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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §3.2 Maximum and Minimum Problems, #58 · Calculus · Optimization

The figure for this exercise is in the book and isn't reproduced…

Fermat's principle says that light travels from $A$ to $B$ on the quickest path. Its velocity above the $x$ axis is $v$ and below the $x$ axis is $w$ .

(a) Find the time $T (x)$ from $A$ to $X$ to $B$ . On $A X$ , time = distance/velocity $= r^{2} + x^{2} / v$ .

(b) Find the equation for the minimizing $x$ .

(c) Deduce Snell's law $(sin a) / v = (sin b) / w$ .

"Closest point problems" are models for many applications.

The figure for this exercise is in the book and isn't reproduced here.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §3.2 Maximum and Minimum Problems, #58

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