Math 231 – Section 004
Calculus I
Week 1
Duo Zhao, PhD candidate
Department of Mathematics
University of North Carolina at Chapel Hill
Before Class
 Complete and Turn in the Placement Form.
 Check your name on the class roster
 Name not listed? Write down your PID and name below
the roster table and visit
http://math.unc.edu/for-undergrads/closed_courses
for more information
Duo Zhao, PhD candidate
Department of Mathematics
Administrata
 Lecture Time: MWF 2:00 ~ 2: 50pm
 Office Hour: 3:00~5:00pm (Math Help Center)
3:00~4:00pm (PH 405, Office)
 Email: duo.zhao@unc.edu,
duozhao@gmail.com,
The Dot In Your
Gmail Address
Doesn’t Matter
 HW( WebAssign, class key: unc 8827 3622)
 Tests & Final (paper test):
Feb 06, Mar 06, Apr 10, May 1(4~7pm)
 Calculators are necessary for HWs, but are not allowed
for all three tests and the final exam.
Grading
2.2 The Limit of a Function
 Demo
f ( x) = x2 - x + 2
lim f ( x ) = 4
x®2
 Notation for lim
lim f ( x ) = L
x®a
f ( x ) ® L as x ® a
2.2 The Limit of a Function
 One-side limit
lim- f ( x ) = L
x®a
lim+ f ( x ) = L
x®a
 Infinite limit
lim- f ( x ) = ¥
x®a
The category of a limit
 From an input point of view
 What is input? Dummy variable (does x or y matter?)
 Approaches to a particular value (a real number)
 Approaches to infinity
 Connection/Conversion
 limit
lim f ( x )
 Left limit
lim- f ( x )
 Right limit
x®a
lim f ( x )
x®¥
x®a
x®a
lim+ f ( x )
x®a
lim f ( x ) vs f (a)
x®a
lim f ( x )
In general, nothing to do with
each other by definition, but if
they do (continuous), exploit
the nice property (substitution)
Graphical Interpretation
 The circumvention/detour
x -1
lim
x®1 x -1 x +1
( )( )
1
= lim
x®1 x +1
( )
1
1
=
=
( x +1) x=1 2
 e.g 4
The category of a limit
 From a result/output point of view
 The limit approaches to a number,
The limit approaches to ∞,
The limit approaches to +∞,
The limit approaches to −∞
The limit does not exist.
----Not a good categorization (flat but overlapped)
 hierarchical category (e.g.)
Limit Laws
 The limit is an operator (what’s the operand?)
 Distributive (+, −, ×, ÷), apply to each operand
 Commutative (power, root, polynomial, rational
function)
 Squeeze Theorem (≤,≥ as operator, distributive law)
Note: what happens to lim (f < g) or lim (f > g)
 E.g 6, 11
Example:
the existence of limits
 The limit does exist
if () and only if ()
both (1)the left limit and the (2) right limit exist
and (3) they are equal [e.g. 7, 9]
 Play with the necessary and sufficient condition with this
statement
 The limit exists implies the left limit exists
(to proof)
 The right limit does not exist implies
the limit does not exist either.
(to disproof)
 The left limit is not equal to the right limit (to disproof, e.g. 8)
Central Theme in Calculus I
 Curve Sketching
 Maximum and Minimum Value
 Increasing/decreasing function
v.s. positive/negative derivative
 Techniques to compute derivative
(distributive-like, commutative-like)
 Concave/convex function
(n, u) v.s. positive/negative 2nd derivative
 Linear approximation
 Newton’s method for root finding