MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
MAC2311C: Calculus with Analytic
Geometry I
Recitation Section 0017, Week 2
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
Seok-Young Chung
University of Central Florida
January 21, 2022
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Office Hour
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
I Virtual Office: Monday & Wednesday 9:00am-11:00am
– Zoom meeting with the following link
[Click here for the link]
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Construction of Recitation Section
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
1. Review this week (20 mins): Revisit contents
Office Hour
Construction of Recitation
Section
Correction
2. Worksheet (30 mins): Solve about 10 problems, possible to
talk each other, would be graded mainly based on completion.
(10 mins Break)
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
3. Quiz (15 mins): Solve 2 problems, impossible to talk each
other, concrete answer.
4. Preview next week (30 mins): A brief preview, e.g, simple
exercise.
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Correction for Exercise 32
MAC2311C: Calculus
with Analytic
Geometry I
Solve for y , state the domain restriction for x :
x +3
x=
.
(y + 1) (y ≠ 1)
2
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
2
Note that (y + 1) (y ≠ 1) = y ≠ 1. By multiplying y ≠ 1 on
both sides, we have
!
"
x y 2 ≠ 1 = x + 3.
Here, under the restriction x ”= 0, we divide both sides by x as
follows
x +3
y2 ≠ 1 =
,
x
x +3
2x + 3
which is equivalent to y 2 =
+1=
.
x
x
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Correction for Exercise 32
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Thus by taking square root, we have
y=
Ú
Correction
Contents Revisited
2x + 3
,
x
2x +3
x
provided that
Ø 0, that is, x (2x + 3) Ø 0 so that x Æ
x Ø 0. Then the domain restriction for x is given by
3
x Æ≠ ,
2
x > 0.
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
≠ 32
or
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
The Slope of Scant Line
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
Let m be the slope of scant line. Then
f (x ) ≠ f (a)
.
x æa
x ≠a
I Slope of Secant Line at a: m = lim
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Limit vs Value of function
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
What is the limit of a function?
We say the limit of f (x ) as x approaches a is equal to L if
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
we can make the values of f (x ) arbitrary close to L by taking x to
be sufficiently close to a.
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
I We write
lim f (x ) = L
x æa
I The limit lim f (x ) is not equal to f (a) in general.
x æa
A Typical Example (1)
MAC2311C: Calculus
with Analytic
Geometry I
Consider the following piecewise-defined function
I
2x + 3, x Æ 0
f (x ) =
x 2,
x > 0.
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
Evaluate lim f (x ) and f (1).
2.2 The Limit of a Function
x æ1
2.3 The Limit Laws
Preview next week
2.4 Continuity
鼠
驗
拗
I 1
List of continuous functions
3.1 Defining the Derivative
ftp.T.g
A Typical Example (2)
MAC2311C: Calculus
with Analytic
Geometry I
Consider the following piecewise-defined function
I
2x + 3, x Æ 0
f (x ) =
x 2,
x > 0.
Evaluate lim f (x ) if it exists.
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
X
x æ0
Seok-Young Chung
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
it
List of continuous functions
3.1 Defining the Derivative
Ii
Ii
fifa
fifa
s
J o
3
fifty DIVE
Doesnotexist
One-Sided Limits
MAC2311C: Calculus
with Analytic
Geometry I
We say the limit of f (x ) as x approaches a from the left (right) is
equal to L if
we can make the values of f (x ) arbitrary close to L by taking x to
be sufficiently close to a with x < a (x > a).
I We write the left-hand limit as
lim f (x ) = L
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
I We write the right-hand limit as
lim f (x ) = L
x æa+
I Note that
lim f (x ) = L if and only if lim+ f (x ) = L and lim+ f (x ) = L
x æa
Introduction
Preview next week
x æa≠
x æa
Seok-Young Chung
x æa
A Typical Example (3)
Consider the following piecewise-defined function
I
2x + 3, x Æ 0
f (x ) =
x 2,
x > 0.
Evaluate lim f (x ) and lim f (x ).
x æ0≠
x æ0+
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
fins
List of continuous functions
ff
3
3.1 Defining the Derivative
The limit could be infinity
it
1
Consider the function f (x ) = .
x
Evaluate lim f (x ) and lim f (x ).
x æ0≠
x æ0+
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
iii
ii
劒刈
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
fN
so
if
if
t
No
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
List of the Limit Laws
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
1. lim (f (x ) + g(x )) = lim f (x ) + lim g(x )
x æa
x æa
x æa
x æa
x æa
x æa
2. lim (f (x ) ≠ g(x )) = lim f (x ) ≠ lim g(x )
3. lim (cf (x )) = c lim f (x )
fi. l
x æa
x æa
x æa
x æa
ya
ii
Correction
Contents Revisited
2.1 A Preview of Calculus
2.3 The Limit Laws
Preview next week
x æa
2.4 Continuity
f (x )
limx æa f (x )
=
if lim g(x ) ”= 0
5. lim
x æa g(x )
limx æa g(x ) x æa
1
2n
n
6. lim (f (x )) = lim f (x )
x æa
x æa
Ò

7. lim n f (x ) = n lim f (x )
x æa
Construction of Recitation
Section
2.2 The Limit of a Function
4. lim (f (x )g(x )) = lim f (x ) · lim g(x )
x æa
Office Hour
鄕 徘 楡叩
if bio
List of continuous functions
3.1 Defining the Derivative
Tips
Consider the combination of Law 1,2,3,6 and 7.
