Did You Ace the Calculus Calculation Contest as a Sophomore?

[JWHooper]

I'll work on it, and I'll show my time.

 

[JWHooper]

65 seconds.

 

[Crosson]

I voted for 10 seconds. Using binomial coefficients this derivative is just a matter of writing the answer:

http://en.wikipedia.org/wiki/Leibniz_rule_(generalized_product_rule)

Admittedly, I didn't simplify it, in which case I would fall into 10-60 seconds.

 

[Gib Z]

Is there a "I think it's a waste of time" option?

 

[ice109]

is there a book somewhere of all these cute little tricks like the generalized product rule? and don't go telling me that i should be able to figure em out on my own.

 

[Crosson]

I don't know about an entire book devoted only to tricks, but I first learned that one from Mary Boas' Mathematical Methods for the Physical Sciences. I think you will find physicists use these kinds of tricks most often; it has become common in the math department to say "here are the basic rules, specific examples are a waste of time."

Is there a "I think it's a waste of time" option?

Be careful, teaching introductory calculus is the main source of employment for math PhDs (from the perspective of university administration). Calling any part of math "a waste of time" is a slippery slope, since it quickly becomes hard to justify any of it as time well spent.

 

[Gib Z]

Finding the fourth derivative of some product is really quite pointless though no? The vast majority of geometrical and physical applications require only up to the 2nd or 3rd derivative at most.

PS. It takes me 20 seconds if you let me leave the answer in series form. Don't ask me why I did it.

EDIT: ice109, here's one: https://www.physicsforums.com/showthread.php?t=206039

 

[ice109]

Finding the fourth derivative of some product is really quite pointless though no? The vast majority of geometrical and physical applications require only up to the 2nd or 3rd derivative at most.

PS. It takes me 20 seconds if you let me leave the answer in series form. Don't ask me why I did it.

EDIT: ice109, here's one: https://www.physicsforums.com/showthread.php?t=206039

do you read the posts in the threads you post in? did you not see the link to the leibniz identity like 3 posts back?

 

[BrendanH]

Does anyone dare give their answer?

 

[Gib Z]

do you read the posts in the threads you post in? did you not see the link to the leibniz identity like 3 posts back?

Do you read the links posted? That identity is in terms of lower derivatives, but derivatives still. What i meant was the write the sine term in its series form, multiply through by the x term and finding the 4th derivative of the resulting series.

 

[Crosson]

Finding the fourth derivative of some product is really quite pointless though no? The vast majority of geometrical and physical applications require only up to the 2nd or 3rd derivative at most.

PS. It takes me 20 seconds if you let me leave the answer in series form. Don't ask me why I did it.

The irony is that the direct way to calculate coefficients of a power series is by evaluating higher order derivatives! Of course this is almost never done in practice since we keep working with the same 10 functions who p-series we know by heart, but Liebniz's formula is very helpful once we venture out of the familiar functions, e.g.

Prove:

$$\sqrt{\frac{\pi }{2 x}} J_{\frac{1}{2} (2 n+1)}(x)=x^n \left(-\frac{x^{-1} d}{ dx}\right)^n \frac{\sin (x)}{x}$$

where the bessel function $J_p(x)$ is given by:

$$J_p(x)=\sum _{n=0}^{\infty } \frac{(-1)^n \left(\frac{x}{2}\right)^{2 n+p}}{\Gamma (n+1) \Gamma (n+p+1)}$$

If I remember correctly I spent ~2 hours working on this problem as a sophomore, and along the way I used a lot of tricks; Liebniz's rule was essential, and the most difficult part was getting the factorial terms to match up.

 

[daveyinaz]

As a sophomore? Man, you must have gone to a better school than I did.

 

[Crosson]

As a sophomore? Man, you must have gone to a better school than I did.

Not really, I went to a fourth tier state school in my home town. Fortunately I had a good physics professor, and there was another good student at that time as well. That problem was given to us as part of a take-home test in the math methods course that used the Boas book I mentioned earlier. Most classmates showed that the first few terms of the two series were equal, and that received full credit! What a strange education I've had...