Degrees: Pharmacy and Biotechnology

Date: Jan 10, 2017

## Question 1

The rate of growth of certain bacteria population is the square root of the number of bacteria in the population. How much will the population have increased after 1 hour from the beginning of the growth? How long will it take until the population is four times the population at the beginning?

## Question 2

The temperature of a chemical process depends on the amounts $x$ and $y$ of two substances, according to the function $T(x,y)=4x^3+y^3-3xy$. Determine the local extrema and saddle points of the temperature function (recall that the amounts $x$ and $y$ cannot be negative).

## Question 3

An ecological model explains the number of individuals in a population through the function \[f(x,y)=\dfrac{e^t}{x},\] where $t$ is the time and $x$ the number of predators in the area. Give an approximation of the number of individuals at $t=0.1$ and $x=0.9$ using the second order Taylor polynomial of function at point $(1,0)$.

## Question 4

The position of a moving object in space is given by the function $f(t)=(e^{t/2}, \sin^2(t), \sqrt[3]{1-t})$.

- Compute the velocity and acceleration vectors at time $t=0$.

Remark: velocity is the variation of space with respect to time, and acceleration is the variation of velocity with respect to time. - Compute an equation of the plane normal to the trajectory at time $t=0$.