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Week 2 Lecture Notes for Math 127

by: Monica Weisenbach

Week 2 Lecture Notes for Math 127 MATH127

Marketplace > University of Massachusetts > Mathematics (M) > MATH127 > Week 2 Lecture Notes for Math 127
Monica Weisenbach
GPA 3.819
Calculus I for Life and Social Science Majors
Thurlow Cook

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Hey, this is week 2 of notes for Math 127 taught by Prof. Thurlow Cook. In it, we touch upon rates of change, finally getting to that staple of calculus, the derivative. Enjoy!
Calculus I for Life and Social Science Majors
Thurlow Cook
Class Notes
Umass Amherst, UMass, math 127, Math
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This 3 page Class Notes was uploaded by Monica Weisenbach on Friday February 6, 2015. The Class Notes belongs to MATH127 at University of Massachusetts taught by Thurlow Cook in Spring2015. Since its upload, it has received 82 views. For similar materials see Calculus I for Life and Social Science Majors in Mathematics (M) at University of Massachusetts.


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Date Created: 02/06/15
Week 2 Note on Sincos graphs For the equations fxAsinBx and fxAcosBx A is the amplitude and 2TIB is the penod For example fx 23in3x the amplitude is 2 it goes up to 2 and down to 2 while the period is 2113 Remember for conversion purposes that radians 11180 degrees Rate of Change Average vs Instantaneous The average rate of change for the above graph a s x would be fx faxa What about when x approaches a Considering slope formula shouldn t that approach 00 That can t be done Take y x2 and choose a1 Looks like it should be 0 right with x1 fx fa fx f1 x2 1 but it can be x1x1 simples into x1 and 112 xa x 1 x 1 x1 so lim x21 2 xgtgta x1 Remember that the derivative is the instantaneous rate of change Note that as x gets closer to a the secant line which connects a and x gets steeper In other terms the secant line becomes more and more of a tangent line The slope of a tangent line is the derivative The derivative here is lim flx fla Xgtgta Xamp In general terms see graph to side lim flxh flx hgtgtO h f x is another way of writing derivative lim Ay d is yet another way of AxgtgtO Ax dx expressing derivative Let s test this formula on a linear line since the derivative should be identical to the slope Use fx 3x7 with a slope of 3 fXhfX 3Xh73X7 3X3h73X7 simpliesto 3 h h h h Galileo once found by dropping stuff off buildings that y12gt2 or that distance traveledhalf of gravitationtime squared G is either 8ms or 32 fts When t1 distance fallen is 53212 16 feet When t2 distance fallen is 53222 64 feet When t3 distance fallen is 53232 144 feet What is the speed of the falling object at those seconds You need the derivative dy lim 5gth2 5gt 5gt22thh2t2 5g2thh2 5gh2th 2 gt dx tgtgtO h h h h 1 Using this formula When t1 instantaneous speed 321 32 ftsec When t2 instantaneous speed 322 64 ftsec When t3 instantaneous speed 323 96 ftsec i I 3 l T I


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