Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 3)
Calculus I
4. Let us call a function f : R R satisfying the properties
1. Explain why the function
(
x
f (x) =
0
i. f (x) is continuous at x = 0,
if x 6= 1,
if x = 1;
ii. f (x + y) = f (x) + f (y) for all x, y R
a lovel
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 4) Solutions
1. Let
f (x) =
x1
x+1
2
,
g(x) =
Calculus I
1
1.
x2
Find (f g)0 (1).
Answer:
2
x1
=1
, so by the Chain Rule,
x+1
x+1
2
d x1
x1
2
4(x 1)
0
f (x) =
=2
=
.
2
dx x + 1
x+1
(x + 1)
(x + 1)3
On the other hand, g(1)
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 1)
1. Determine whether each of the following statements is true or false.
If it is true, prove it. If it is false, give a counterexample.
Calculus I
3. Determine whether each of the following statements is true for all
fu
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 2) Solutions
Calculus I
1. The statement
As x approaches 1, f (x) gets closer to 2.
does not imply that lim f (x) = 2. Why not? Explain.
x1
Answer: Consider the function f (x) = x2 . As x approaches 0, f (x) gets closer to
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 6)
1. (a) What does the notation
Calculus I
3. Evaluate the following definite integrals geometrically.
Z 0
(a)
16 x2 dx
Z
f dx
4
mean? What is it called?
Z
8
(b)
(b) What exactly does the + symbol in the formula
Z
Z
Z
f +
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 1) Solutions
Calculus I
1. Determine whether each of the following statements is true or false. If it is true, prove it. If it is
false, give a counterexample.
(a) If a nonempty subset of Z is bounded from above, then it h
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 6) Solutions
Calculus I
1. (a) What does the notation
Z
f dx
mean? What is it called?
Answer: It denotes the set of all antiderivatives of f (x). It is called the indefinite integral
of f (x).
(b) What exactly does the + s
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 4)
1. Let
f (x) =
x1
x+1
2
,
Calculus I
5. Consider the function defined on [0, 1] by
(
x+2
if 0 x < 1,
f (x) =
2
if x = 1.
1
g(x) = 2 1.
x
Find (f g)0 (1).
Determine whether f (x) has an absolute maximum on [0, 1]. Does
t
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 7) Solutions
Calculus I
1. Using calculus, verify that the circumference of a circle of radius r is 2r.
(Hint. Use the arc length formula and recall
d
dx
sin1 x =
1
.)
1x2
Answer: Firstly, we will find the length L of the
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 7)
1. Using calculus, verify that the circumference of a circle of radius r is
2r.
(Hint. Use the arc length formula and recall
d
dx
sin1 x =
Calculus I
6. Let S be the shaded region shown below:
1
.)
1x2
(a) Find the are
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Mth 111, Calculus I
Chapter 3. Differentiation3
Division of General Studies
UNIST
Sparing, 2017
3.4 Curve Sketching
Corollary (3.4.1)
Suppose that a function f (x) is continuous on [a, b] and
differentiable on (a, b).
(1) If f (x) > 0 on (a, b), then f i
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 2)
1. The statement
Calculus I
6. The graph of f (t) is shown below:
As x approaches 1, f (x) gets closer to 2.
does not imply that lim f (x) = 2. Why not? Explain.
x1
2. Prove the following.
Determine whether each of the
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 5) Solutions
Calculus I
1. Find an example of a continuous function f (x) where its concavity changes at a point (c, f (c) on
its graph, and yet (c, f (c) is not an inflection point.
Answer: Consider the continuous functio
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 3) Solutions
Calculus I
1. Explain why the function
(
x
f (x) =
0
if x 6= 1,
if x = 1;
is not continuous at x = 1. Is the discontinuity there removable? That is, can f (1) be redefined
so that f (x) is continuous at x = 1
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Spring 2017
Discussion Set (Week 5)
1. Find an example of a continuous function f (x) where its concavity
changes at a point (c, f (c) on its graph, and yet (c, f (c) is not an
inflection point.
