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: \[ \log(\sqrt{x^2+y^2}) = y. \]

  1. Compute the equations of the tangent and normal lines to the graph of $y$ as a function of $x$ at the point $(1,0)$.
  2. 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 $(1,1,1)$.

  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?
  2. 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.

  1. 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 $y$.
  2. 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 $f(x,y)=e^y(y^2-x^2)$.