Problem 1. Let r : [0, 1] R n be a differentiable vector function. Assume that kr(t)k is constant on [0, 1]. Show that r 0 (t) is perpendicular to r(t) for every t [0, 1].

Problem 2. Let f : R n R be a function. Let a R n. Define what it means for f to be differentiable at a.

Problem 3. True or false? No justification required.

(a) Let f : R n R be a function such that all of its partial derivatives exist at some point a R n. Then, the function f is differentiable at the point a.

(b) Let f : R n R and g : R n R be functions that are continuous at some point a R n. Then, the function 3f(x) 2 + 4g(x) is continuous at the point a.

(c) Let f : R n R be a differentiable function and let a R n. Assume that f(a) = 0. Then, the function f has either a local minimum or a local maximum at the point a.

Problem 4. Let f : R R be a function of one variable and let g : R 2017 R be a function of 2017 variables. Assume that f and g are both differentiable. Let a R 2017 be a point such that g(a) = 1. Assume that f 0 (1) 6= 0 and that g x1 (a) = 1. Determine whether or not a is a critical point of the function f g.