In this activity, we will look at level surfaces created by two three-variable functions and consider the direction in which the output is increasing or decreasing as fast as possible.
Write a couple of sentences to describe shape and characteristics will the level surfaces of \(f\text{.}\) Be sure to include all possibilities for different types of level surfaces. Compare what is the same or different about these level surfaces and relate the differences to the level value, \(k\text{.}\)
If you are at the point \((1,1,-1)\text{,}\) in what direction should you change the input of \(f\) 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.
Write a couple of sentences to describe shape and characteristics will the level surfaces of \(g\text{.}\) Be sure to include all possibilities for different types of level surfaces. Compare what is the same or different about these level surfaces and relate the differences to the level value, \(k\text{.}\)
If you at the point \((2,0,-2)\text{,}\) in what direction should you change the input of \(g\) to see the greatest increase in the output of \(g\text{?}\) You should think about how a change in each variable will change the output of \(g\) and talk about this in your reasoning.
