Math 255
Winter 2012
Review Sheet for Second Midterm Exam
Solutions
1. Let r(t) = cos t, sin t, t . Find T( ), N( ), and B( ).
.
r ( t) =
sin t, cos t, 1 ,
1
r
= sin t, cos t, 1 ,
T( t ) =

r
2
T
N( t ) =
= cos t, sin t, 0 ,
T 
1
B(t) = T N = sin t,
Winter 2012
Math 255
Problem Set 2
Due on Tue, Jan 17
Section 11.4, Pages 719720:
14) Sketch the curve r = 2 + cos 2 and nd the area it encloses.
42) Find all the points of intersection of r2 = sin 2 and r2 = cos 2.
48) Find the exact length of the polar
Winter 2012
Math 255
Problem Set 1
Due on Tue, Jan 10
Section 11.1, Pages 692695:
2) Sketch the curve x = 2 cos t, y = t cos t, t [0, 2 ], by using parametric equations to plot points. Indicate with an arrow the direction
in which the curve is traced as t
MATH 255 SECTION 003
FINAL
April 19, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
One side of a US letter size paper (8.5" 11" ) with notes is OK.
NO CALCULATOR, BOOKS, or OTHER NOTES.
Problem Points Score
MATH 255 SECTION 003
MIDTERM 2
March 21, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
One side of a US letter size paper (8.5" 11" ) with notes is OK.
NO CALCULATOR, BOOKS, or OTHER NOTES.
Problem Points Sc
MATH 255 SECTION 003
MIDTERM 2 (alternative)
March 22, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
One side of a US letter size paper (8.5" 11" ) with notes is OK.
NO CALCULATOR, BOOKS, or OTHER NOTES.
Pro
MATH 255 SECTION 003
MIDTERM 1
February 1, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
One side of a US letter size paper (8.5" 11" ) with notes is OK.
NO CALCULATOR, BOOKS, or OTHER NOTES.
Problem Points
Math 255
Winter 2012
Moment of Inertia
1
Denition
The moment of inertia is explained in 16.5 of the textbook. The moment
of inertia is related to the rotation of an object about an axis. For a particle
with mass m, the moment of inertia is given by mr2 ,
Winter 2012
Math 255
Limit
(15.2) Let f be a function of two variables whose domain D
includes points arbitrarily close to (a, b). Then we say that the limit of
f (x, y ) as (x, y ) approaches (a, b) is L and we write
lim
(x,y )(a,b)
f (x, y ) = L
if for
Math 255, Winter 2010
Review excercises for Exam 2
1. Let
(x2 + y 2 ) log(x2 + y 2 ) for (x, y ) = (0, 0)
.
0
(x, y ) = (0, 0)
f (x, y ) =
(a) Is f (x, y ) continuous at (0, 0)?
(b) Is f (x, y ) dierentiable at (0, 0)?
Solution:
(a) In polar coordinate, f
MATH 255
Applied Honors Calculus III
Winter 2011
Midterm 1 Review Solutions
11.1: #19
Particle starts at point (1, 0), traces out a semicircle in the counterclockwise direction, ending
at the point (1, 0).
11.1: #21
Particle starts at point (0, 3), traces
Math 255, Winter 2010
Solutions for Midterm Exam 2
1. (10pts): Show that the function y (x, t) = f (x + 4t) + g (x 4t),
where f and g are twice continuously dierentiable functions,
satises the wave equation
ytt = 16yxx .
Solution 1:
Let u = x + 4t and v =
,
M ath 2 55
T est 2
W inter 2 008
T here a re 6 p roblems f or a t otal o f 5 0 p oints.
N ame:SOLUTIONS
N otes :
1. 011 t his t est, v ecto rs a r e w ritten w ilh a n a rro w:
n ot ation, o r you will lose p oints.
v E V3 a nd s ca lars w ithout:
v E l
Math 255 Midterm Exam 1wi"t.r 2oLr
Instructor/Section:
Notes:
1. There are 6 problems for a total of 100 points.
2. On this test, vectors are written with an arrow: rf e R3 and
use correct notation, or you will lose points.
scalars
3. Notes and calculator
Winter 2012
Math 255
Problem Set 3
Due on Tue, Jan 24
Section 13.2, Pages 841842:
16) Find the sum of vectors 1, 0, 2 and 0, 4, 0 , and illustrate geometrically.
