1. Math 231 – Section 004 Calculus I Week 1 Duo Zhao, PhD candidate Department of Mathematics University of North Carolina at Chapel Hill

2. 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 http://math.unc.edu/for-undergrads/closed_courses Duo Zhao, PhD candidate Department of Mathematics

3. The Dot In Your Gmail Address Doesn’t Matter 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,duo.zhao@unc.eduduozhao@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.

4. Grading

5. 2.2 The Limit of a Function Demo Notation for lim

6. 2.2 The Limit of a Function One-side limit Infinite limit

7. 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/Conversion limit Left limit Right limit

8. Graphical Interpretation

9. The circumvention/detour e.g 4

10. 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

11. 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) E.g 6, 11

12. 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)

13. 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 2 nd derivative Linear approximation Newton’s method for root finding