February 8,2011 Marco Becerril Unit 1, Topic: What is Calculus? Text Refernce: Grade 12 Functions course. See handouts. Idea: An introduction to Calculus.
- What is Calculus? (1:32) http://www.youtube.com/watch?v=ismnD_QHKkQ
- Calculus in 20 Minutes! (20 mins) http://www.thinkwell.com/a/calculusin20minutes?utm_source=youtube&utm_medium=info&utm_campaign=calcin20min
http://www.thinkwell.com/a/calculusin20minutes?utm_source=youtube&utm_medium=info&utm_campaign=calcin20min
- 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
HOMEWORK; Due Friday February 11,2011: Sometimes Life is like a Rollercoaster
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
-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
-MPQ = 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
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
Review: Differences between instantaneous rate of change and average rate of change. 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
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 Thinking about Limits: The Delinquent Teacher (handout)
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.
Finding the Limit: The Fibonacci Sequence (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...
asdf
Example. What happens to the function f(x) = x*x - 2 around x = 3 ?
xfad
fd2fa
fd2.5
fd2.9
fff2.99
ff2.999
fa3fff
ff3.001
ff3.01
ff3.1
ff3.5
ff4
f(x)fa
fd2
fa4.25fa
fa6.41fa
fff6.9401f
ff6.9940f
fffff7
ff7.0060f
ff7.060f
ff7.61f
ff10.25f
ff16f
--- approaching from the left/negative ----> ffffffffffffffffffffff<---- from the right/positive ---- fffffffffffffffffffffffffffffffffffffffffffffff
>>> 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)
Example. 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.
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; there can still be a value to the limit.*
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.
February 8,2011
Marco Becerril
Unit 1, Topic: What is Calculus?
Text Refernce: Grade 12 Functions course. See handouts.
Idea: An introduction to Calculus.
- What is Calculus? (1:32)
http://www.youtube.com/watch?v=ismnD_QHKkQ
- Calculus in 20 Minutes! (20 mins)
http://www.thinkwell.com/a/calculusin20minutes?utm_source=youtube&utm_medium=info&utm_campaign=calcin20min
http://www.thinkwell.com/a/calculusin20minutes?utm_source=youtube&utm_medium=info&utm_campaign=calcin20min
- 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
HOMEWORK; Due Friday February 11,2011: Sometimes Life is like a Rollercoaster
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
-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
-MPQ = 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
- Average and instantaneous rate of change
http://www.youtube.com/watch?v=GUVFX5-SaTA&feature=relmfu (Part 1)
http://www.youtube.com/watch?v=ctV8MOiaOfA&NR=1 (Part 2)
http://www.youtube.com/watch?v=9-rnCKY98gI&feature=relmfu (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 WeiReview: Differences between instantaneous rate of change and average rate of change.
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)
asdf
LIMITS
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
Thinking about Limits: The Delinquent Teacher (handout)
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.
Finding the Limit: The Fibonacci Sequence (handout)
After finding a couple of more bn, it appears that the number gets closer and closer to 1.618
i.e.
asdf
Example. What happens to the function f(x) = x*x - 2 around x = 3 ?
ffffffffffffffffffffff
>>> 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)
http://www.youtube.com/watch?v=W0VWO4asgmk&feature=player_embedded
originally from Calculus, Khan Academy http://www.khanacademy.org/video/introduction-to-limits?playlist=Calculus
*Khan Academy is a very useful website, teaching from a very shallow and understandable basis
Example.
Since the limits are different from the two sides, therefore the limit as x approaches 0 does not exist.
Going through some other simple examples from Khan Academy...
Part1 http://www.khanacademy.org/video/limit-examples--part-1?playlist=Calculus
2 http://www.khanacademy.org/video/limit-examples--part-2?playlist=Calculus
3 http://www.khanacademy.org/video/limit-examples--part-3?playlist=Calculus
CONTINUITY
If the left-hand limit and the right-hand limit are the same, then the function is said to be continuous.A function is continuous, if...
1) could be drawn without taking the pencil off the paper
2)
3)
Homework: intro of limits handout
Reviewing Limits
Monday, February 14, 2011, by Joel ThamA 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; there can still be a value to the limit.*
*For a limit to exist, if we approach from the left or the right, we must get identical answers.*
The following page has an animation that explains when limits exist: http://www.calculus-help.com/when-does-a-limit-exist/
This site has more videos explaining concepts in this chapter: http://www.calculus-help.com/tutorials
The 2 magic words of calculus:
*approximation*approach
Limits and Discontinuity:
For a function to be continuous:
Homework:
Ski Boat Limits Assignment (due Wednesday, February 16)