# Homework 4 - Due Feb 21

You may work in groups, but the homework must be done individually.

Linearize \(y=x^3-5x^2+14x+1\) about \(x=3\).

In class we practiced linearizing simple equations. You can also linearize non-linear differential equations (see Example 2.27 in your textbook). Study Example 2.27 and then work problem 53 in chapter 2 of your textbook.

Sketch a control system representation of the following physiological system: When the blood glucose concentration becomes elevated, the excess glucose acts directly on the islets of Langerhans of the pancreas to increase their secretion of insulin. This increased insulin secretion causes the cells of the body to become more permeable to glucose. Glucose is then transported into the interior of the cells and the blood glucose level falls.

Blood glucose can also be regulated by the sympathetic nervous system. A low blood glucose level excites the sympathetic hypothalamic nuclei, which then send impulses through the sympathetic nervous system to cause epinephrine release by the adrenal medulla and norepinephrine release by both the adrenal medulla and the sympathetic nerve endings. These hormones, especially epinephrine, exert a direct effect on the liver cells to increase the rate of glucose production (glycogenolysis—the splitting of glycogen into glucose molecules). The glucose then diffuses into the blood to elevate the blood glucose concentration.

In class we discussed how the equilibrium of a physiological system is equal to the intersection point of two curves. The specific curves we looked at were alveolar ventilation \(\dot{V}_A\) and arterial carbon dioxide partial pressure \(P_{aCO_2}\). Assume that you have performed experiments and that you have determined the relationship between \(\dot{V}_A\) and \(P_{aCO_2}\). When \(\dot{V}_A\) is the independent variable and \(P_{aCO_2}\) is the dependent variable the equation is

\(P_{aCO_2}=\frac{4.15}{\dot{V}_A}\)

When \(P_{aCO_2}\) is the independent variable and \(\dot{V}_A\) is the dependent variable the equation is

\(\dot{V}_A=2.41P_{aCO_2}-3.82\)

Find the equilibrium values for \(\dot{V}_A\) and \(P_{aCO_2}\).

(a) Write the differential equations for the concentrations of each compartment in the system shown below. The variables of the system are \(C_A(t)\), \(C_B(t)\), and \(C_C(t)\). (b) Assume \(k_1=1\), \(k_2=2\), \(k_3=3\), \(k_4=4\), \(k_5=5\). Using Matlab, find the transfer function \(G(s)=C_A(s)/\dot{M}_{in}(s)\). Hint: after solving for the transfer function G symbolically, you can substitute numbers for the symbols with the command

`subs(G,[k1,k2,k3,k4,k5],[1,2,3,4,5])`

. Note that this command may be “buggy” and give you an unsimplified answer (i.e., be sure to simplify your answer by canceling common factors in the numerator and denominator before writing down your final answer).Write the differential equations for each compartments in the system shown below. The variables of the system are \(C_A(t)\), \(C_B(t)\), and \(C_C(t)\).

Write the differential equations for the following set of chemical reactions.