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.
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I liked how you added colors and that one picture, to make your post a bit more interesting. Also, I thought your response to number 7 was a good point, because while some people are able to miss class and do their work on the internet, there are definitely some cases,(like your own), where people just have to be in class in order to learn the material,and/or not fall behind.
ReplyDeleteHey Tubby, a few quick questions.
ReplyDelete1. In number 1, what are a and b?
2. In number 2, do you mean f inverse?
3. What does it mean to be a minimalist? I'm fascinated.
a few quick answers
ReplyDelete1) fixed (im pretty sure)
2)i think thats what i meant, my packet ha what looked like a 1 in parenthesis with f, i think thats what your commenting on (i can be a little dense though, that is what you mean right, haha) should it not be there?
3)ah, we great minimalists are sometimes mistaken as lazy, but in fact i would say that we are time efficient. we finish what needs to get done, but only further pursue the things that make us most happy (skiing, hockey, music etc.) hence minimalists manage their time to ensure maximum awesomeness, unfortunately, for me calculus is not something i further pursue on my own, which is why i find it is best for me to attend class.
thanks for all the good advice!
2) I think the one should be a negative one (that's how we write f inverse f to the -1).
ReplyDelete3) being a minimalist sounds like a good idea. in fact, its something a lot of mathematicians try hard to be. one aspect of a "good proof" in math is if it is elegant - if it is short and sweet and gets to the point gracefully. sounds like you're on the right track.
2) yes your right it should be a negative one.
ReplyDelete3)why thank you very much : )