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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 11.8.3 .
In this activity, we will look at the calculation and interpretation of the gradient for
\(f(x,y,z)=x+y-z\) and
\(g(x,y,z)=z-x^2-y^2\text{.}\)
(a) Calculate
\(\nabla f\) and
\(\nabla g\text{.}\)
(b) Sketch the level surfaces of
\(f\) for the values
\(k=\{-2,-1,0,1,2\}\text{.}\) Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces change in terms of the value of
\(k\text{.}\)
(c) Write a few sentences about how the direction and magnitude of
\(\nabla f\) is related to the level surfaces from the previous part.
(d) Sketch the level surfaces of
\(g\) for the values
\(k=\{-2,-1,0,1,2\}\text{.}\) Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces change in terms of the value of
\(k\text{.}\) You may find it helpful to notice that each of the level surfaces can be expressed with
\(z\) as a function of
\(x\) and
\(y\text{.}\)
(e) Write a few sentences about how the direction and magnitude of
\(\nabla g\) is related to the level surfaces from the previous part.
