Need a little help from gifted math people...

Hi all,

I've tried solving the following example, I think I'm on the right track but I need reassurance and help simplifying the result, because my ever so strict professor picks random people to show their solution at the blackboard, and I absolutely dread this.

Here it is:

You have the function f(x,y,z)= x^2+2yz+y^2

Imagine that you travel in the xyz-room among the following lines

x(t)= 0.5 - (sqrt(3)/2)t

y(t)=sqrt(3)/2+0.5t

z(t)= 1

During your travels you continously get the values of the function f.

a) State the function that shows how the functionvalues of f depend on t

*I think that means I should put the equations of x(t), y(t) and z(t) into *f(x,y,z)= x^2+2yz+y^2

Am I correct? This gives me:

f(x,y,z)=(0.5 - (sqrt(3)/2)t)^2 + 2(sqrt(3)/2 + 0.5t) + (sqrt(3)/2 + 0.5t)^2=

t^2+ t+ 1+ sqrt(3)

Is this right..?

In the next question he asks how fast the function value increases/decreases when we pass the point that is prevailing when t=0

So, when t = 0: x= .05, y=sqrt(3)/2, z=1

They're looking for the "direction-derivative", how do I find the answer to how fast the function value increases/decreases when we pass the point that is prevailing when t=0..?

Please help me out, I dont want to make a fool out of myself at the blackboard!!

Thanks so much