Lesson+Summaries+-+Rates+of+Change


 * February 7, 2011**
 * By: Gillian**

If it's been a while since you've been in math, you might want to practice some basic algebra and factoring skills. Try the attached sheet.

- What is Calculus? (1:32) http://www.youtube.com/watch?v=ismnD_QHKkQ - Calculus in 20 Minutes! (20 mins) [] media type="custom" key="8411770" - How to enter data in a graphing calculator: a) STAT ===> Enter ===> L1 (x-values) ... L2 (y-values) b) How to graph: 2nd + y= ==> enter ==> enter ==> zoom + 9 c) How to find the curve of best fit: STAT ==> right arrow ==> scroll down ==> enter ==> enter
 * February 8,2011**
 * Marco Becerril**
 * Unit 1, Topic: What is Calculus?**
 * Text Refernce: Grade 12 Functions course. See handouts.**
 * Idea: An introduction to Calculus.**

HOMEWORK; Due Friday February 11,2011: Sometimes Life is like a Rollercoaster

-Rate of change of f (x) between P and Q is the same as the slope of the secant joining P and Q. - Coordinates of P= ( a, f (a) ) -Coordinates of Q = ( a+h, f (a+h) ) - h= 0.001 -//**M**PQ// = __f (a+h) - f (a)__ = __f (a+h) - f (a)__ .................a+h-a ...............h - Ex. Find the instantaneous rate of change of f (x)= x^2 at x=7. (f (a+h) - f (a))/ h = (f (7+0.001) - f(7))/0.001 = (49.012001-49)/0.001 = 14
 * February 9, 2011**
 * Marco Becerril**
 * Unit 1, Topic: Solving Rate Of Change Problems Algebraically**
 * Text Reference: Grade 12 Fuunctions. See handouts.**
 * Idea: To solve Rate of Change Problems Algebraically**

- Average and instantaneous rate of change [] (Part 1) [] (Part 2) [] (Part 3)

Homework: Handouts: a) Algebra Review: Getting Ready b) 1.2 Rates of Change using equations Sometimes Life is like a rollercoaster

__﻿INTRO TO LIMITS__
Fri. Feb.11, 2011. by Mengdi Wei

//Instantaneous rate of change is how a function changes at a specific point, represented by the slope of the point's tangent.// //Average rate of change is the change between points; it is often the slope of a secant.// Instantaneous Rate of Change - A Dilemma (Handout) //The instantaneous rate of change of the function cannot be determined at point P, because as approaching from the two sides of P, the rates of change, or the slopes of the tangents, are conflicting and P does not "swing" between the two.// asdf
 * __Review:__** Differences between instantaneous rate of change and average rate of change.

**__LIMITS__**
asdfasdfasdfasfasfasfasfasf (read //as the limit as h approaches 0 of the function...// --->) When calculating instantaneous rate of change, we try to get h as close as possible to zero, forcing h to approach zero. This is called taking the ** limit **. f If the teacher keeps on changing her mind like this, after 1, 10, 100, etc. years, //eventually she would end up somewhere near the middle.// e.g. as time approaches infinity, there would be a definite position, the middle in this case. i.e.
 * Thinking about Limits: The Delinquent Teacher (handout)**

After finding a couple of more bn, it appears that the number gets closer and closer to 1.618 i.e. //*1.618, , phi, the golden ratio!! More on Monday... //
 * Finding the Limit: The Fibonacci Sequence (handout)**

asdf

__**Example.**__ What happens to the function f(x) = x*x - 2 around x = 3 ? ﻿ --- approaching from the left/negative > f f f f f f f f f f f f f f f f f f f f f f                     < from the right/positive f f f f f f f f f f f f f f f f f f f f f f                      f f f f f f f f f f f f f f f f         f f f f f f   f f f
 * x fad || fd 2 fa || fd 2.5 || fd 2.9 || f f f  2.99 || f f  2.999 || fa 3 f f f   || f f  3.001 || f f  3.01 || f f  3.1 || f f  3.5 || f f  4 ||
 * f(x) fa || fd 2 || fa 4.25 fa || fa 6.41 fa || f f f  6.9401 f || f f  6.9940 f || f f f f f     7 || f f  7.0060 f || f f  7.060 f || f f  7.61 f || f f  10.25 f || f f  16 f ||

** >>> For a limit to exist, it must be the same when approached from both the negative and positive sides <<< **
This last example seems pretty obvious and the limit seems kinda useless, but if the situation changes a little, this could change: Here's a very simple introduction to limits (emphasis on the second example) media type="youtube" key="W0VWO4asgmk?rel=0" height="327" width="416" [] //originally from Calculus, Khan Academy// []
 * Khan Academy is a very useful website, teaching from a very shallow and understandable basis

what's the limit of f(x) as x approaches 0? //Since the limits are different from the two sides, therefore the limit as x approaches 0 __does not exist.__//
 * __Example.__**

//Going through some other simple examples from Khan Academy...// Part1 [] 2 [] 3 []

**__CONTINUITY__**
If the left-hand limit and the right-hand limit are the same, then the function is said to be ** continuous **. Three types of discontinuity: ** infinite ** (asymptotes), ** jump **, and** removable **(a hole).

A function is continuous, if... 1) could be drawn without taking the pencil off the paper 2) exists. 3) if f(x) always exists (no undefined, no hole).


 * __Homework:__** intro of limits handout

=**__Reviewing Limits__**= Monday, February 14, 2011, by Joel Tham

A limit is a tool in math that allows us to figure out or predict the behaviour of a function as x approaches some value.



In the above equation: //a// represents a real (ℝ) number (ie. 2, -5, 0) //L // is the value of the limit *To exist, a limit must have a ℝ answer* If the limit is ±∞, we say that the limit approaches (or goes to) infinity, but it doesn’t actually exist. Example:  If we’re lucky, we can find the limit of a function by straight substitution. That means that the function exists at the value of //x // = //a.// Example: Example: *A function does not need to exist at //x// = //a//<span style="font-family: 'Cambria Math',serif;">; there can still be a value to the limit.*

<span style="font-family: 'Cambria Math',serif;">*For a limit to exist, if we approach from the left or the right, we must get identical answers.* <span style="font-family: 'Cambria Math',serif;">The following page has an animation that explains when limits exist: <span style="background-clip: initial; background-origin: initial; background-position: 100% 50%; cursor: pointer; padding-right: 10px;">[] <span style="font-family: 'Cambria Math',serif;">This site has more videos explaining concepts in this chapter: []

<span style="font-family: 'Cambria Math',serif;">The 2 magic words of calculus:
<span style="font-family: 'Cambria Math',serif;">*approximation <span style="font-family: 'Cambria Math',serif;">*approach

<span style="font-family: 'Cambria Math',serif;">Limits and Discontinuity:
<span style="font-family: 'Cambria Math',serif;">For a function to be continuous: <span style="font-family: 'Cambria Math',serif; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">

Homework:
Ski Boat Limits Assignment (due Wednesday, February 16)