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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 - PowerPoint PPT Presentation
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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: [email protected],
[email protected], 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