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# Calculus 1, Week 1 Notes 21-111

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This 4 page Class Notes was uploaded by Monica Chang on Tuesday September 6, 2016. The Class Notes belongs to 21-111 at Carnegie Mellon University taught by Deborah Brandon in Fall 2016. Since its upload, it has received 12 views. For similar materials see Calculus 1 in Mathematical Sciences at Carnegie Mellon University.

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Date Created: 09/06/16

Week 1 BASICS REWIEW Arithmetic: a+b=b+a a+b +c=a+(b+c) ab+c =ab+ac ab=ba ab)c=a(bc) Multiplying Fractions: a c ac × = b d bd Dividing Fractions: a÷ = ad b d bc Adding Fractions: a+ = ad+cb= ad+bc b d bd bd bd Exponent Basics: an means multiply a by itself n times 0 =0wheren>0 0 a =1wherea≠0 0 isindeterminate a = 1 wherea≠0 an becau0e n−n n −n 1=a =a =a a 1=a a−n 1 =a−n an Laws of Exponents: Given m & n are integers and a & b are real numbers: a a =a m+n m a m−n n =a wherea≠0 a a a m (¿¿n) n mn (¿¿ m) =a =¿ ¿ n n n (ab) =a b a n an ( ) = nwhereb≠0 b b −n n n ( ) =( ) = b b a an Root Basics: 1 n n √a=a na =( a) m √ √ 3 o Ex. Calculate 2 . 8 3 2 3 2 We would rather do (√8) than do √8 . Root Laws: 1 1 1 √ab=(ab) =a b = a √ √n 1 1 n n na =( ) = a = √a √b b 1 √b bn Factoring Formulas: 2 2 x −y =(x−y)(x+y) 2 2 2 x +2 xy+y =(x+y) x −2xy+y =(x−y) 2 x −y =(x−y)(x +xy+y ) 2 x +y =(x+y)(x −xy+y )2 You can expand the following equations easily using Pascal’s Triangle to 0ind the coefficients of each term: (x+y) =1 (x+y) =x+y 2 2 2 (x+y) =x +2 xy+y (x+y) =x +3x y+3 x y +y Here are the starting rows of Pascal’s Triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 … Rationalizing: x −y =(x−y)(x+y) We call (x−y) and (x+y) conjugates. o Ex. Rationalize √a− √ . (√ √ b ) a−b √a− √= √a− √ )× a+ b = a+ b (√ √ ) √ √ Note: In calculus, we do not care about having radical signs in the denominator. Completing the Square: Use the following idea: 2 2 2 x +2 xy+y =(x+y) x +2 xy=(x+y) −y2 o Ex1. Solve for x: x +8x+3=0 . x +8x+3=0 2 x +8x=3 x +8x+16=3+16 (x+4) =13 x+4=± 1√ x=± √3−4 o Ex2. Complete the square: F(x)=x −7x+3 2 F(x=x −7x+3 2 −7 −7 2 −7 2 F(x=x +2 (2) ( ) 2 ) − 2 +3 2 2 F(x=(x− ) +3− −7 2 ( 2 7 2 37 F(x=(x− ) − 2 4 Writing an equation in this form can make it so much easier to graph Quadratic Formula: For equations in the form a x +bx+c=0 , you can use also the quadratic formula to solve for x: −b± b√−4ac x= 2a Methods: Now you know 3 methods to solving equations in the 2 form a x +bx+c=0 . o Factoring o Completing the Square o Quadratic Formula

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