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

- 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

- 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

(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.

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.

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

- 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.
- Vocab - Mirror points, means if you have points (a,b) the mirror points are (b,a).
- Always check your answers, either on nDeriv or just look at the slides from class.
- Or we could just say "Look before you leap..."

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

This scribe post was clear and concise . I chose to be a virtual student on wednesday and this post made the derivative of the inverse function understandable. The colors emphasized key words/details and I also enjoyed the cartoon at the bottom. Ill definitely check my answers from now on.

ReplyDeleteBeston,

ReplyDeleteYou have done an incredible job on this Scribe Post. Your classmates who chose a Virtual Classroom will feel as if they had been in class with you. They will certainly benefit from what you have scribed here! Your summary of the lesson is clear, complete, and concise. Your use of fonts and colors to highlight important points is both visually stimulating and strategically placed. Your creativity is apparent throughout the post (photo, cartoon, and vocab. quiz) and adds to the reader’s interest and understanding of the lesson. I hope that you had half the fun writing this as I had reading it. This Scribe Post will be tough to top!

This scribe post helped me a lot since I took virtual class on Wednesday. It provided clear and concise description of what Bru covered in the class. Also it was hard for me to figure out what should be filled in the blank of our work sheet, and this Scribe post you posted helped me in doing the work. Pictures are interesting and fits perfectly into what we have been planned about virtual class as a class.

ReplyDeleteThis was a fantastic post. It covered what needed to be covered and did so in a way that was both easy to understand and follow. I like the use of pictures to really drive the important points home. I only have two possible suggestions. First, for people who were not in classy, maybe it would be helpful to describe what the horizontal line test is (and also the vertical). And the use of equation editor could make things clearer. But other than that, great work! (ps, i got the quiz right!!!)

ReplyDeleteHey Beston,

ReplyDeleteThis was a very helpful scribe post for all who chose to take the virtual class last Wednesday. Your scribe post at first appeared quite daunting due to the length, but it proved to be a rather quick read packed with all of the necessary information. Your post was informative and light hearted (always a nice touch in calculus if I may say so myself). As dammitmmad said earlier, it would be great if you could use the equation editor throughout your post, although I know it can be time consuming. I struggle with it myself quite often.

Great job Beston!

mc Casper

This post was awesome and super helpful! You gave a great description of everything we did in class, the review helped me a lot on the homework. what i loved the most was your homework tips, they helped me out a lot while i was finishing up the packet. I could have used a little more help on some of the more challenging problems, maybe an example, but the tips were a great idea. Otherwise loved the pictures and the comic!

ReplyDeleteHi Beston,

ReplyDeleteYou did an awesome job on your scribe post. The post is separated into two different parts. The first part is review of the inverse; and the second part is the some real calculus. The entire post is really well organized. All the important points were highlighted with different colors. All the awesome pictures provide us a great sense of the concept. I was in that class on Wednesday. So I don’t think you missed anything that we had talked about in the class. Great job.

However, don’t you feel this post is really, really strong at the beginning, but really weak at the end? What I mean is that the introduction is awesome, but the conclusion is not consistent with the introduction. Everything is just incredibly good beside the final. After I read the post, I just don’t feel that the post is ended. I felt there was something missed. I think if you added the definition of derivative of the inverse of a function, your post would be arbitrarily close to perfect, since nothing is perfect. It is not hard to find the definition of derivative of the inverse of a function. And it has people to check whether their conjectures are right or wrong.

Over all, I like this post a lot.

This was a wonderful scribe post. As somebody who chose to go the virtual class, this scribe post was a very helpful tool in my understanding of the derivative of the inverse of a function. I liked how you first referred to where I could find the exploration, so that I had a reference. Your use of colors and pictures added a creative aspect to your post, one that made it more interesting and engaging to read. Your post was very well organized, which also made this topic easier to understand. I like the way you first reviewed inverses and then went into finding the derivative of the inverse. The example was also a very helpful one. Overall this scribe post is a great review of the lesson, and a fun, creative, and helpful tool.

ReplyDeleteThis Scribe Post is great! The two images you included were quite intriguing and added to post, making it more visual. I like how you organized the post, dividing it into parts and using colors to accent important concepts. The 4 things at the end really helped me especially on the homework. As someone who came to class on Wednesday, I found this very helpful in terms of review. My only suggestion would be to use equation editor, but other than that, awesome job!

ReplyDeletehi,

ReplyDeletethank you all so much for commenting and suggesting some corrections. I will keep it in mind for my next posts that I should use more examples and the equation editor. Special thanks to YDplusSB, i knew there was something missing but I couldn't place it. You are right on about putting the definition of the derivative of the inverse of a function, I will fix that immediately!

thanks,

Beston

Hi Beston,

ReplyDeleteYour post is detailed and very clear. I agree with the previous posts that your use of color was helpful, and that the post would have been even more professional looking if you used equation editor.

Also, there may be a small error in the beginning of your post. In your review explanation of inverse functions, you state that "Roots are the inverses of Squared Roots". I'm not sure what you mean here, but I'm guessing that you want to relate the operation of squaring a number with the operation of taking the square root. Is that correct? If I misunderstood your intent, then it might help to clarify your terminology in that line.

Great work!

SKS

Hey there,

ReplyDeleteIs your name Beston because your are indeed the best!? Great scribe post :). I like the picture at the top of your scribe post too, very interesting and creative. Overall a masterpiece indeed, is there a lourve for mathematics? Because this belongs there.