22) Find a, a + b, a b, 2a, and 3a +4b when a = 3i 2k and b = i j + k.
32) Ropes 3 m and 5
Winter 2012
Math 255
Problem Set 4
Due on Tue, Jan 31
Section 13.5, Pages 865867:
12) Find parametric equations and symmetric equations for the line of intersection of the planes x + y + z = 1 and x + z = 0.
26) Find the equation of the plane through the
Winter 2012
Math 255
Problem Set 5
Due on Tue, Feb 14
Section 14.1, Pages 891892:
9) Use Maple to sketch the curve with r(t) = t, cos 2t, sin 2t . Indicate
with an arrow the direction in which t increases.
25) Show that the curve with parametric equations
Math 255003
Winter 2012
(MTuWF 910, 316 Denn)
Instructor: Manabu Machida
Oce: 3836 East Hall
Email: mmachida@umich.edu
Phone: 7349369932
Course Web Site: Go to http:/www.umich.edu/~mmachida/math255w12/. Almost everything about the course can be found o
Math 255
Winter 2012
Trigonometric and Hyperbolic Functions
1
Basic Relations
3
cos = cos 30 =
,
6
2
1
1
= cos 45 = , cos = cos 60 =
4
3
2
2
3
1
1
sin = sin 30 = , sin = sin 45 = , sin = sin 60 =
6
2
4
3
2
2
2
2
cos + sin = 1, cos() = cos , sin() = sin
c
Math 255
Winter 2012
Review Sheet for Final Exam
Solutions
1. Find the moments of inertia Ix , Iy , I0 for the lamina which occupies the
region D that is bounded by y = ex 1/2, y = 0, and x = 0, and has the
density function (x, y ) = y .
.
y dA =
Ix =
0
D
Math 255
Winter 2012
Review Sheet for Final Exam
1. Find the moments of inertia Ix , Iy , I0 for the lamina which occupies
the region D that is bounded by y = ex 1/2, y = 0, and x = 0, and
has the density function (x, y ) = y .
2. Find the moments of iner
Winter 2012
Math 255
Review Sheet for Second Midterm Exam
1. Let r(t) = cos t, sin t, t . Find T( ), N( ), and B( ).
2. At what point do the curves r1 (t) = t, t2 , t3 and r2 (s) = 2s + 1, s2 +
1, es intersect? Find cos , where is the angle of intersectio
Winter 2012
Math 255
Review Sheet for First Midterm Exam
Solutions
1. Describe the motion of a particle with position (x, y ): (a) x = sin t,
2
y = cos t, 1 t 2, and (b) x = 2 cos t, y = 3 tan t, 0 t /4.
.
(a) Using the general relation sin2 = (1 cos 2)/2
Winter 2012
Math 255
Review Sheet for First Midterm Exam
1. Describe the motion of a particle with position (x, y ): (a) x = sin t,
2
y = cos t, 1 t 2, and (b) x = 2 cos t, y = 3 tan t, 0 t /4.
2. Match the parametric equations with Curve I (left), Curve
MATH 255 Applied Honors Calculus III Winter 2012
Lab 2: Quadric Surfaces and Parametric Space Curves with Maple
1
The objective of this lab is to familiarize yourself with the shapes and properties of cylinders,
quadric surfaces (section 13.6), and parame
MATH 255 Applied Honors Calculus III Winter 2012
Lab 1: Parametric and Polar Curves with Maple
1
Preliminaries.
Start Maple by clicking on the icon that appears either on your desktop or the dock. If asked,
select to open a worksheet rather than a documen
Math 255
Winter 2012
Vector Calculations by the LeviCivita Symbol
1
Einstein Summation
Let us consider vectors
a = a1 , a2 , a3 ,
b = b1 , b2 , b3 ,
c = c1 , c2 , c3 .
(1)
We use i, j, . . . to express components of a vector. For example, the dot
product