×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

by: Monica Chang

12

2

4

# Calculus 1, Week 1 Notes 21-111

Monica Chang
CMU

Enter your email below and we will instantly email you these Notes for Calculus 1

(Limited time offer)

Unlock FREE Class Notes

Everyone needs better class notes. Enter your email and we will send you notes for this class for free.

Topics covered include: - arithmetic - fractions - exponents - roots - factoring - rationalizing - completing the square - quadratic formula
COURSE
Calculus 1
PROF.
Deborah Brandon
TYPE
Class Notes
PAGES
4
WORDS
KARMA
Free

## Popular in Mathematical Sciences

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.

×

## Reviews for Calculus 1, Week 1 Notes

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

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

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Bentley McCaw University of Florida

Forbes

#### "Their 'Elite Notetakers' are making over \$1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com