Degrees: Pharmacy and Biotechnology
Date: Jan 19, 2018

## Question 1

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

## Question 2

An organism metabolizes (eliminates) alcohol at a rate of three times the amount of alcohol present in the organism. 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?

## 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

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 $p$.

## 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:

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.