Find the vector equation that represents the curve of intersection of the paraboloid \( z=2 x^{2}+y^{2} \) and the surface \( y=e^{x} \). Write the equation so that one of the functions is simply \( t \). \( x(t)= \) \( y(t)= \) \( z(t)= \)

To find the vector equation representing the curve of intersection, let's parameterize the equation using \( t \) for \( x \). Since \( y = e^x \), we can substitute \( t \) for \( x \): \( x(t) = t \) \( y(t) = e^t \) Now, substituting \( x(t) \) and \( y(t) \) into the paraboloid equation to find \( z(t) \): \( z(t) = 2(t^2) + (e^t)^2 = 2t^2 + e^{2t} \) So, the vector equation becomes: \( x(t) = t \) \( y(t) = e^t \) \( z(t) = 2t^2 + e^{2t} \) Thus, we have accurately expressed the intersection curve in terms of the parameter \( t \).