MH1100_2011_sem1_Course_Review_Nov. 10_2011

MH1100_2011_sem1_Course_Review_Nov. 10_2011 - MH1100...

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Unformatted text preview: MH1100 Calculus I Course Review Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Semester 1 2011/12 (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 1 / 35 Real numbers N ⊂ Z ⊂ Q ⊂ R . Many irrationals (and rationals) between any two distinct real numbers. Algebraic operations: + ,- , × , ÷ . (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 2 / 35 Trichotomy of real numbers Trichotomy of real numbers: a < , a = , a > . Useful properties: ab > ⇐⇒ ‘ a > & b > or ‘ a < & b < . a > & b > = ⇒ ( a + b ) 2 > . Application: Solve inequalities: f ( x ) > 0, f 00 ( x ) < 0. (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 3 / 35 Ordering Ordering a < b , a = b , a > b . a < b & b < c = ⇒ a < c . a ≤ b : a < b or a = b . Intervals. Closed and bounded: [ a , b ] . Open interval: ( a , b ) , (-∞ , b ) , ( a , ∞ ) . Neighbourhood of a : ( a- h , a + h ) , where h > 0. Deleted Neighbourhood of a : ( a- h , a ) ∪ ( a , a + h ) . (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 4 / 35 Real numbers: Absolute Value | x | = ( x if x ≥ ,- x if x < . | x- a | < δ ⇐⇒ a- δ < x < a + δ. | f ( x )- L | < ⇐⇒ L- < f ( x ) < L + . (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 5 / 35 Absolute Value: Useful properties | ab | = | a | · | b | . | a b | = | a | | b | , b 6 = . Triangle Inequality: | a ± b | ≤ | a | + | b | . | a- b | ≥ | a | - | b | . | cos x | ≤ 1 , | sin x | ≤ 1. (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 6 / 35 Functions Domain, codomain, range Rule Basic Functions Algebraic operations & composition (Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University) MH1100 Calculus I Semester 1 2011/12 7 / 35 Injective Functions Definition. f ( x 1 ) = f ( x 2 ) ⇐⇒ x 1 = x 2 . Enough to prove: = ⇒ . (since f is a function.) Increasing (or decreasing) = ⇒ injective. Converse is not true. Theorem: Suppose f continuous on an interval I and injective. Then f is either increasing or decreasing on I ....
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MH1100_2011_sem1_Course_Review_Nov. 10_2011 - MH1100...

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