NS110 Lecture : Early Canadian Nation-State

For each of the following vector fields FF, decide whether it is conservative or not by computing curl FF. Type in a potential function ff (that is, ∇f=F∇f=F). Assume the potential function has a value of zero at the origin. If the vector field is not conservative, type N.

F(x,y,z)=6xi+7yj+k
f(x,y,z)=f(x,y,z)=

equation editor

Equation Editor

D. F(x,y)=(6siny)i+(−6y+6xcosy)j
f(x,y)=

Let b > 0 and f : [0, b] → R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n ∈ N, there exists some c ∈ (0, b) such that cf ′(c) = −nf(c).

Hint: Apply Rolle’s theorem to the function g(x) = xⁿ f(x).