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 rosterName 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 candidateDepartment of Mathematics

The Dot In Your Gmail Address Doesn’t Matter

AdministrataLecture Time: MWF 2:00 ~ 2: 50pmOffice 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 viewWhat is input? Dummy variable (does x or y matter?)Approaches to a particular value (a real number)Approaches to infinityConnection/ConversionlimitLeft limitRight limit

Graphical Interpretation

The circumvention/detour

e.g 4

From a result/output point of viewThe 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 operandCommutative (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 SketchingMaximum and Minimum ValueIncreasing/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 approximationNewton’s method for root finding