In Exercises $37 - 42$ , use the partial derivatives of role="math" localid="1650186824938" $f (x, y) = x^{y}$ and the point $(e, 3)$ specified to
$(a)$ find the equation of the line tangent to the surface defined by the function in the $x$ direction,
$(b)$ find the equation of the line tangent to the surface defined by the function in the $y$ direction, and
$(c)$ find the equation of the plane containing the lines you found in parts $(a)$ and $(b)$ .

Question

Accepted Answer

Part $(a)$ , The equation of the line tangent to the surface defined by the function in the $x$ direction is

$x = e + t, y = 3, z = e^{3} + 3 e^{2} t$

Part $(b)$ , The equation of the line tangent to the surface defined by the function in the $y$ direction is

$x = e, y = 3 + t, z = e^{3} + e^{3} t$

Part $(c)$ , The equation of the plane containing the lines you found in parts $(a)$ and $(b)$ is

localid="1650216415404" $3 e^{2} x + e^{3} y - z = 5 e^{3}$

Short Answer