In this activity, we will look at level surfaces created by two relatively basic three variable functions and consider the direction in which the output is increasing or decreasing as fast as possible.
What shape and characteristics will the level surfaces of \(f(x,y,z)=2x-y+z\) have? You should talk about all of the possibilities for the different types of level surfaces and compare what is the same or different about these level surfaces (relating the differences to the level value, \(k\)).
If you are at the point \((1,1,-1)\text{,}\) in what direction should you change the input of \(f(x,y,z)=2x-y+z\) to see the greatest increase in the output of \(f\text{?}\) You should think about how a change in each variable will change the output of \(f\) and talk about this in your reasoning.
What shape and characteristics will the level surfaces of \(g(x,y,z)=x^2+y^2+z^2\) have? You should talk about all of the possibilities for the different types of level surfaces and compare what is the same or different about these level surfaces (relating the differences to the level value, \(k\)).
If you at the point \((2,0,-2)\text{,}\) in what direction should you change the input of \(g(x,y,z)=x^2+y^2+z^2\) to see the greatest increase in the output of \(f\text{?}\) You should think about how a change in each variable will change the output of \(f\) and talk about this in your reasoning.
