If Statements. Let $x$ be a scalar real number within the interval $(0,50]$ assigned to object x. If the following R code is performed

: if (x > 25) {
: y <- sum( x < 1:50 )
: }else {
: y <- sum( x > 0:49 )
: }

then the object y represents

\displaystyle y

\displaystyle=

\displaystyle\left\{\begin{array}[]{lll}{\bf\mbox{A}}&&\mbox{if}\leavevmode% \nobreak\ \leavevmode\nobreak\ x>25\\ {\bf\mbox{B}}&&\mbox{if}\leavevmode\nobreak\ \leavevmode\nobreak\ x\leq 25\end% {array}\right.

for some statements A and B. Match up A and B from the following list

i)

$\lfloor x\rfloor$
ii)

$\lceil x\rceil$
iii)

$\lfloor 50-x\rfloor$
iv)

$\lceil 50-x\rceil$
v)

$\lfloor 25-x\rfloor$

where $\lfloor x\rfloor$ denotes rounding down to the nearest integer and $\lceil x\rceil$ denotes rounding up. In R these operations can be performed directly using floor(x) and ceiling(x).