Degrees: Pharmacy and Biotechnology

Date: Nov 7, 2016

## Question 1

The amount $x$ and $y$ in mg of two compounds in a certain chemical reaction are related by the following equation: \[ \log(\sqrt{x^2+y^2}) = y. \]

- Compute the equations of the tangent and normal lines to the graph of $y$ as a function of $x$ at the point $(1,0)$.
- Compute the approximate change of the amount $y$ if $x$ changes by 2mg, from the same point $(1,0)$.

## Question 2

The temperature at a point $(x,y,z)$ in three-dimensional space is given by the following function: \[ T(x,y,z)= \frac{e^{xy}}{z} \]

Suppose we are position at $(1,1,1)$.

- In which direction will the temperature decrease the fastest? What will be the rate of that decrease? What is the meaning of your result?
- Compute the directional derivative in the direction where $y$ increases twice as much as $x$, and $z$ increases half of $x$. What is the meaning of your result?

## Question 3

Allometric growth refers to relationships between sizes of different parts of an organism. Suppose $x(t)$ and $y(t)$ are the size of two organs in an organism of age $t$; then the allometric relationship is given by the equation: \[ \frac{1}{y}\frac{dy}{dt} = k \frac{1}{x}\frac{dx}{dt}, \] where $k$ is a positive constant.

- Compute the differential equation that explains $y$ as a function of $x$ (that is, take $x$ as the independent variable and $y$ as the dependent one). Solve the equation for $y$.
- Assume $y$ denotes the mass of a cell, and $x$ its volume, with $k=0.0794$, compute $y$ as a function of $x$ if $x=1000\ \mu$m$^3$ at the age at which $y$ is equal to 1 ng.

## Question 4

Find the local extrema and saddle points of the function $f(x,y)=e^y(y^2-x^2)$.