[MATH2605] Exam 1
Feb. 12. 2007
Problem 1
Consider a vector x=
x
y
z
. Provide an algorithm to compute a vector orthogonal to x. The subroutine should work
for all the following vectors.
x=
0
1
2
,x=
2
0
1
,x=
1
2
0
,x=
2
0
0
,x=
0
1
0
,x=
0
0
1
(1)
Problem 2
Let x0=
1
−1
2
,x1=
3
1
3
,p=
2
3
1
.
Compute the minimum distance between pand the line passing x0and x1.
Compute the minimum distance point q, i.e., compute qsuch that qis on the line and ||p−q|| is minimum.
Problem 3
Compute 3−2
1 0 100.
Problem 4
Let A=VDV−1, where V=
1 2 −1 2
2 1 2 3
1−2−1 1
2 1 1 3
, and let D=
1 1 0 0
0 1 0 0
0 0 −1 0
0 0 0 2
.
Compute the trace and determinant of A.
Problem 5
Let f(x,y) = x4+y3+2xy2.
Compute ∇fand Hf.
Compute the quadratic approximation to f(x,y)at (x,y)=(1,1).
Compute the maximum and minimum curvature at (x,y)=(1,1).
Problem 6
Let F(x,y) = 3x3sin(y2)
cos(xy). Compute the Jacobian JF.
Next exam will be on Mar. 7.
1