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%Martin Pergler's 150s test C questions (hard questions).
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\begin{document}

\begin{center}
\large
C Questions from Martin Pergler's 150s Tests
\end{center}

\normalsize
%Name \ansblank[3.65in] Student Number \ansblank
%\\[4pt]
\begin{Comment}{What is it?}
Part C of all my 150s tests and exams gives a choice of 3
somewhat harder questions requiring students to think about
some aspect of the material they have probably not yet
considered. I also try to make these questions have some
intrinsic mathematical interest.

This is a compendium of C questions that I have used. Feel free
to use the occasional one in class, on homework, on a test, or
for instructor inspiration. Please do not distribute wholesale
to students or use regularly in class or on tests without prior
arrangement with me, since I will reuse some of these questions
with minor modifications in future years.

A couple of questions are marked \dag. These started off life as
C questions, but were simplified to become B questions. They
could be unsimplified...

\end{Comment}

\begin{enumerate}
%
\item Demonstrate that $\Lim_{x\rightarrow 1}{\D x^n-1 \over \D x-1}=n$. You
don't have to use $\epsilon-\delta$.
%Hint: start by expanding
%$(x-1)(1+x+x^2+\dots+x^k)$.
[If you remember the definition of the
derivative, by doing this question you have shown that if $f(x)=x^n$, then
$f'(1)=n$.]
%
\item Recall the function $D(x)=\begin{cases} 0&\text{ $x$ irrational},\\
1/q&x=p/q.\end{cases}$ For what $c$ does $\Lim_{x\rightarrow c}D(x)$ exist
and what is it?
%
\item \begin{enumerate} \item Let $f(x)$ and $g(x)$ be two functions such that $f(x)\leq g(x)$ for
all $x$ except possibly $x=c$. Prove that $\Lim_{x\rightarrow c}f(x)\leq
\Lim_{x\rightarrow c}g(x)$ if both of these limits exist.
\item Give an example to show that we can't replace the $\leq$'s in part (a)
by $<$'s. (A sketch is fine)
\end{enumerate}

\item Let $f(x)=1/(x-1)$. We want to approximate $f(x)$ near $x=0$ by
polynomials.
\begin{enumerate}\item Find a linear polynomial $p_1(x)$ such that
$p_1(0)=f(0)$ and $p_1'(0)=f'(0)$.
\item Find a quadratic polynomial $p_2(x)$ such that
$p_2(0)=f(0)$, $p_2'(0)=f'(0)$, and $p_2''(0)=f''(0)$.
\item Explain what process you could follow to find an $n$th degree
polynomial $p_n(x)$ such that $p_n^{(k)}(0)=f_1^{(k)}(0)$ for $k=0,1,
\dots n$.
\item Why do we know that no such $p_n(x)$ can approximate $f(x)$ well
near $x=1$?
\end{enumerate}
%
\item Suppose $f$ is a function which is a product of several other
functions, say $f(x)=g_1(x)g_2(x)\dots g_n(x)$. Prove that
$$
\frac{f'(x)}{f(x)}=\frac{g_1'(x)}{g_1(x)}+\dots+\frac{g_n'(x)}{g_n(x)}
$$
(provided $f(x)\neq 0$). This often provides a convenient method of
calculating $f'(x)$. [Hint: use induction.]
%
\item \begin{enumerate} \item
Suppose $\frowney(x)$ is a truly abysmally awful function, but $|\frowney(x)|\leq
1$. Show that $g(x)=x^2\, \frowney(x)$ is differentiable at $x=0$ and compute the
derivative there. [Warning: $\frowney$ is probably not differentiable itself, so
don't just use product rule.]
\item Show by a picture or by an example that $k(x)=x\,\frowney(x)$ may not be
differentiable at $x=0$.
\end{enumerate}
%
\item \dag You skydive out of an airplane 10,000 feet above the ground. (10
points)
\begin{enumerate}
\item Assume there was no air resistance. At what speed (ft/s) would you
``gently land'' on the ground? Take acceleration due to gravity to be
32 ft/s$^2$. [If you don't have a calculator, write down right numbers in an
equation and then roughly estimate the answer.]
\item Fortunately, air resistance is substantial. You will speed up for the
first 10s or so, then hurtle down at a constant speed (called, somewhat
unfortunately in this case, ``terminal velocity'' by physicists) of about
170 ft/s (120
mph) until you open your parachute. That will slow you down so that you
truly gently land on the ground.

Sketch graphs of velocity versus time and position versus time for the trip
(with air resistance and eventual parachute opening).
\end{enumerate}

\item (Zeros of polynomials)
 Suppose $f(x)$ is a polynomial. The value $a$ is called a zero of order $k$
 if $f(x)=(x-a)^k g(x)$ where $g(x)$ is a polynomial such that $g(a)\neq 0$
 (in other words, $(x-a)$ appears ``$k$ times" if you factor $f(x)$
 completely.)
\\[3pt]
 (a) If $a$ is a zero of order $1$, prove $f'(a)\neq 0$.
\\[3pt] (b) If $a$ is a zero of order $2$, prove $f'(a)=0$ but $f''(a)\neq 0$.
\\[3pt] (c) If $f(a)=0$ and $f'(a)=0$, then $a$ is a zero of order at least $2$.
\\[3pt] (Incidentally, $a$ is actually a zero of order $k$ if and only if
  $f^{(j)}(a)=0$ for $j=1,...,k-1$ but $f^{(k)}(a)\neq 0$.)

