Question

Use a computer to investigate the family of surfaces \(f(x,y) = {e^{c{x^2} + {y^2}}}\)how does the shape of the group depend on c?

Short answer

As c increases, the graph of \(f(x,y) = {e^{c{x^2} + {y^2}}}\) is stretched towards the X-axis.

Step-by-step solution

Graph the function  

\(f(x,y) = {e^{c{x^2} + {y^2}}}\), when \(c = 0.5,1,2\)

Take\(c = 0.5\)

The graph of \(f(x,y) = {e^{\frac{1}{2}{x^2} + {y^2}}}\) can be given below:

Take\(c = 1\)

The graph of \(f(x,y) = {e^{{x^2} + {y^2}}}\) can be given below:

Take\(c = 2\)

The graph of \(f(x,y) = {e^{2{x^2} + {y^2}}}\) can be given below:

Graph the function  

\(f(x,y) = {e^{c{x^2} + {y^2}}}\), when \(c = 3,10\)

Take\(c = 3\)

The graph of \(f(x,y) = {e^{3{x^2} + {y^2}}}\) can be given below:

Take\(c = 10\)

The graph of \(f(x,y) = {e^{10{x^2} + {y^2}}}\) can be given below:

Hence, as c increases, the graph is stretched towards the X-axis.