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(b). A mathematical biologist created a model to administer medicine reaction (measured in change of blood pressure or temperature) with the model given by R=m² (C /2- m/3). where c is a positive constant and m is the amount of medicine absorbed into the blood. The sensitivity to the medication is defined to be the rate of change of reaction R with respect to the amount of medicine m absorbed in the blood. (1) Find the sensitivity, R'.

(ii) Find the amount of medicine that is being absorbed into the blood when the reaction is maximum.

(iii) Find the instantaneous rate of change of sensitivity with respect to the amount of medicine absorbed in the blood.

(iv) Find the amount of medicine that is being absorbed into the blood when the sensitivity is maximum.

[Verify your answer by MATHEMATICA and attach the printout of the commands and output

B A Mathematical Biologist Created A Model To Administer Medicine Reaction Measured In Change Of Blood Pressure Or Temperature With The Model Given By Rm C 2 M3 class=

Sagot :

LammettHashidn

(i) If R(m) = m² (c/2 - m/3) = cm²/2 - m³/3, then differentiating with respect to m gives the sensitivity function,

R'(m) = cm - m²

In Mathematica, define a function R by

R[m_] := m^2*(c/2 - m/3)

Then you can get the derivative with either R'[m] or D[R[m], m].

(ii) Find the critical points of R, i.e. the points for which R' is zero or undefined. Since R' is a polynomial, it is defined everywhere, so we only care about the first case. We have

R'(m) = cm - m^2 = m (c - m) = 0

with solutions m = 0 or m = c.

Compute the second derivative of R :

R''(m) = c - 2m

Check the sign of R'' at each critical point:

  • R''(0) = c - 0 = c > 0, which indicates a minimum at m = 0
  • R''(c) = c - 2c = -c < 0, which indicates a maximum at m = c

At the second critical point, we have a maximum value of

R(c) = c³/2 - c³/3 = c³/6

In M, we get the same result via

Maximize[R[m], m]

which should return a list like

{c^3/6, {m -> c}}

where the first element is the maximum, and the second element is a list with a rule pointing the target variable m to the critical point.

(iii) We've done this in part (ii) already while carrying out the second derivative test above,

R''(m) = c - 2m

In M, this is accomplished by R''[m] or D[R[m], {m, 2}].

(iv) We carry out the derivative test like in part (ii), but this time for R'. We have critical points when

R''(m) = c - 2m = 0

which occurs for m = c/2. The second derivative of the sensitivity function is the third derivative of R,

R'''(m) = -2

which is negative for all values of m, which means R' is in fact maximized at m = c/2. At this point of maximum sensitivity, we have

R(c/2) = c(c/2)²/2 - (c/2)³/3 = c³/8 - c³/24 = c³/12

In M, run

Maximize[R'[m], m]

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