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Calculus 1, Week 1 Notes

by: Monica Chang

Calculus 1, Week 1 Notes 21-111

Monica Chang
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About this Document

Topics covered include: - arithmetic - fractions - exponents - roots - factoring - rationalizing - completing the square - quadratic formula
Calculus 1
Deborah Brandon
Class Notes




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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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