SAM P L E EX A M
Final Examination GE 85-111 Page 1 of 10
(10 Marks) 1. (a) By the method of 'oints determine the bar forces in bars AB, AH,
BH, HG, BG and BC. (Note that the reactions are given.)
5 m
425 RN 400 kN 200 kN 100 RN 275 k
Examples:
1. Find all critical points of f (x, y) = 2x2 + y 2 + 8x
6y + 20.
solution: We nd the partial derivatives and set them equal to 0
fx = 4x + 8 = 0
fy = 2y
These imply that x =
6=0
2 and y = 3. So f has one critical point at ( 2, 3).
2. Find all c
This is a disk of radius 4 centered at the origin.
For another example, let
f (x, y) =
(x
4
2)(y
3)
This function is dened unless x = 2 or y = 3. So the domain will be entire xy-plane except
for the following two lines
Level Curves
Let z = f (x, y) be a f
0
0.1
Precalculus Review
Real Line and Order
When discussing order on the real number line, we use the following symbols:
<
>
less than
less than or equal to
greater than
greater than or equal to
We use the following notation for intervals:
x 2 (a, b)
7
Functions of Several Variables
Suppose you run a company which produces two types of television. Let x1 be the quantity
of type 1 and x2 be the quantity of type 2. Then the usual functions in which we are
interested (prot, revenue, and cost) will depend
UNIVERSITY OF WINDSOR
Civil and Environmental Engineering
GE 85-111
E . . H l . I
Mid-Term Test
October 22, 1997
Students Name: ___________
Students ID. No.: ___________________
Group No.: ___________________
Your Instructors Name: ___________
NOTE:
1
Examples: Find the partial derivatives of the following functions.
1. f (x, y) =
p
x2
y 2 = (x2
y 2 )1/2
solution: For each of the partial derivatives we use the general power rule:
1
fx = (x2
2
y2)
1/2
(2x),
1
fy = (x2
2
y2)
1/2
( 2y)
2. f (x, y) = 4x2 y