\item(More on cubic splines)
We talked in class about cubic splines through two specific points
with specified derivatives at those points. Here we'll combine two cubic
splines
to pass through 3 given points, $P(-1,-1)$, $Q(0,0)$, and $R(1,3)$. We'll do
this by finding a cubic $f(x)$ that passes through $P$ and $Q$ and
another cubic $g(x)$ that passes through $Q$ and $R$, and make sure
they link up without any corner or bump at $Q$.
\begin{enumerate}
\item I claim we can assume $f(x)=ax^3+bx^2+cx$
   and $g(x)=dx^3+ex^2+fx$ for some coefficients $a,b,c,d,e,f$. Why do
   we not need constant terms (something times $x^0$)?
\item To avoid a corner at $Q$, we insist that $f'(0)=g'(0)$. What does this
   imply about $c$ and $f$?
\item We actually also insist that $f''(0)=g''(0)$ (to make the transition
   smoother). What does this imply about the remaining coefficients?
\item The cubic splines shouldn't have a funny "hook" at $P$ and $R$. We can
   ensure this by forcing $f''(-1)=g''(1)=0$. Use this to express $d$ and
   $a$ in terms of $b$.
\item You should only have two unknowns left now. Use the fact that
   $f(x)$ passes through $P$ and $g(x)$ passes through $R$ to solve for
   these remaining unknowns, and hence find exactly what $f$ and $g$
   should be.
\end{enumerate}
(A computer could of course do this in a flash even if the numbers
were nastier, and given $n$ points, we could find $n-1$ cubics which would
connect them "smoothly" in a similar manner.)

\item (Superior and inferior limits)
A number $U$ is called an upper bound of $f(x)$ near $c$ if
 given any $\epsilon>0$, there exists a $\delta$ such that $0<|x-c|<\delta$
 implies $f(x)\leq U+\epsilon$.
Similarly, a number $L$ is called a lower bound of $f(x)$ near $c$ if
 given any $\epsilon>0$, there exists a $\delta$ such that $0<|x-c|<\delta$
 implies $f(x)\geq L-\epsilon$.
\begin{enumerate}
 \item Prove that if $U$ is an upper bound of $f(x)$ near $c$, then so is
   any $U'>U$. (It's also true that if $L$ is a lower bound of $f(x)$
   near $c$, then so is any $L'<L$, but you don't have to prove that.)
 \item Prove that if $L$ and $U$ are lower and upper bounds for $f(x)$ near
   $c$, then $L\leq U$.
\item What does it mean if $f(c)$ is both an upper and lower bound for $f(x)$ near $c$?
\item What are the lower and upper bounds near $0$ of $f(x)=\sin (1/x)?$
\end{enumerate}
(The smallest possible $U$ is called the superior limit of $f(x)$ at $c$,
 and the largest possible $L$ is called the inferior limit.)

\item (\dag Related rates and falling objects)
\begin{enumerate}\item
    You are painting a second storey window, standing at the top of
    a 15' ladder leaning against your house. Suddenly you feel the foot
    of the ladder slowly sliding away from the wall at 1 inch/sec.
    At what rate is the
    top of the ladder moving downwards when the base is 5 feet away from the
    wall?
    (Hint: Don't forget to convert 1 inch/sec to feet/sec) (5 points)
\item The top of the ladder scrapes off a little stone in the stucco
    at the instant mentioned in part (a). Neglecting air resistance, how
    long does it take the stone to hit the ground? (5 points)
\end{enumerate}
 [When I first typed the test, part (a) read ``At what rate are you
  moving downwards when the base....''. But on rereading it, I figured
  the only reasonable answer to that would be ``as quickly as possible''.]

\item (\dag Chemical kinetics)
  When a solution of substance $A$ is exposed to light, a chemical reaction
  takes place in which two molecules of $A$ combine to form one molecule of
  $B$.
  At first, the reaction takes place at a fairly rapid rate as molecules
  of $A$ are plentiful in the solution. However, the rate of the reaction
  gradually gets slower and slower as the concentration of $A$ (labelled
  $[A]$
  by chemists) gets
  smaller and smaller and it happens less and less often that 2 molecules of
  $A$ manage to ``find'' each other. ($6+4=10$ points)
\begin{enumerate}
  \item Draw quick approximate sketches of $[A]$ and $d[A]/dt$
    as a function of time, based on the above information.
  \item When $[A]$ is measured by experiment, the data lie on a curve
     $[A] = 1/k(t+C)$, where $t$ is time and $k$ and
     $C$ are some constants. Check that this formula for $[A]$ satisfies the
     relation $-d[A]/dt = k[A]^2$. [This is predicted by the theory of
     chemical kinetics. The square reflects that 2 molecules of A are
     needed to produce one molecule of B and is called the order of the
     reaction]
\end{enumerate}

