Math 1502
Final Tuesday April 27
Tom Morley

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

Problem 1

a)   Find the Taylor series WITH general term for ∫_0^x (sint^(1/2))/t ... in various ways . You may assume that x, , >, , 0.}], ScriptLevel -> 0.], , }] Null

b) Find a series to compute e^(1/5). Using one of the error estimates, find out how many terms you need
to get the answer to within .005.

Problem 2 (15 points)

1) Find all solutions to the equation A x = b, where,

A =( 2 3 4 5 1 ) 0 3 2 1 1 2 6 6 6 2 , b = ( 3 ) 1 4.


2)   Find a basis for the image of A.


3) Find a basis for the kernal of A.



Problem #3  

Let  A = ( 1 -2 ) -2 1. Find A^102x, where x = ( 3 ) 5.


Part II: A is as above. Let x = x(t).  Solve x'(t)  = Ax(t), where x(0) =  ( 3 ) 5

Problem 4 (5 points)

Ricky Dale Crain  would like you to solve the least squares prblem Min || A x - b || , where
A = QR,    Q = 1/2^(1/2)   ( 1 0 -1 ) -1 0 1 0 -1 0 0 +1 0 , R = ( Sqrt[2] Sqrt[2] Sqrt[2] ) 0 2 Sqrt[2] 4 Sqrt[2] 0 0 4 Sqrt[2]. Work directly with Q and R.  Here  b =( 1 ) 2 3 4

Problem 5 (10  points)

Find the matrices of:

a) Rotation (in the plane) by 60 degrees counterclockwise
b) Reflection about y axis
c) Reflection about the line x=y
Find the matrix of the combined opertion gotten by doing a) b) c) in that order

[Graphics:HTMLFiles/index_14.gif]

⁃Graphics⁃

Draw (together with the intermediate steps) , that happens to the House:

Created by Mathematica  (May 2, 2005)