TRIUMPHS-Derivatives of Sine and Cosine-GROUP 2 by Roser Gine https://padlet.com/rgine16/187lpkfzhgxtrjfj en-us 2021-04-09 23:37:41 UTC 2025-11-15 01:35:23 UTC hello@padlet.com TASK 1 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761298 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761298 TASK 2 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761299 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761299 TASK 3 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761300 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761300 TASK 4 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761301 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761301 TASK 5 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761303 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761303 TASK 6 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761304 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761304 TASK 7 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761305 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761305 TASK 8 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761308 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761308 TASK 9 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761309 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761309 TASK 10 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761310 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761310 TASK 11 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761314 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761314 TASK 12 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761317 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761317 EULER Excerpt 1 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761318 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761318 EULER Excerpt 2 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761319 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761319 EULER Excerpt 3 rgine16 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761320 2021-04-09 23:37:41 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1401761320 "Finally, it was Hardy who, in 1908, defined a function as a relation between two variables π‘₯ and 𝑦 such that β€œto some values of π‘₯ at any rate correspond values of 𝑦.” https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407545581 2021-04-12 14:28:58 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407545581 Calculus III Textbook: https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407547475 A function maps each input x in a subset D of the real numbers to a unique real number y. The set D is the domain of the function, and the range is the set of all real numbers y that have at least one input x in D such that f(x)=y.]]> 2021-04-12 14:29:20 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407547475 (b) https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407582132 2021-04-12 14:35:26 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407582132 (a) Euler's Goal maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407663968
  • To derive a definition for differential calculus
  • Giving context to the definition provided at the end of the paragraph
    • ]]>     2021-04-12 14:48:59 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1407663968     https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414330437 2021-04-13 23:50:53 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414330437 maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414387452 A function f defines from a set A to a set B is a rule that associates with each element of the set A one, and only one, element of the set B.]]> 2021-04-14 00:17:22 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414387452 maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414390801 A function f from a set D to a set Y is a rule that assigns a unique (single) element f(x) ∈ Y to each element x βˆˆ D.]]> 2021-04-14 00:18:49 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414390801 (b) maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414403303 Comments:

    Questions: what does differential calculus have to do with the analysis of the infinite? ]]>     2021-04-14 00:23:47 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1414403303     maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416141912 I think that Euler believed 'transcendentals' to mean any quantity that is subject to change or that could be determined]]> 2021-04-14 12:49:27 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416141912 (a) using limits maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416300783 "ratio" -> fraction
    "w gets smaller" -> limit as w heads to 0 ]]>     2021-04-14 13:22:04 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416300783     (b) maria_moccio https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416309770 Compare: ]]> 2021-04-14 13:23:39 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416309770 Comparing https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416476810 The beginning of the project uses h to represent the increase to x, as opposed to w. Both definitions give the ratio of the increase of x, h or w, to the change in the function of x (shown in the top of the fraction in the first definition). The definition in the beginning, however, takes into the account the limit of this ratio as the increase in x goes to 0.]]> 2021-04-14 13:55:04 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1416476810 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418510320 2021-04-14 21:01:24 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418510320 Graph Key https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418510764 grey=sin(x)
    green=sin(x+dx)]]>     2021-04-14 21:01:35 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418510764     (a) using work from task 3 without limits https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418513407 2021-04-14 21:02:39 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1418513407 Key is in differentiating omega=0 from the limit as omega approaches 0 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1421425677 Allows the ratio to remain 2x:1 despite "vanishing increments"]]> 2021-04-15 14:56:32 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1421425677 Task 8: Single Graph Approach https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1421563231 This was my approach for Task 8, which uses two points on a single graph. Does this work seem correct?]]> 2021-04-15 15:21:47 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1421563231 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442602603 2021-04-21 13:55:48 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442602603 Task 9 Work https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442728089 2021-04-21 14:20:11 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442728089 Comments https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442780184 As the distance gets smaller and smaller, the sin approaches the angle itself]]> 2021-04-21 14:29:55 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442780184 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442801159 2021-04-21 14:33:59 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442801159 b) https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442803069 start to diverge around pi/2, which is approximately 1.57]]> 2021-04-21 14:34:21 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442803069 (b) https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442877464 dy -> change in sine *Task 8*]]> 2021-04-21 14:48:21 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442877464 (a) https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442879458 To Euler, "vanishing terms" mean extremely small terms in a series that will head to 0 as they shrink. From Task 8 we saw that as angle x gets smaller (approaches 0) it will approach itself aka 0.]]> 2021-04-21 14:48:38 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1442879458 (a) Euler's Goal https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443010313
  • Setting the stage by moving from polynomials to transcendental functions
  • Introducing sine and cosine
  • Euler's goal was to differentiate sin x 
    • ]]>     2021-04-21 15:13:02 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443010313     (b) https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443023508 Comment: 

    Question:]]>     2021-04-21 15:15:30 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443023508     https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443069986 dy = dx * cos x
    -> dy/dx = cos x]]>     2021-04-21 15:24:03 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443069986     https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443126137 2021-04-21 15:34:26 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1443126137 a) sin (0.1) = 0.0998334166 https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1449877142 0.1-(0.1^3)/6+(0.1^5)/120= 0.0998334167]]> 2021-04-23 00:38:13 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1449877142 once you get to the cos(x+dx)=cosx-dxsinx step, you can subtract cosx from both sides. Then since cos(x+dx)-cosx=dy, you get dy=-dxsinx, which matches definition of derivative of cosx https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1450221519 2021-04-23 02:43:15 UTC https://padlet.com/rgine16/187lpkfzhgxtrjfj/wish/1450221519