I spy with my little eye...MATH!

I looked and looked until I saw
what to my eye looked like a little fly,
but when I took a second look
the functions opened like a book,
This here is what I found
after a lot of looking around;
A function hiding in front of my face,
A function hiding with little trace...

There was one above my head, And it was invertible!




There was one hiding beneath the bed, And it was not a one-to-one function!





There was one hiding on top of her head, And it was continuous but not differentiable!





There was one hiding on his head, And it was differentiable and continuous!




There was one hiding in the lead, And one that was not continuous or differentiable!




And that is all I saw that day,
Which very much led me astray...

Choices, choices, choices...

Howdy,

So my choice tonight was...to do my homework! Starting with...

1. For any point (a,b) on an invertible function, the derivative of the inverse of the function evaluated at b is equal to the reciprocal of the derivative of the function evaluated at a.
2. (Sorry but I couldn't figure out how to do a -1 for the inverse)

3. Yes, volume is an invertible function.

4.


5.

black=(4/3)3.1415x^3
blue=(x/(4/3(3.1415)))^(1/3)
red=12.566(x-1)+4.189
green=(1/12.566)(x-4.189)+1

6. When graphing this the slopes of the tangent lines were 12.566 and (1/12.566), these are the red and green lines on my graph.
7. I learn much better visually and socially, that is why I decided to come to class. I don't think that I could learn as effectively on my own, so if there is ever a choice again I bet you $5 that I will be in class, unless I'm sick.

Scribe Post: Derivative of the Inverse of a Function

To get the exploration for the class go to the website to "Student Handouts" and you can find it under chapter 4, it is called "Exploration 4-5A_Derivative of the Inverse of a Function_10.pdf".

So in the last class, which was Virtual Wednesday, we began exploring the Derivative of the Inverse Function. There was a choice of whether to be in class or not, so how did your virtual calculus class go?

From: http://teamsubmarine.files.wordpress.com/2009/07/virtual-reality-helmut.jpg

I hope it went well and if it didn't then I hope I can help!

Our Objective: Discover the unique relationship between the derivative of a function and the derivative of the inverse of the function.

Part I.
To begin the lesson we reviewed.
Inverse of a real number -

• An additive inverse, known as the opposite of the number. ex. -3 and 3
• A multiplicative inverse, known as the reciprocal of the number. ex. ⅓ and 3

Inverse Operations - an Inverse Operation is used to "undo" the other operation.

• Subtraction is the inverse of Addition
• Roots are the inverses of Squared Roots
• Sin-¹(x) is the inverse of sin(x)
• Multiplication is the inverse of division
• And logarithms are the inverses of power functions

And Inverses of a Function! - How to determine if a relation is the inverse of a given function.

• Numerically - the X and Y values are interchanged in a table
• Graphically - The reflection over the line y=x
• Algebraically - By using the inverse operations

So now vocab test...If a function's inverse is also a function it is _______
(look at the bottom for the answer)

That means that it is a one-to-one function. So remember how a function is "for every x there is a unique y"? A one-to-one function just switches the x and y, so it is "for every y there is a unique x".
If you are ever wondering if a function's inverse will be a function just do the horizontal line test, which will tell you if the function's inverse will or won't be a function.

Part II.
So the second part to all of the review of inverse functions is the derivative of the inverse of a function.

The problem that I found most helpful for learning this is the first on that part of the exploration:

y=3+4x

now find it's inverse function,
to do that interchange the y and x, x=3+4y
now solve for the new y, y=(x-3)/4 or y= (1/4)x-(3/4)

Since we now have the inverse function we can find the derivative and the derivative of the inverse function.

f '(x)= 4 and (f -¹)'(x)= (1/4)

And these are...RECIPROCALS!!!
after doing many more like these we came up with our own conjectures of what this all means...

My Conjecture: The derivative of a function is the reciprocal of the derivative of the inverse function.

General Formula for the Derivative of the Inverse of a Function:
In symbols: If f(x) is an invertible function then for any point (a,b) on f(x)
In words: For any point (a,b) on an invertible function, the derivative of the inverse of the function evaluated at b is equal to the reciprocal of the derivative of the function evaluated at a.

Here in the slide show is the general formula for the derivative of the inverse of a function. I think this is really helpful to go over just to get it in your head.

CRMS Calculus 2010 January 20, 2010

View more documents from Colorado Rocky Mountain School.

This is what we did in class, the remainder of the exploration was homework and here are a few things just to help out,

1. To write the equation of the tangent line use point slope form: y=m(x-h) + k where (h,k) are your points and m is your slope.
2. Vocab - Mirror points, means if you have points (a,b) the mirror points are (b,a).
3. Always check your answers, either on nDeriv or just look at the slides from class.
4. Or we could just say "Look before you leap..."
   

From:http://victorysugar.sulekha.com/mstore/victorysugar/albums/default/look%20before%20u%20leap.JPG

Because if you don't that fish could be laughing at you.

Great, well I hope this was helpful for everyone, please leave me comments on what worked and what didn't!

Beston

Oh and the next scribe is Flying Slug :)

vocab answer= invertible

Cusping Things

1. In the introduction to our weblog it was suggested that because of Web 2.0, “Anyone can learn anything, from anyone, anywhere, at anytime”. Do you agree or disagree with this statement? If you agree, then what role do schools and formal education play in your learning? If you disagree, then what gets in the way of learning?

I agree with the statement “Anyone can learn anything, from anyone, anywhere, at anytime”, as a general statement. I think there are exceptions but that may be because I have not taken advantage of the internet completely. Since the internet does have the ability to screen people and hide them behind pseudonyms, how can anyone really trust anyone else? The human relationships that are built with face to face contact have a substantial and real feel about them. The relationships formed online though are doubtful. That lack of commitment on the internet reminds me of one of the reasons I enjoy school, the relationships, saying "hi" on the path, seeing people you know and care about. The internet doesn't give you that. I don't think it ever can. The commitment of being somewhere at a certain time and place to be with certain people and interacting on many levels is something that I don't think the internet can ever achieve. School is not only social though, I think that in school I understand most concepts and difficult problems through my peers. When they show me a problem by going over each step with me, I learn. This is something that the internet can have as well, but if people were not interacting in this way in the real world, then they would not know how to communicate on the internet. Motivation and competition are more qualities that school holds over the internet. Without the beginning nudge from our teachers I doubt that I would research what I do and if my peers weren't there we would not have vocabulary competitions in Spanish. All in all I feel that the internet can be anywhere, for anyone, at anytime or place, however if we do not hold up the day to day interactions that we have with each other we would lose a great deal.

The Product Rule

So the online source that I found was from HMC and was short, succinct, and easy to read. The examples were really easy for me to understand and the explanations of the Product Rule made sense. I like this better than the textbook which seemed to me to be longer than it needed to be. The "fairly complicated algebra" passed me by and I found that the Property section was all I needed to understand the rule. I definitely feel that the online source was more useful to me.
If I had wanted all the long algebra they gave the link, just in case.

This definition and example come from

http://www.math.hmc.edu/calculus/tutorials/prodrule/

Definition:

h(x)=f(x)g(x)=f(x)g(x)+f(x)g(x)

Example:

If h(x)=x2sinx then

h(x) = =2xsinx+x2cosx = (x2)sinx+(x2)(sinx)

COOL Mind Map