Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
+UNIST CALCULUS II T/F Questions
[Date]
[Group NO]
Line Integral
1.
[Student ID]
[Name]
8. Let M (x , y, z ) , N (x , y, z ) , and P (x , y, z ) be
differentiable functions with the region
If and are two onetoone
parametrizations of the same curve and F
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
1
2014 Calculus II
Final Exam, Dec. 18, 2014
Name
ID
Supervisor
(sign)
Direction : This exam contains 10 problems. Write all relevant works clearly and
concisely in English. You should justify your answers for all problems except True or
False problems. Y
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
A PPLIED L INEAR A LGEBRA (MTH10304, 10305)
Midterm Exam. 16 Oct. 2014 19:00  21:00
1. Determine whether the statement is true of false. (2pts. each)
(1)
0 1 1 0
0 0 0 1
0 0 0 0
is in reduced rowechelon form.
Answer] True.
(2) If the sizes of matrices
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Differential Equations Instructor :
MTH 20101 /20102/20103 Student ID :
October 21, 2015 Name :
Midterm Exam
Show all your work
1. (20 points) Consider the following autonomous equation.
dy _ 1 2
dt _ y my
(a) (5 pts) Find out all equilibrium soluti
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Tarjeta 5 min: 25 Increbles datos sobre el cuerpo humano.
El cuerpo humano es una increble mquina humana, un complejo aparato con millones
de clulas, venas, huesos, rganos entre muchas otras cosas.
Es por eso que aqu encontraras los 25 datos ms increbles
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
+UNIST CALCULUS 1 Discussion SET WEEK 2
[Name]
[Student ID]
1. (a) What does it mean for f to be continuous at a?
(b) What does it mean for f to be continuous on the interval (,)?
What can you say about the graph of such a function?
[course No.]
At which
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
144
Chapter 7.
Infinite Sequences and Series
P
Example 7.3.5 (1) The series
n=1 an with an =
1
comparing it with bn = n . Indeed,
2n+1
n2 +2n+1
diverges by
an
2n2 + n
= lim 2
= 2.
n bn
n n + 2n + 1
lim
P
1+n ln n
diverges: Indeed, since an
(2) The series
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
+UNIST CALCULUS 1 Discussion SET WEEK 2
[Name]
[Student ID]
[course No.]
1. Explain what each of the following means and illustrate with a sketch.
(a) () =
(b) + () =
(c) () =
(d) () =
(e) () =
10. Suppose an object moves along a straight line with p
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
158
Chapter 7.
Infinite Sequences and Series
P
1
n n
2
3
Example 7.4.7 Let f (t) =
n=0 (1) t = 1 t + t t + = 1+t , which
is convergent on (1, 1). Then
x
Z x
t2 t3 t4
1
ln(1 + x) =
dx = t + +
2
3
4
0 1+t
0
2
3
x
x
x4
= x
+
+ , 1 < x < 1.
2
3
4
For x = 1,
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
+UNIST CALCULUS 1 Discussion SET WEEK 1
[Name]
[Student ID]
1. (a) What is a function? What are its domain and range?
Find the domain of y =
x +3
4  x2  9
[course No.]
4. Give an example of each type of function.
(a) Linear function
(b) Power function
(
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 13
Appendix: The Gamma
Function
There is an important function in many different parts of mathematics, called
the gamma function. The theory of the gamma function was developed
in connection with the problem of generalizing the factorial function
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 7
Infinite Sequences and Series
7.1
Sequences
Everyone knows how to add a finite number of real numbers. However, in
many occasions in real life, one encounters with problems asking to add
infinitely many numbers. Question is how we can do that. I
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 3
Differentiation
In the real life, people encounter many problems for maximizing, or minimizing certain values depending on some other quantities. Such a problem
is usually called an optimization problem, for which the differentiation of
a functi
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
21
2.2. CONTINUITY OF FUNCTIONS
(2) f (x) =
(3) f (x) =
(4) f (x) =
x 5, L = 2, x0 = 9, = 1.
1
x,
x2
L = 14 , x0 = 4, = 0.05.
5, L = 11, x0 = 4, = 0.5.
6. Prove the following limit statements:
(1) limx0 4 x = 2.
