MATH 411 Lecture : MATH411_BOYLE-M_SPRING2012_0101_FINAL_SOL
Document Summary
Math 411 spring 2012 boyle final exam solutions. Points in rn are column vectors (even if they are typed horizontally): (a) (15 pts. ) De ne what it means for f to be di erentiable. For every x in rn, there exists an m n matrix a such that. F (x + h) f (x) ah. ||h|| (in this case a is the derivative of f at x. The matrix a must be the matrix df (x) whose i, j entries are the partial derivatives. Suppose f : rn r all partial derivatives are continuous functions. We will show that the derivative of f at x is the 1 n matrix df (x) (the transpose of the gradient vector). Mean value theorem, given h there is a point ph which is a convex combination of x and h such that f (x + h) f (x) = df (ph)(h). ||f (x + h) f (x) df (x)h||
