Pharmacy exam 2017-11-06

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 ±0.01mm, what will be the error in the measurement of the volume?

Note: The volume of a sphere of radius r is equal to 43πr3.

Rate of growth of the volume: 250π mm³/s.
Error in the volume: π mm³.

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?

Let t the time and w(t) the weight of the baby at time t.

  1. Differential equation: w=kw
    Particular solution: w(t)=7.6t+10.89.
    w(12)=10.1 kg.
  2. At 7 months.
  3. No, because the function is always increasing.

Question 3

The function f(x,y)=yex212y2 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 x0 and y0.
  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).

  1. f has a local maximum at (0,1) and the maximum value is z=f(0,1)=1/e.
  2. Directional derivative of f at (1,0) along the direction of v=(2,1): fv(1,0)=1e5.
  3. Pf,(1,0)2(x,y)=2xy+3ye.

Question 4

Given h(t)=(tcos(t),cos(t),ln(t2+1)), compute the tangent line and normal plane to the trajectory determined by h at the point (0,1,0).

  1. Tangent line: (t,1,0).
  2. Normal plane: x=0.
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