Sample Final Exam Math 1502  - Tom Morley

Open Book and Notes.  Carefully explain your proceedures and answers. Calculators allowed, but answers mush be exact.

Problem 1 (15 points)


Find a series for ((∫ ) _0)^x     t^2sin(t^(1/2))/t^(1/2) t.  Hint: start with a series for sin(t) and manipulate.

Math 1502

Problem 2:  (10 points)

a) Set up the equations to   find the best fit by a linear function  to the data:

x y
1 -1
2 3
3 1
4 2
5 2
6 4


and
b) Set up the equations to  find the best fit to same data by a quadratic ax^2 + bx + c.

Math 1502

Problem 3  (20  points)

Find a basic for ker(A), ker(A^t) , the image of A, and the image of A^t,
where A is:  ( 1 4 3 ) 2 -1 -3 -1 -1 0 -2 3 5.

Math 1502

Problem 4  (10  points)



Let A be the matrix ( 2 1 ) 1 2. Find the eigenvalues and eigenvectors of A. Use this to find a formula for A^125x, where x = ( 3 ) 4.  The problem asks for a formula involving the eigenvalues and eigenvectors of A, a computed numerical answer is not sufficient.

Problem 5  (10  points)

Let L be the line though the points {1,2,3} and {-1,0,1}. Let P be the plane determined by the equation 2x -y + z + 4.  
a) Find the intersection of the line L and the plane P
b) Find the closest point on the plane P to the point {1,0,2}.
c) Find the closest point on the line L to the point {1,0,2}.

Created by Mathematica  (December 5, 2005)