# Calculus problems, where to begin?

Hello! I have a math problem (mostly proofs) that im stuck
on, partially because I do not know where to begin and partially
because I believe I dont even fully understand the problem. I
was wondering if any of you would be kind enough to show me what to
do? Thank you.

1. Suppose f(x,y) is differentiable for all (x,y), f(x,y)=17 on the
unit circle x^2+y^2=1 and grad f is never zero on the unit circle. For
any real number K, find a unit vector parallel to grad
f(cos(k),sin(k))....grad f stands for the gradient of f. But isnt it contradicting what its saying? It says f(x,y)=17 on the unit circle x^2+y^2. How the...?

I'm just supposing f(x,y) is differentiable for all (x,y), f(x,y)=17 on the unit circle x^2+y^2=1 and grad(f) is never zero on the unit circle(?) So you just find a unit vector parallel to grad f(cos(k),sin(k)), for k real, right?

PS- Do level curves apply to this problem?

Hello Jeebus,

You're being asked to find the direction of &nabla;f on the unit circle (k is just an angle). I think it's easier if we use polar coordinates (r,&theta;), and their corresponding unit vectors r and &theta; (don't know how to make the little hats yet). If we look at the gradient on the circle, and dot it with &theta;: &theta; dot &nabla;f, we get the rate of change of f in the &theta; direction. But what is the rate of change of f in the &theta;-direction on the unit circle? And what can you conclude about the direction of &nabla;f from this?

Hope this helps,
dhris

HallsofIvy