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
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