Calculus Final Exam (1) Find the Jacobi matrix for
6%
f(x,y) =
·exsin y e−ycos x
¸
(2) Find an equation of the tangent plane to the graph of the equation z=x2+4y2at the point(−2,1,8).
6%
(3) Find all the local maxima, local minima, and the saddle points of f(x,y) =4x2−2y3+4y2+4xy.
10%
(4) Evaluate the following integrals 20%
(a) Z∞
−∞e−x2dx (b)
Z2
0 Z 1
y/2
sin x x dx d y (5) Find the solution of the equation 10%
X0(t) =
µ1 −5 4 −3
¶ X satisfying the initial value condition X(0) =
µ1 1
¶
.Also, find the limit of X(t)as t→∞.
(6) Find the solution of the equation 10%
X0(t) =
µ 1 9
−1 −5
¶ X satisfying the initial value condition X(0) =
µ 1
−1
¶ .
(7) Let X be a continuous random variable with the density function 10%
f(x) = (
2e−2x for x≥0 0 for x<0 Find E(X)and var(X).
(8) Given the three points(0,2),(1,0), and(2,1), use the method of least squares to find the least square 8%
line.
(9) Find and analyze all equilibria of the following system 20%
dN1
dt = N1(1−N1−N2) dN2
dt = N2(0.5−0.25N2−0.75N1)
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