!
"
I lim x 3 ≠ 8x + 9
x æa
!Ô
Ô "
I lim 2 x ≠ 4x 2 + 6 ≠ 3 x
x æ3
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
i
2.2 The Limit of a Function
2.3 The Limit Laws
价
想 想
乂t
P
왮 癎 뺎9
이 檄 怡fimi ta
R
8 at 9
by I and 2
시뺎9
8
一
Preview next week
2.4 Continuity
List of continuous functions
by 3
9
by 6
3.1 Defining the Derivative
Tips
Consider the combination of Law 1,2,3,6 and 7.
!
"
I lim x 3 ≠ 8x + 9
x æa
!Ô
Ô "
I lim 2 x ≠ 4x 2 + 6 ≠ 3 x
x æ3
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
ii
2.2 The Limit of a Function
2.3 The Limit Laws
一
癎
s3
惻州 楗
一
B 4.5t6 F
B
S
36t6.3B
30t B
V3
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
The Squeeze Theorem
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
If f (x ) Æ g(x ) Æ h(x ) when x is near a except possibly at a and
lim f (x ) = lim h(x ) = L.
x æa
Then
x æa
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
lim g(x ) = L.
2.3 The Limit Laws
x æa
Preview next week
2.4 Continuity
List of continuous functions
Note that
3.1 Defining the Derivative
L = lim f (x ) Æ lim g(x ) Æ lim h(x ) = L
x æa
which implies that
x æa
x æa
L Æ lim g(x ) Æ L.
x æa
Need to know
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
sin x
I lim
= 1.
x æ0 x
i
1 ≠ cos x
I lim
= 0.
x æ0
x
1 ≠ cos x
I lim
= 1.
1 2
x æ0
x
2
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Definition: Continuity
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
A function f (x ) is said to be continuous at a point a if
Construction of Recitation
Section
Correction
1. f (a) is well-defined. Es a is inthe domain off
2. lim f (x ) exists.
fix
Contents Revisited
3. lim f (x ) = f (a).
Preview next week
x æa
f뱂
f뱂
x æa
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
I Recall that lim f (x ) exists if lim f (x ), lim f (x ) exist and
+
x æa
lim f (x ) = lim≠ f (x ).
x æa+
x æa
x æa
x æa≠
I We say f is discontinuous at a point a if it fails to be
continuous at a.
List of continuous functions
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
The following functions are continuous on their domains:
I Polynomials
e.g. f (x ) = x 2 + 2x + 3
I Exponential functions
e.g. f (x ) = e 2x +1
I Logarithmic functions
e.g. f (x ) = log (x + 2)
I Trigonometric and inverse trigonometric functions
e.g. f (x ) = sin(x ), g(x ) = tan≠1 (x )
I Root functions
Ô
e.g. f (x ) = 3x ≠ 1
I Rational functions
2
+3
e.g. f (x ) = 2x x≠x
+1
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Continuous Function
MAC2311C: Calculus
with Analytic
Geometry I
!
3
Let us consider the limit lim x ≠ 8x + 9
x æa
Since
必改 9
is
"
Seok-Young Chung
Introduction
Office Hour
line
continuous on thereal
Construction of Recitation
Section
Correction
Contents Revisited
we can use the direct substitution property as follows
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
i
8 at 9
3.1 Defining the Derivative
Exercise
The graph of f (x ) is shown below. Which of the following
statements are true?
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Exercise
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
jump
i
i
6
removable
infinite
i
s
i
i
i
i
X1.2.
Correction
I
l
Contents Revisited
6
I
i
i
q
continuous
f (x ) has a removable discontinuity at x = 6.
f (x ) has a jump discontiuity at x = 6.
3. f (x ) has an infinite discontiuity at x = 6.
4. f (x ) is continuous at x = 6.
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Table of Contents
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation Section
Correction
Introduction
Office Hour
Construction of Recitation
Section
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
Definition: Differentiation
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Introduction
Office Hour
Construction of Recitation
Section
Correction
Let f be a function. The derivative function f Õ is defined by
f (x + h) ≠ f (x )
f Õ (x ) = lim
hæ0
h
if the limit exists.
I If f (x ) is differentiable at a, then f (x ) is continuous at a.
I The converse does not hold true.
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative
A Typical Example
MAC2311C: Calculus
with Analytic
Geometry I
Seok-Young Chung
Find the derivative of the function f (x ) = x 2 ≠ 2x .
Introduction
Office Hour
Construction of Recitation
Section
HhF2xthD 1x2.2x
ftp.fingl
fi
h
t2hxt5X2h
if
Hath
2
H
if
12Xth 2
2
2
Tt
絿州
涵啖喙
一
sa 2 IT
一如
Correction
Contents Revisited
2.1 A Preview of Calculus
2.2 The Limit of a Function
2.3 The Limit Laws
Preview next week
2.4 Continuity
List of continuous functions
3.1 Defining the Derivative