2
x , x 6= 2
(2) limx2 f (x) = 4 for f (x
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
32
Chapter 3.
3.1.1
Differentiation
Exercises
1. Prove the following formulas:
(tan x) = sec2 x,
(sec x) = sec x tan x,
(cot x) = csc2 x,
(cscx) = cscx cot x.
2. Find the first and second derivatives of the functions:
1
1
2
2
(1) y = 3x . (2) y = (x + 1)
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 4
Integration
4.1
Indefinite Integrals
Many real problems require to find a function F for a given function f such
that F = f . If such a function F exists, it is called an antiderivative of f .
Definition 4.1.1 A differentiable function F is an a
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
98
Chapter 5.
Calculus of Transcendental Functions
Example 5.3.2 Use LHopitals Rule to show that limx0+ (1 + x)1/x = e.
Solution: This is an indefinite form of type 1 . If f (x) = (1 + x)1/x , then
1
ln(1 + x)
1+x
ln lim+ f (x) = lim+ (ln f (x) = lim+
= l
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 2
Limits and Continuity
Mathematicians of seventeenth century were very much interested in studying motions of moving objects and the motions of planets or stars. This
study involved both the speed and the directions of motion at any instant.
They
Ulsan National Institute of Science and Technology
Calculus
MATHEMATIC 101

Spring 2016
Chapter 5
Calculus of Transcendental
Functions
5.1
Inverse Functions
We are now interested in finding the derivatives of inverse functions when
the functions are invertible. Recall that a differentiable function f : I J
from an interval I onto J is invert
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Differential Equations
( MTH 20102 )
March 10, 2015
Jung Eun Kim
Ch 2.4: Differences Between Linear and Nonlinear Equations
Recall that a rst order ODE has the form y' = f (t, y), and is
linear if f is linear in y, and nonlinear if f is nonlinear in y.
Ex
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Differential Equations
( MTH 20102 )
Spring 2015
Jung Eun Kim
Chapter 2. First order Differential equations
Section 2.1 Linear Equations; Method of Integrating
Factors
A linear first order ODE has the general form
dy
!
= f (t , y )
dt
!
where f is linear
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Ch 7.7: Fundamental Matrices
Suppose that x(1)(t), x(n)(t) form a fundamental set of solutions f
or x' = P(t)x on
<t< .
The matrix
x1(1) (t )
x1( n ) (t )
(t )
,
(
xn1) (t )
(
xnn ) (t )
whose columns are x(1)(t), x(n)(t), is a fundamental matrix for th
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Ch 10.1: TwoPoint Boundary Value Problems
In many important physical problems there are two or more independent vari
ables, so the corresponding mathematical models involve partial differential e
quations.
This chapter treats one important method for sol
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Midterm Exam MA201
2006, Spring
1. (10 points) Solve the given dierential equation by nding an appropriate integrating factor
(3xy + y 2 )dx + (x2 + xy)dy = 0.
2. (10 points) Solve the given dierential equation
3
y + x2 y = (ex sinh x)
1
,
3y 2
y(0) =
3
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Pa
i&'(MA201)eK
2006. 6. 15.(3) 19:00 22:00
q
l
cfw_:
r
:
<
:
s2:
":
f
S q q Gz '< S.
: K
?
>
< >
 +a 6 .
< 0
B
 T _ Uq ~ ^; [ M #.
m < m < c #
r @
@ >
E
 S0U S <_z NF #.
< K A
C
>
B E
 bT +T + +U .
>
< m< > m:
"
?
 4M D M a U #.
:< 7
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Differential Equations
( MTH 20102 )
March 12, 2015
Jung Eun Kim
Review: Autonomous equations
dy/dt = f (y) is called an autonomous equations where the
independent variable t does not appear explicitly.
The roots of f (y)=0 are called critical points. (
Ulsan National Institute of Science and Technology
differential
MATHEMATIC 101

Spring 2015
Differential Equations
(MTH 20102)
Jung Eun Kim
March 23, 2015
Review
Wronskian
W = W (y1 , y2 )(t) =
y1 (t) y2 (t)
= y1 (t)y2 (t) y1 (t)y2 (t).
y1 (t) y2 (t)
W is called the Wronskian determinant, or more simply, the
Wronskian of the solutions y1 and y2