CARROLL COLLEGE
Department of Mathematics, Engineering, and Computer Science
Name: K «E: *1 _
MA 189 Mathematics for Engineers II May 2, 2011
Final Exam John L. Scharf
There are 10 problem worth 10 points each, plus a 15 point bonus problem. Show all o
CARROLL COLLEGE
Department of Mathematics, Engineering, and Computer Science
Name: 63:;
45
MA 189 Mathematics for Engineers II May 4, 2010
Final Exam John L. Scharf
Show all of your work for every problem. If you get a result from the calculator, ex
Cody Nosigner and Nick Lumetta
Lab 5 MA233
Contents
Quadratic Examples
Critical Points
4 dimensional optimization
Quadratic Examples
1. the rst has a saddle point because you can see as the change in y is
concave up the change in x is concave down
2. T
Cody Nosinger and Nick Lumetta
Lab 4 MA233
Contents
Review Problem
Linear Approximation $ f(x,y) = (1/(x2+y2)+.5 $
Linear Approximation of $ g(x,y) = cos(x2)+y2).5) $
Linear Approximation of $ g(x,y) = cos(x2)+y2).5) $ at dierent
points for P, Q, and R
C
Cody Nosinger and Nick Lumetta
Lab 3 MA233
0.1
0.2
Contents
Exploration of f (x, y ) = x2 y 2 + x y
Geometry of the Gradient Vector
Gradient Field Plots
MATLAB Questions
Exploration of f (x, y ) = x2 y 2 + x y
1. The rst lines establish the symbolic varia
Double and Triple Integrals
Rectangular Coordinates
Regions dened by functions in rectangular coordinate in the xy -plane.
The dierential region is given by dA = dxdy .
Polar/Cylindrical Coordinates
Regions that are circular in the xy -plane.
The di
Double and Triple Integrals
Rectangular Coordinates
Regions dened by functions in rectangular coordinate in the xy -plane.
The dierential region is given by dA = dxdy .
Polar/Cylindrical Coordinates
Regions that are circular in the xy -plane.
The di