Brain Teaser - Area enclosed by a parabola and a line

[Thomas Klemm]

(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.

Quote:Thanks for posting this interesting problem!

1+

Cheers
Thomas