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\topmatter
\title Oblig 2 for MA 001 \endtitle
\rightheadtext{John Rognes}
%\author John Rognes \endauthor
%\date October 2000 \enddate
\address Department of Mathematics, University of Oslo \endaddress
\endtopmatter

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\document

Each student registered for the fall 2000 exam in MA 001 must submit
acceptable solutions to two obligatory problem sets.  This is the second
of those.  Each student prepares and hands in an individual solution to
his or her ``group'' teacher, who then corrects and grades the
solutions.  Those qualifying for a passing grade (4.0 or better) are
considered accepted.  Remember to write your full name and the
number of your MA 001 colloquium group on the solution you hand in.

The absolute deadline for handing in solutions to both obligatory problem
sets is Friday November 10th 2000.

Give a mathematical justification for all answers.  Give exact answers
when possible.

\head Problem 1 \endhead

Student A and student B are both going to drink a cup of coffee.
The coffee is served black, with volume
$V_k = 150$ ml and temperature $T_k = 68 \degC$.
Both students want a dash of milk in the coffee, with volume $V_m =
10$ ml.  The milk is stored cold, and has temperature $T_m = 4 \degC$.
Student A pours milk in her coffee, but then the mobile phones 
of both students ring, and student B does not pour milk in his cup.

(a)
When the milk is poured in the coffee, the mixture gets temperature
$$
T_b = {V_k T_k + V_m T_m \over V_k + V_m} \,.
$$
Find the mixed temperature in student A's cup of coffee with milk.

(b)
As the conversations drag out, the coffee temperature (with or
without milk) decreases exponentially towards the room temperature,
which is constant at $T_r =
20 \degC$.  After $t$ minutes the temperature is given by the expression
$$
T(t) = T_r + (T_0 - T_r) \left(1\over2\right)^{t/h} \,.
$$
Give a precise interpretation of the constants $T_0$ and $h$.

(c)
After 10 minutes ($t = 10$) the temperature in student A's coffee (with
milk) has fallen to $42 \degC$.  Find the constant $h$.  What is the
temperature in student A's coffee after another 10 minutes ($t = 20$) ?

(d)
After
20 minutes student B is finished with the phone call.  What is then
the temperature in student B's coffee (without milk) ?

(e)
Student B then pours milk in his coffee.  Find the mixed temperature
for the coffee with milk added after 20 minutes.  Which cup of coffee
with milk is warmest after 20 minutes ?

\head Problem 2 \endhead

(a)
Let $a$, $b$, $k$ and $\ell$ be positive constants.  Show that
the function
$$
F(t) = -(a/k) t e^{-kt} - (a/k^2) e^{-kt} - (b/\ell) e^{-\ell t}
$$
is an anti-derivative to the function
$$
f(t) = a t e^{-kt} + b e^{-\ell t} \,.
$$

After a rainfall, water runs into a water reservoir.  After $t\ge0$ hours
the incoming flow rate is $f(t) \ \text{m}^3/\text{h}$, where $f(t)$
is given as in (a) and $a = 2.0 \cdot 10^5$, $b = 2.0 \cdot 10^4$, $k =
1.0$ and $\ell = 0.1$.

(b)
Find $f(0)$ and $\lim_{t\to\infty} f(t)$.

(c)
Let $V(t) \ \text{m}^3$ be the total water volume that has run into
the reservoir from time $0$ to time $t$.  Then $V'(t) = f(t)$.  Find a
formula for $V(t)$ for $t\ge0$.

(d)
Find $V(0)$ and $\lim_{t\to\infty} V(t)$.

(e)
Water is also let out of the reservoir at a constant outgoing flow
rate $c = 2.0 \cdot 10^4 \ \text{m}^3/\text{h}$
for $t\ge0$, until the reservoir is empty.  Show that the reservoir is
emptied within $t=20$ hours.

\head Problem 3 \endhead

A wave in the $xy$-plane is at time $t$ given as the graph of the
function
$$
f_t(x) = 4 \cos x + 3 \cos(x-t) \,.
$$
This is the sum of two waves in the $xy$-plane: one at rest with amplitude
4, and one with amplitude 3 moving to the right as $t$ grows.

(a)
Show that $f_t(x) = (4 + 3\cos t) \cos x + (3 \sin t) \sin x = C \cos(x -
x_0)$ where $C = \sqrt{25 + 24 \cos t}$ and $\tan x_0 = 3 \sin t/(4 + 3
\cos t)$.


(b)
Let $g(t) = 3 \sin t/(4 + 3 \cos t)$ for $0 \le t \le \pi$.  Find $g'(t)$
and decide where $g$ is increasing and where $g$ is decreasing.

(c)
Find the maximum point and the maximal value for $g(t)$.  Give exact
answers.

(d)
At time $t$ the top of the wave (wavecrest) closest to the $y$-axis
has $xy$-coordinates $(x_0, C)$.  Find the $xy$-coordinates of the
wavecrest when $g(t)$ has its maximal value.

\head Problem 4 \endhead

(a)
The equation $y = c x^r$ is valid for $(x, y) = (4, 5)$ and $(x, y) =
(25, 2)$.  Find $c$ and $r$.  Is the graph of $y$ as a function of $x$
a straight line in an orthogonal linear coordinate system, in a singly
logarithmic coordinate system, or in a doubly logarithmic coordinate
system ?

(b)
Use L'H{\^o}pital's rule (possibly several times) to find
$$
\lim_{h\to0} {1 - \cos 2h \over h^2} \,.
$$

(c)
Find the definite integral
$$
\int_{-\pi/2}^{\pi/2} \sin^4 \! x \cos x \, dx
$$
by using the substitution $u = g(x) = \sin x$.

\enddocument
