Brain Teaser - Area enclosed by a parabola and a line

[Gerson W. Barbosa]

(09-16-2015 05:12 AM)Thomas Klemm Wrote:  

(09-16-2015 02:06 AM)Gerson W. Barbosa Wrote:  The equation of the normal line shouldn't be difficult to derive, but it's ready for use on page 70:

\[y-y_{1}=-\frac{1}{\frac{dx}{dy}}(x-x_{1})\]

There's a typo: \(dx\) and \(dy\) got switched. The correct formula is:

\[y-y_{1}=-\frac{1}{\frac{dy}{dx}}(x-x_{1})\]

But then this notation can be misleading as the derivative should be evaluated where \(x=x_1\).
We could use \(\frac{dy}{dx}\Big|_{x=x_1}\) however I prefer to use \(f'(x_1)\).

Quote:For the case y = x^4, we have

\(y-y_{1}=-\frac{1}{4x^{3}}(x-x_{1})\)

The variable in the derivative should be \(x_1\):

\(y-y_{1}=-\frac{1}{4x_1^{3}}(x-x_{1})\)

Quote:\((x_{1},y_{1})=(u,u^{4})\)
\(y-u^{4}=-\frac{1}{4u^{3}}(x-u)\)

Now we're back on the same road.

Thanks, Thomas, for pointing these out!
The example and the illustration in the book make it quite clear, but by looking at the formula alone that notation indeed seems to be misleading. In my hand-written solution above I used a capital X instead of x1, but that might be a bit confusing. I prefer your lower-case u as no subscript is required.
I'd revised the formula before posting, but I missed the dy and dx swap. Well, I typed that with the keyboard on my lap, the mouse sliding on top of a pile of books and the monitor far on on the floor, a totally anti-ergonomic way. (My wife reclaimed the sewing-machine stand I had been using for months, thus disturbing the progress of science )

Cheers,

Gerson.