Choices

Derivative of the Inverse of a Function:

1. If f(x) is an invertible function, then for any point 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. If f(x) is an invertible function, then for any point (a,b) of f(x):

3. The volume of a sphere is a function of its radius:

Is volume an invertible function?

”Why yes indeed it is”

4. Inverse Function

5. Graph of:

(2) Choose any point (1,4.188) on V(r) and graph the tangent line through this point (Blue)(1) The volume function, V(r) (black)

(3) The inverse function r(V) (red)

(4) The tangent line through the “mirror” point on the graph of the inverse function. (Green)

6. OH MY! The tangents are reciprocals!

Function of the tangent line for V(r) at point (1,4.188)

Function of the tangent line for r(V) at point (4.188,1)

7. I am a strong believer in attending Calculus. I like having the opportunity to ask questions an see what specifically I have trouble understanding as well as listening to a variety of explanations from Bru and other students. I also seem to struggle with the Internet and technology in general so good old-fashioned pencil and paper helps me take in the material much better. Lastly, I am a procrastinator/ minimalist, so having the time to sit down in class provides a much more productive learning experience for me.

A Very Late Choice Indeed

Derivative of an Inverse Function

The property is simply the reciprocal of the derivative of the inverse of the original function. In layperson terms, it is simply one divided by the derivative of the inverse function. Here it is in symbols to hopefully make it clearer.


This is the inverse of the volume function:

Below is the picture and description of the following functions.
(1) The inverse volume function r(V);
(2) The volume function, V(r);
(3) x=1 on V(r) and graph the tangent line through at this point.
(4) The tangent line through the “mirror” point on the graph of the inverse volume function.

The first function is the inverse volume function which is the gray function in the image of the graph. The second function is the original volume function which is the red function in the image of the graph. The third function is the tangent line to the volume equation at the point x=1. The forth function is the tangent line to the inverse volume function at the mirror point.

To find the function of the tangent line to point x=1 for the volume function you can simply type in the volume function into your calculator and use nDeriv(y1, x, x) to find the derivative at that point. Use the point slope form of a linear equation to form the tangent line at the point x=1 for the volume function. Use the derivative at the point x=1 for the volume function as the slope of the tangent line. Then insert the values of the volume function for x and y when x=1 into the point slope form as x and y. Aha, now you have the slope of the tangent line at the point x=1 for the volume function!

The function of the tangent line of the mirror point for the inverse volume function is found similarily to the previous tangent line. Because we know that this function is the inverse to the volume function the slope of the tangent line through the mirror point will be the inverse of the slope of the linear function of the tangent line to the volume function when x=1. Also we know that the x and y value at the point x=1 will be switched around for the tangent line to the inverse volume function at the mirror point. This is all that you have to do to find the function of the tangent line at the mirror point while utalizing point slope form.

I choose not to go to class and to take the virtual class because I wanted to try something new. Honestly I am quite biased and believe that you learn much better in the classroom instead of on your own by using the internet. I wanted to see if the classroom actually did have a big advantage.

Choices

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. In other words, the derivative of the inverse of a function is the reciprocal of the derivative of the function.
2. In symbols:


3. The function for volume of a sphere is below:

Yes, volume is an invertible function.

4.The inverse of this function:

5.


6. The slope of the tangent of the red function is 11.36, and I found the slope of the green function is .088, which is equal to 1/11.36.
7.I chose not to go to class. I've never done a virtual class before, so I wanted to see if I could effectively learn the class material without actually going to class. Also, by taking the virtual class I was able to sleep in, (this was really just a nice bonus). By utilizing my tools on the internet, I was able to learn the subject matter. The scribe post was also a very helpful tool. Overall, I think I was able to learn just as well in the virtual class, but had this been a more complicated, difficult topic, it's definitely helpful to have a teacher who knows what they are talking about.

Forum Post: Choices

1. For any function that is invertible, the derivative of the inverse of a function at a point is equal to one over the derivative of inverse of the original function.

2.




3.





Is volume an invertible function?
Yes Volume is an invertible function.

4.





5. Using the online graphing tool (right sidebar) or another graphing tool, plot the following four graphs:

(1)





(2)




(3)





(4)





















6. I found the derivative of (1,4/3pi) and then (4/3pi,1), yielding slopes of 12.566 and 1/12.566. The slopes of the two tangents are recipricols of one another.


7. I decided to come to class due in large part to the fact that I felt that I would benefit the most from being able to participate in my education in person and would be able to activitly pursue and ask questions of what we were doing and how to do them. As well I felt if necessary the web material, the virtual classroom would be avialable to me after class to furhter expand upon my knowledge.

