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6 Problems on Calculus III for Computer Science - Practice Exam 1 | MATH 2605, Exams of Mathematics

Material Type: Exam; Class: Calc III for Comput Sci; Subject: Mathematics; University: Georgia Institute of Technology-Main Campus; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

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[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 ||pq|| is minimum.
Problem 3
Compute 32
1 0 100.
Problem 4
Let A=VDV1, where V=
1 2 1 2
2 1 2 3
121 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

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[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 =

 (^) , x =

 (^) , x =

 (^) , x =

 (^) , x =

 (^) , x =

Problem 2

Let x 0 =

, x 1 =

, p =

Compute the minimum distance between p and the line passing x 0 and x 1. Compute the minimum distance point q , i.e., compute q such that q is on the line and || pq || is minimum.

Problem 3

Compute

Problem 4

Let A = VDV −^1 , where V =

, and let^ D^ =

Compute the trace and determinant of A.

Problem 5

Let f ( x , y ) = x^4 + y^3 + 2 xy^2. Compute ∇ f and 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 ) =

3 x^3 sin( y^2 ) cos( xy )

. Compute the Jacobian J F.

Next exam will be on Mar. 7.