2. Let f (x) = x
2/3
Calculus I
(h) Apply the Second Derivat
Ulsan National Institute of Science and Technology
Calculus II
MTH 112

Fall 2016
Mth 111, Calculus I
Chapter 3. Differentiation2
Division of General Studies
UNIST
Spring, 2017
3.2 Differentials
Let y = f (x) be a differentiable function on an interval I . For a
fixed a I and a point x I near a, the value f (x) can be
approximated by
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Review for Sections 2.1 & 2.2
Solving 1st order linear ODEs by integrating
factor method: dy p(t ) y g (t ), (t ) e p (t ) dt .
dt
: After multiplying and simplifying, integrate
and solve for y
Solving 1st order separable ODEs:
M ( x)dx N ( y)dy 0
: Dir
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.1: 2nd Order Linear Homogeneous
EquationsConstant Coefficients
A second order ordinary differential equation has the
general form
y f (t, y, y)
where f is some given function.
This equation is said to be linear if f is linear in y and y':
y g (t )
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 4.1: Higher Order Linear ODEs:
General Theory
An nth order linear ODE has the general form
dny
d n1 y
dy
P0 (t ) n P1 (t ) n 1 Pn1 (t ) Pn (t ) y Gt
dt
dt
dt
We assume that P0, Pn, and G are continuous realvalued
functions on some interval I = (, )
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 5.2: Series Solutions Near an Ordinary Point,
Part I
In Chapter 3, we examined methods of solving second order
linear differential equations with constant coefficients.
We now consider the case where the coefficients are functions
of the independent
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Review of Ch.1
DEs are equations with derivatives.
Classification of DEs by the following :
ODE vs PDE
Order of DEs
Linear vs nonlinear
Issues for solutions of DEs: existence,
uniqueness and how to find it.
Direction field, equilibrium solutions, syst
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 5.4: Euler Equations;
Regular Singular Points
Recall that the point x0 is an ordinary point of the equation
d2y
dy
P ( x ) 2 Q( x ) R ( x ) y 0
dx
dx
if p(x) = Q(x)/P(x) and q(x)= R(x)/P(x) are analytic at at x0.
Otherwise x0 is a singular point.
Thu
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.5: Nonhomogeneous Equations: Method of
Undetermined Coefficients
Return to the nonhomogeneous (2nd order linear) ODE
y p (t ) y q (t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 5.1: Review of Power Series
Finding the general solution of a linear differential equation
depends on determining a fundamental set of solutions of the
homogeneous equation.
So far, we have a systematic procedure for constructing
fundamental solution
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 3.6: Variation of Parameters
Recall the nonhomogeneous equation
y p(t ) y q(t ) y g (t )
where p, q, g are continuous functions on an open interval I.
The associated homogeneous equation is
y p(t ) y q(t ) y 0
In this section we will learn the varia
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 5.2: Series Solutions Near an Ordinary Point,
Part I
In Chapter 3, we examined methods of solving second order
linear differential equations with constant coefficients.
We now consider the case where the coefficients are functions
of the independent
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.5: Homogeneous Linear Systems with Constant
Coefficients
We consider here a homogeneous system of n first order linear
equations with constant, real coefficients:
x1 a11x1 a12 x2 a1n xn
x2 a21x1 a22 x2 a2 n xn
xn an1 x1 an 2 x2 ann xn
This system c
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.9: Nonhomogeneous Linear Systems
The general theory of a nonhomogeneous system of equations
x1 p11(t ) x1 p12 (t ) x2 p1n (t ) xn g1 (t )
x2 p21(t ) x1 p22 (t ) x2 p2 n (t ) xn g 2 (t )
xn pn1 (t ) x1 pn 2 (t ) x2 pnn (t ) xn g n (t )
parallels that
Ulsan National Institute of Science and Technology
Differential equation
MTH 201

Fall 2016
Ch 7.7: Fundamental Matrices
Suppose that x(1)(t), x(n)(t) form a fundamental set of
solutions for x' = P(t)x on < t < .
The matrix
x1(1) (t ) x1( n ) (t )
(t )
,
x (1) (t ) x ( n ) (t )
n
n
whose columns are x(1)(t), x(n)(t), is a fundamental ma