Pharmacy exam 2018-01-19

Degrees: Pharmacy and Biotechnology
Date: Jan 19, 2018

Question 1

Find an equation of the tangent plane to the surface $S: e^xy-zy^2+\frac{x^4}{z}=-1$ at the point $P=(0,1,2)$.
Find the tangent line to the curve obtained by the intersection of $S$ and the plane $z=2$ at the given point $P$.

Tangent plane: $x-3y-z+5=0$.
Tangent line: $(3t,1+t)$ or $y=\frac{x}{3}+1$.

Question 2

An organism metabolizes (eliminates) alcohol at a rate of three times the amount of alcohol present in the organism per hour. If the organism does not have alcohol at initial time and it starts to get alcohol at a constant rate of 12 cl per hour; how much alcohol will be in the organism after 5 hours? What will be the maximum amount of alcohol in the organism? When will that maximum amount be achieved?

Solution

Let $y$ be the alcohol in the organism and $t$ the time.
Differential equation: $y’=12-3y$.
Solution: $y(t)=4-4e^{-3t}$.
$y(5)=3.99$ cl.
The maximum amount of alcohol will be 4 cl and it will be achieved at $t=\infty$.

Question 3

Three alleles (alternative versions of a gene) $A$, $B$ and $O$ determine the four blood types $A$ ($AA$ or $AO$), $B$ ($BB$ or $BO$), $O$ ($OO$) and $AB$. The Hardy-Weinberg Law states that the proportion of individuals in a population who carry two different alleles is

$$ p(x,y,z)=2xy+2xz+2yz $$

where $x$, $y$ and $z$ are the proportions of $A$, $B$ and $O$ in the population. Use the fact that $x+y+z=1$ to compute the maximum value of $p$.

Solution

There is a local maximum at $(\frac{1}{3},\frac{1}{3})$ and $f(\frac{1}{3},\frac{1}{3})=\frac{2}{3}$.

Question 4

Three substances interact in a chemical process in quantities $x$, $y$ and $z$. At equilibrium, the three quantities are related by the following equation:

$$ \ln z - \frac{x^2y}{z}=-1 $$

Assume $z$ is an implicit function of $x$ and $y$; compute the variation of $z$ when $x=y=z=1$ and $y$ decreases at the same rate as $x$ increases.

Solution

Directional derivative of $z$ in $(1,1,1)$ along $\mathbf{v}=(1,-1)$: $z’_\mathbf{v}(1,1,1)=\frac{1}{2\sqrt{2}}$.

Exam

Last updated on Sep 15, 2020