Degrees: Pharmacy and Biotechnology
Date: Nov 6, 2017

## Question 1

Adenoma is a benign tumor, which grows usually in spherical shape. Suppose the rate of growth of the radius of a certain adenoma is equal to half the size of the radius per second; compute the rate of growth of the volume of the tumor when the radius is 5mm.

If the measurement of the radius has a possible error of $\pm 0.01$mm, what will be the error in the measurement of the volume?

Note: The volume of a sphere of radius $r$ is equal to $\frac{4}{3}\pi r^3$.

## Question 2

The weight of a baby during the first few months of life grows at a rate proportional to the reciprocal of the weight. Suppose a baby’s weight was 3.3 kg at birth, and 4.3 kg a month later.

1. What will be the weight of the baby one year after birth?
2. When will the weight be equal to 8 kg?
3. Is this model of the weight good to determine the weight of a person during his whole life?

## Question 3

The function $f(x,y)=ye^{-x^2-\frac{1}{2}y^2}$ gives the quantity $z=f(x,y)$ of a substance during a chemical process, depending on the quantities $x$ and $y$ of two other substances.

1. Compute the maximum value of $z$ assuming that $x\geq 0$ and $y\geq 0$.
2. What will be the variation of $z$ at $x=1$ and $y=0$ when $x$ increases twice as much as $y$?
3. Compute the second degree Taylor polynomial of $f$ at the point $(1,0)$.

## Question 4

Given $h(t)=(t\cos(t), \cos(t), \ln(t^2+1)),$ compute the tangent line and normal plane to the trajectory determined by $h$ at the point $(0,1,0)$.