MATH 411 Final: MATH411 BOYLE-M SPRING2013 0101 FINAL SOL

• School: University of Maryland
• Department: Mathematics
• Course: MATH 411
• Professor: All

Document Summary

Math 411 spring 2013 boyle final exam solutions. Solution. f (p + tu) f (p) t lim t 0 (b) use the chain rule to prove duf (x) = u f (p). De ne g : r r2 by g(t) = p + tu. = df (p))dg(0) (by the chain rule) u2(cid:19) (p)(cid:17)(cid:18)u1. We have ||u|| = r. by the cauchy-schwarz inequality. |u f (p)| ||u|| || f (p)|| = r|| f (p)|| . If t > 0 is a scalar and u = t f (p), then t = r/|| f (p)|| and. |u f (p)| = |t f (p) f (p)| = r|| f (p)|| and the upper bound is achieved. || f (p)|||| f (p)||2: de ne f : r2 r by setting f (x, y) = ex+xy+y2. Using the power series for the exponential, we have f (x, y) = 1 + (x + xy + y2) +