Calculus Midterm Exam (1) Solve the following differential equations.
24%
(a) dy
dx− xy = x.
(b) (x + y)dx + (x − y)dy = 0.
(c) dN
dt = 2N (N − 10) µ
1 − N 100
¶ .
(2) Suppose that N(t) denotes the size of a population at time t. The population evolves according to the 12%
logistic equation but, in addition, predation reduces the size of the population so that the rate of change is given by
dN dt = N
µ 1 − N
50
¶
− 9N 5 + N. Find all equilibria, and discuss their stability.
(3) Find the length of the curve 8%
y = − ln(cos x) from x = 1 to x =π3.
(4) Suppose that a the amount of time T that a butterfly spends feeding on the nectar of a flower is a random 10%
variable whose density is given by
f (x) = (
3e−3x for x > 0
0 otherwise
Show that f (x) is a density function and compute the mean of T . (5) (a) Find the eigenvalues and eigenvectors of the matrix
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A =
·−1 −3
2 4
¸
(b) Find A30
· 4
−2
¸
without using a calculator.
(6) Let 8%
(AB)−1=
·−1 3 0 2
¸
and
B =
·0 −2
1 3
¸
Find A.
(7) Given the Leslie matrix 12%
L =
·1 3 0.5 0.5
¸
find the growth rate of the population, and determine its stable age distribution.
(8) Let A be a matrix and v1 and v2be two eigenvectors of A, with eigenvalues λ1andλ2, respectively.
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Assume thatλ16=λ2, prove that v1and v2are linearly independent, i.e. there is no constant c satisfying that v2= cv1.
