ME 201 ASSIGNMENT #5 Minima and Maxima of Multidimensional Functions 1. Find all critical points for the following functions and classify each as yielding a relative maximum, minimum, saddle (inflection) point or none of these a. b.
f ( x, y ) = 3 x y " x
ME 201 ASSIGNMENT #11 Surface Integrals, Divergence and Stokes Theorems 1. Evaluate the surface integrals of the following scalar functions: a.
" ( x
S
2
+ y 2 ) z dS where S is that part of z = x + y cut out by x = 0, y = 0, x + y = 1
ANSWER 2 3 15
!
b.
ME 201 ASSIGNMENT #10 Line Integrals and Greens Theorem 1. Evaluate the line integrals of the following scalar functions: a.
CCW
" (x
2
2
+ y 2 ) ds once around the square C in the xy-plane with vertices (-1,-1), (-1,1),
ANSWER 32 3
(1,-1) and (1,1) b.
!
ME 201 ASSIGNMENT #7 Romberg and Double Integration 1. Use Romberg integration starting with N = 2 to evaluate the following definite integrals. Perform a first level extrapolation to eliminate the 1 N 2 terms and a second level extrapolation to eliminate
ME 201 ASSIGNMENT #8 Triple Integrals and Volumes 1. Evaluate the following triple integrals over the given region: a. b.
" x y dV
G G
where G is bounded by z = 1" x 2 " y 2 , z = 0 ANSWER 0 where G is bounded by x = 0 , x = 1, z = 0 , y + z = 2 , y = z
!
ME 201 ASSIGNMENT #9 Centroids and Vector Fields 1. Find the centroid of the 2D region bounded by the curves: a.
y = 8 " 2x 2 , y + x 2 = 4
ANSWER (0, 24 5) ANSWER (177 85, 9 17)
b. x = 4 y " 4 y 2 , x = y + 3 , y = 1, y = 0
2. Find the second moment of i
ME 201 ASSIGNMENT #1
Vectors and Vector Operations
r
r
r
1. If u = (1, 3, 6) , v = ("2, 0, 4 ) and w = ( 4, 3, "2) , solve the following vector operations, expressing
your answers in triple notation.
r
r
a. 2 w + 3v
ANSWER (2, 6, 8)
! rr
!
r!
b. v v " 2 v
ME 201 ASSIGNMENT #6 Lagrange Multipliers, Least Squares and Numerical Integration 1. Use Lagrange multipliers to find minimum and maximum values of the function subject to the constraint(s) given: a. b.
f ( x, y,z) = 5x " 2y + 3z + 4 subject to x 2 + 2y
ME 201 ASSIGNMENT #2 Vector Calculus 1. Find the area of the following a. Triangle with vertices (1,2,3) , ( 3,5,10) and ("3," 4,"11) b. Parallelogram with vertices (1," 2,4 ) , ( 3,5,7) , ( 4,6,8) and (2,"1,5) 2. Find the shortest distance between the fo
ME 201 ASSIGNMENT #3 Derivatives and Integrals of Vectors 1. Find parametric equations, position vector and plot the following curves: a.
x 2 + y 2 = 2 , z = 4 directed so that y increases in the first octant
b. z = x 2 + y 2 , y = x directed so that y in
ME 201 ASSIGNMENT #4 Partial Derivatives, Directional Derivatives and Tangents 1. Solve the following partial derivatives: a. b.
"3 f 2x + 3x 3 y 4 if f ( x, y ) = 3 "y y
"2 f if f ( x, y,z) = x y z e x +y +z " y"z
! !
!
c.
"2 f if f ( x, y,z) = ln x 2 +
ME 201 ASSIGNMENT #1 Vectors and Vector Operations
r r r 1. If u = (1,3,6) , v = ("2,0,4) and w = (4,3,"2) , solve the following vector operations, expressing your answers in triple notation. r r a. 2 w + 3v r ! rr ! b. v v " 2 v w ! r r v "w ! c. r r v+w
Assignment #3 Solutions
1 a)
b)
c)
1
Cylinder, radius = 1, offset by 1
in positive y-direction
d)
2
2 a)
3
b)
4
5
3 a)
b)
6
4)
7
5)
6)
8
7)
8)
9
9 a)
b)
c)
10
d)
10 a)
11
b)
11 a)
12
b)
c)
12 a)
b)
c)
d)
13