1. The equilibrium of the graph seems to not changing in between t=7 and t=8
and population is about 168 to 169 billion individuals. But in the table, the
real t where the stock has no change is when t=65 and population stays at
171.4 billion individuals.

1.
[ ]
1
8
1
8
0
0
1
8
1
8
0
0
1
10
[][]
x 0
y +b
z 0
2. The derivative of x(t), y(t) and z(t) means the instantaneous rate of change of morphine in the
ventrogluteal muscle, the blood, and the brain compartments of a patient with respect to time
(in hour

1. We can look at the graph and figure out the local maximum and local minimum. Or we can use
the first derivative test. First find first derivative of the equation and set it to zero. Then try to
plug in the equation and determine whether its a local max

Homework
1. (i) D =
E=
F=
G=
(
(
(
(
)
)
3 4 5
3 6 1
5 6 6
3 2 3
1 0 3
3 2 0
)
1
17 6
2
5
7
19 10 21
2 2 16
2
14 13
16 3 11
)
(ii) Determinant of A = -110
Determinant of B = 33
Determinant of C = 0
A and B matrix have an inverse because they are all squar

1.
k
aP
)
k
f ( P )=(1+
lim
(
k = 1+
aP
k
k
)
lim
(
ln 1+
k =e
e
aP
k
k
)
lim kln (1+
k
ln (1 +
lim
aP
)
k
aP
)
k
1
k
k
e
ln (1 +
lim
aP
)
k
1
k
k
e
lim
ap
k =e
Thevaluesof,aandc
3. yes, we have to put in specific values which means locally and

1. (i) D =
E=
F=
G=
(
(
(
(
)
)
3 4 5
3 6 1
5 6 6
3 2 3
1 0 3
3 2 0
)
1
17 6
2
5
7
19 10 21
2 2 16
2
14 13
16 3 11
)
(ii) Determinant of A = -110
Determinant of B = 33
Determinant of C = 0
A and B matrix have an inverse because they are all square matrix,

Homework
1.
k
aP
)
k
f ( P )=(1+
lim
(
k = 1+
aP
k
k
)
lim
(
ln 1+
k =e
e
aP
k
k
)
lim kln (1+
k
ln (1 +
lim
aP
)
k
aP
)
k
1
k
k
e
ln (1 +
lim
aP
)
k
1
k
k
e
lim
ap
k =e
Thevaluesof,aandc
Homework
3. yes, we have to put in specific values which means