\item (Regression) Suppose you are measuring a physical quantity $y$
which depends on another physical quantity $x$ via the formula
$y=mx$. You want to find $m$. Your measurements may have
experimental error.
\begin{enumerate}
\item If you only make one measurement $(x_1,y_1)$, what is your best
guess for the value of $m$?
\item If you make 2 measurements, $(x_1,y_1)$ and $(x_2,y_2)$,
the values of $m$ you get from
each individual measurement differ. How do you obtain a ``best guess''?
One approach is to consider the ``errors'' $y_1-mx_1$ and $y_2-mx_2$
and choose $m$ so as to minimize the sum of the squares of these errors
(the squaring makes big errors ``worse''). Do this and find
the best $m$ in terms of $x_1,x_2,y_1,y_2$. (This is called ``the principle
of least squares'').
\item Generalize the above to the case when you make $n$ measurements,
$(x_1,y_1),\dots,(x_n,y_n)$. (This whole idea is called ``finding the line
of best fit'').
\end{enumerate}
%
\item Let $0<r<1$ be a rational number. Consider the function
$f(x)=(1+rx)-x^r$.
\begin{enumerate}
\item Find the absolute minimum ($x$ and $y$ values) of $f(x)$ over the
interval $[1,\infty)$. (How can you be sure it's an absolute minimum?)
\item Use part (a) to prove $1\leq x^r \leq 1+rx$.
\item Use part (c) to prove that the limit, as $r\rightarrow 0^+$ through
rational numbers, of $x^r$ is $1$.
\end{enumerate}
%
\item Let $p(x)$ be a polynomial.
\begin{enumerate}
\item Prove that between any two roots of $p$ is a root of $p'$.
\item Prove that between any two successive roots of $p'$ is at most one
root of $p$.
\item For what degrees of polynomial must $p(x)$ have a point of
inflection? (i.e., for what $n$ is the following statement true: ``Any
polynomial of degree $n$ has a point of inflection.'')
\end{enumerate}

\item \dag My farm (ranch?) in Bahrain raises prize racing camels. I believe I
can sell $n$ camels at a price of $300000-3000n$ rials \emph{each}. The total cost of
raising $n$ camels is $600000+190000n$ rials. How many camels should I
raise to maximize my profit (=total revenue - cost)? [Warning: only
whole camels can be raised, or raced for that matter. Suggestion:
divide everything by 1000]. (8 points)
%

\item (A difficult integration)
You are kidnapped by a five-headed alien. The first head promises
that you will be set free when you evaluate $\D \int_0^1 \frac
{dx}{(x^2+1)^2}$.
The second head rather likes you, and so whispers in your ear to try the
substitution $\tan u=x$. The third head pities you, and so whispers in your
other ear that $\sin \pi/4 = \cos \pi/4$. The fourth head dislikes you and wants you
to go away as soon as possible, and so bellows at you to try dividing the
identity $\sin^2 x+\cos^2 x=1$ by something. The fifth head gives in to
the pressure of its peers to help you and reminds you of the double angle formula, $\cos 2y=2 \cos^2
y-1$.

Solve the integral.

\item (Smoothing and filters)
Suppose you have a function with discontinuities. One way of ``smoothing
it out'' to get a continuous function involves averaging it by integration
near each point. More precisely, given a function $f(x)$ and a number $a>0$,
define the $a$-smoothing
$$
S_af(x)=\frac1{2a}\int_{-a}^{a}f(x+t)\,dt.
$$
\begin{enumerate}
\item Let $H(x)$ be the ``step'' function which is 1 for $x\geq 0$ and
0 for $x<0$. Calculate and sketch $S_1H(x)$.
\item Let $\lfloor x\rfloor$ be the greatest integer $\leq x$. Sketch the
graph of $S_{0.1}\lfloor x\rfloor$.
\item Sketch the graph of $S_{1.1}\lfloor x\rfloor$.
\item What happens to the graph of $S_a\lfloor x\rfloor$ if $a$ is very
large?
\end{enumerate}
[In practice, the average used is a weighted average with more weight
close to the point in question and less as you get farther away. Such
filters are used in signal processing.]

\item (Sketching a special function)
The Fresnel sine-integral function is defined as $S(u)=\int_0^u\sin
\big(\frac \pi 2 s^2 \big)\,ds.$ This integral cannot be expressed in
terms of elementary functions.
\begin{enumerate}
\item What are the critical points of $S(u)$?
\item Estimate the values of $S(u)$ at the critical points in the
interval $[-2,2]$ by approximating the appropriate areas under the
curve below with triangles as marked.

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\end{picture}

\item Draw a sketch of $S(u)$ for $u$ in the interval $[-4,4]$.
\end{enumerate}
[This function and its cousin involving $\cos$ arise in optics
(diffraction) and in freeway design.]

\end{enumerate}

\end{document}




