math 231 – section 004 calculus i week 1 duo zhao, phd candidate department of mathematics...
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Math 231 – Section 004Calculus IWeek 1
Duo Zhao, PhD candidateDepartment of MathematicsUniversity of North Carolina at Chapel Hill
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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
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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: [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.
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Grading
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2.2 The Limit of a FunctionDemo
Notation for lim
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2.2 The Limit of a FunctionOne-side limit
Infinite limit
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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
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Graphical Interpretation
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The circumvention/detour
e.g 4
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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
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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
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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)
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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
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