Choices

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. In symbols this is:

3. The volume of a sphere is a function of its radius:

This is an invertible function because it passes the horizontal line test.

4. The inverse of this function is:


5. Here is a graph. The Black line is the original function, f(x) for the volume of a sphere. The Red line is a line tangent to f(x) at 1. The Blue line is the inverse of the original function, f^-1(x). The Green line is a line tangent to f^-1(x) at 4.189.


6. The slope of the tangent of f(x) is 12.566 and the slope of the tangent of f^-1(x) is 0.079 because 0.079=1/12.566

7. I came to class on virtual Wednesday because I really value a classroom environment for learning. I find i rely heavily on Bru and other classmates to be able to fully understand important parts of a lesson. I felt that in a virtual section I would get very frustrated because when I had questions I would not be able to get them answered to my satisfaction.

Derivative of the Inverse of a Function!!!

Derivative of the Inverse of a Function

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:

Is volume an invertible function?

Yes volume is an invertible function because its inverse is also a function:

Tangent line of volume function: 12.566(x-1)+4.1888

Tangent line of inverse volume equation: (1/12.566)(x-4.1888)+1

(they are reciprocals!!)

I chose not to attend class on Wednesday for two reasons, one, because I wanted to sleep in (sorry Bru!). But more importantly, because after some thought, I realized that most of what we do in class is just taking notes and filling in blanks in our explorations, and since I could see all the notes from class online, I thought it would be just as easy to fill out the exploration myself at home. Without the help of my classmates, I had some trouble with the equations and understanding exactly what was going on, but after I read the scribe post of that day I understood everything. I feel like I completely grasp this concept now. I really feel like both options are equally educational, going to class you get more audio and interaction if you need help, but the virtual route provides a little more of a challenge for the student to figure out the math on their own. The struggle of making sure I was doing everything right helped me learn the material better.

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.

Choices

1) The Derivative of the Inverse of a Function says that for any point (a,b) on an invertible function, the derivative of the function's inverse evaluated at point b is equal to the reciprocal of the derivative of the function evaluated at point a.
2)



3) Volume is an invertible function
4)



5)


a)(4/3)*3.141592654x^3
b)12.566375(x-1)+4.1887902
c)(x/(4*3.141592654/3))^(1/3)
d).09403455(x-4.1887902)+1






6) My point (1, 4.1887902)
slope of tangent line at f(1)=12.566375
slope of tangent line at f'(2.3562)= 0.09403455
12.566375*0.09403455=1

7)I chose to come to class on Virtual Wednesday because I feel as though I learn better in a classroom environment. It provides structure and an atmosphere in which it is easier for me to focus. The resources of Bru and my classmates are of great help when I don't completely understand a concept. Learning from Bru and my classmates in a classroom as opposed to an online classroom is a much more reliable method of learning for me and is a more concrete way to learn the material. I have the opportunity to ask questions and am provided with a structured and complete lesson.

Derivatives and Inverse Derivatives

1. For the point (a,b) on the x,y plane the derivative of the inverse function derived at point b is the reciprocal of the derivative of the function at point a.

2. If F(x) is an invertible function, then for any point (a,b) on F(x):


3. The volume equation is an invertible function because its inverse is a function.

4. This is my thought process:




5.
Black is the original volume function, red is the inverse of that, blue is the derivative of the original function at x = 1, and green is the derivative of the inverse at point y = 1
The functions I used were as follows:
Y = (4/3)*pi*x^3
Y^-1 = ((3x)/(4*pi))^(1/3)
Y' = 4*pi*(x-1)+((4/3)*pi)
Y'^-1 = ((1/(4*pi))(x-((4/3)*pi)))+1

6. 1/(4*pi) = the inverse of 4*pi

7. I came to class on Wednesday because I would not have learned the same amount on my own. I needed the review that we did in class and I learn better with interaction. I also probably don't have the initiative to do the work on my own, so it is good that I came to class because otherwise I might not have studied thoroughly.

Choices, Choices

1. Derivative of the Inverse Function: 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. In Symbols:

3. Volume of a Sphere:

Invertible? YES!

4. Inverse of a Sphere:

5.

Red:

Blue:

Green:

Orange:

6. (1/4pi) is the inverse of 4pi

7. Why Class? I chose to come to the class room because I know that otherwise I wouldn't have had the complete focus I have in class room setting. I know that, me personally, I need to have visual immersion in a subject, which class can do, and I even sometimes lose that focus in class, without all the distractions the online world has.