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10.06 - Algebra review(preparing for chapter 2(a2 b2(a b =...

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Algebra review (preparing for chapter 2): ( a 2 b 2 )/( a b ) = a + b ( a 3 b 3 )/( a b ) = a 2 + ab + b 2 ( a 4 b 4 )/( a b ) = a 3 + a 2 b + ab 2 + b 3 ( a 5 b 5 )/( a b ) = a 4 + a 3 b + a 2 b 2 + ab 3 + b 4 etc. ( a + b ) 2 = 1 a 2 + 2 ab + 1 b 2 ( a + b ) 3 = 1 a 3 + 3 a 2 b + 3 ab 2 + 1 b 3 ( a + b ) 4 = 1 a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + 1 b 4 ( a + b ) 5 = 1 a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + 1 b 5 etc. The coefficients in the polynomials to the right of the equal signs are given by the entries of an array of numbers discovered by the ancient Chinese, called Pascal’s triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 etc.
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Chapter 2 Main point of 2.1? … ..?.. Limits of the form lim h 0 ( f ( x + h ) – f ( x ))/ h , called derivatives , play a unifying role; they let us talk about things like “slope of the tangent line” and “instantaneous velocity” in a single common framework. Example 1: Let P be the point ( a , f ( a )) on the curve y = f ( x ), and Q be the nearby point ( a + h , f ( a + h )). Then the slope of the secant PQ is ( f ( a + h )– f ( a ))/(( a + h )– a ) = ( f ( a + h )– f ( a ))/ h , so as Q approaches P
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