Math 231 – Section 004 Calculus I Week 1 Duo Zhao, PhD candidate Department of Mathematics...

Math 231 – Section 004Calculus IWeek 1

Duo Zhao, PhD candidateDepartment of MathematicsUniversity of North Carolina at Chapel Hill

Before ClassComplete 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_coursesfor more information

Duo Zhao, PhD candidate

Department of Mathematics

The Dot In Your Gmail Address Doesn’t Matter

AdministrataLecture 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,

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 FunctionDemo

Notation for lim

2.2 The Limit of a FunctionOne-side limit

Infinite limit

In general, nothing to do with each other by

definition, but if they do (continuous), exploit the nice property (substitution)

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/Conversionlimit

Left limit

Right limit

Graphical Interpretation

The circumvention/detour

e.g 4

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.)

The category of a limit

Limit LawsThe 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 existand (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