Calculus Project by Jack Raisides

Calculus Project by Jack Raisides https://padlet.com/jraisides/37v7a7wksmr4km4h en-us 2020-11-30 19:03:12 UTC 2024-06-05 11:48:00 UTC hello@padlet.com What is Calculus jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984392301

Calculus is a branch of math that you use to calculate the instantaneous rate of change. It differs from other branches of math because it does not just use one formula or equation to solve a problem, you have to solve for the formula, and then solve the problem.

]]> 2020-12-03 16:19:58 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984392301 What is a limit jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984438675 In mathematics, a limit is the value a function 'approaches' as the input value becomes increasingly closer to a certain value]]> 2020-12-03 16:29:01 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984438675 Limit Notation jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984464898 F(x) = 25
Limx->8

"Limx->8" Is the limit statement, and it means that you are approaching x = 8 from both sides. "f(x)" is the function of the graph you are using. "25" is the y-value of the limit.]]> 2020-12-03 16:34:00 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984464898 Solving Graphically and with tables jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984476774 Graphically: Determine the y value on the graph by estimating as function 'approaches' a certain x value
With tables: Plug in values that become increasingly closer to the desired value (from both sides) ie. lim x-->4 then plug in 3.9, 3.99, 3.999 and 4.1, 4.01, 4.001 etc. if the function seems to indicate they all relate to the same value, then the limit exists]]> 2020-12-03 16:36:18 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984476774 If f(2)=4, can you conclude anything about the limit of f(x) as x approaches 2? jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984561474 You can not conclude if 4 is the limit or not because there is not enough information. If the line is continuous then 4 is the limit, but if it is not continuous then there could be a hole at 4.]]> 2020-12-03 16:52:09 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984561474 If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984565373 If f(x) as x approaches 2 = 4, then f(2)=4]]> 2020-12-03 16:52:55 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/984565373 limit Properties jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/994205203 2020-12-07 14:43:10 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/994205203 Factoring jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/995079506 2020-12-07 17:30:57 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/995079506 Defining Continuity jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/995138745 There are three parts of defining continuity:
1. The function is not defined at x = c
2. The Limit of F(x) does not exist at x = c
3. The limit of F(x) exists at x = c, but it is not equal to F(c)]]> 2020-12-07 17:42:34 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/995138745 Removable Discontinuities smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006835283 Point/removable discontinuity is when the two-sided limit exists, but isn't equal to the function's value. These types of discontinuities present themselves as holes.
Example: ( x^2-16 / x^2+4 ) has a removable discontinuity (hole) at x=4]]> 2020-12-10 16:30:07 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006835283 Non-Removable Discontinuities smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006928158 There are two types of non-removable discontinuities
1. Jump discontinuity is when the two-sided limit doesn't exist because the one-sided limits aren't equal.
2. Asymptotic/infinite discontinuity is when the two-sided limit doesn't exist because it's unbounded.]]> 2020-12-10 16:48:25 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006928158 Non-Removable Example smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006970976 Example: any function that is fully simplified that has a denominator of x+4 would have a Non-Removable discontinuity at x= -4]]> 2020-12-10 16:57:19 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1006970976 Conjugate Pairs jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1017178997 2020-12-14 18:10:26 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1017178997 Polynomial Function jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026012286 2020-12-16 23:01:10 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026012286 Rational Function jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026013385 2020-12-16 23:01:51 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026013385 Piecewise Function jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026013750 2020-12-16 23:02:05 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026013750 IVT example jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026747530 Click on link below]]> 2020-12-17 06:00:13 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026747530 Rational Functions and Vertical Asymptotes jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026755131 Not all rational functions have vertical asymptotes, in order to have a vertical asymptote the denominator must be set equal to zero and solved. In some cases they can not be solved. For example f(x) = 1/(x^2 + 4) you set x^2 +4 = 0 then subtract 4 so you get x^2 = -4, now you have to take the square root of x^2 and -4, but you can not take the root of -4, meaning that this can not be solved and does not have a vertical asymptote.]]> 2020-12-17 06:05:39 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026755131 Derivatives jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026756013 2020-12-17 06:06:18 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026756013 Tangent Line Derivative jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026757392 2020-12-17 06:07:18 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026757392 g'(8) = 10 jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026758350 F'(x) is derivative notation, and is read as "F prime of x." g'(8) = 10 means that g(8) is differentiable at 10]]> 2020-12-17 06:08:00 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026758350 Differentiable Meaning jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026758672 A function is differentiable at a point when there's a defined derivative at that point. Meaning the slope of the tangent line of the points from the left is approaching the same value as the slope of the tangent of the points from the right.]]> 2020-12-17 06:08:16 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026758672 How to Show Differentiability jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026759359 In order to prove if a function is differentiable, first you must determine if it is continuous. If it is discontinuous then it cannot be differentiable. If it is proven continuous then you find the limit from the left and right. If they are equal then the function is differentiable. Example: ]]> 2020-12-17 06:08:47 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026759359 Short Cut Rules and Examples jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026760390 2020-12-17 06:09:34 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1026760390 Real World Use jraisides https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1028103311 https://docs.google.com/document/d/1YLU_19suRvOehz32HIjKsh34-gEGyZdvXAoGu_ZaQ6c/edit]]> 2020-12-17 15:30:48 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1028103311 Derivative Example smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030315882 2020-12-18 04:48:32 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030315882 Evaluating limits flowchart smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030376509 2020-12-18 05:37:19 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030376509 Solving the Example Graphically smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030473788 When evaluating a limit graphically, it is important to consider the y value that corresponds to the function as it 'approaches' a certain x value]]> 2020-12-18 07:01:33 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1030473788 Explanation of IVT smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032817240 The intermediate value theorem describes a key property of continuous functions: for any function that's continuous over the interval [a,b], the function will take any value between f(a) and

f(b) over the interval.

]]> 2020-12-18 22:55:28 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032817240 Rational function smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032818941 function with vertical asymptotes at x=6 and x=-2, and with a zero at x=3:

(x-3)
---------------
(x-6)(x+2)]]> 2020-12-18 22:57:17 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032818941 pg. 253 smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032838567 2020-12-18 23:19:31 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1032838567 Pg. 272 smithgrady https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1033024125 2020-12-19 03:48:34 UTC https://padlet.com/jraisides/37v7a7wksmr4km4h/wish/1033024125