Consider the function f(x, y) = e^(y - x^2 - 1). Find the equation of the level curve of f that passes through the point (2, 5). Also sketch this curve.

Steps to Solve

1. Evaluate the function at the given point

   We need to find the value of the function $f(x, y) = e^{y - x^2 - 1}$ at the point $(2, 5)$.
   Substitute $x = 2$ and $y = 5$ into the function: $$ f(2, 5) = e^{5 - 2^2 - 1} $$
   This simplifies to:
   $$ f(2, 5) = e^{5 - 4 - 1} = e^0 = 1 $$

2. Set the function equal to the computed value

   The level curve corresponds to points where the function takes the value computed: $$ e^{y - x^2 - 1} = 1 $$ We can rewrite this as:
   $$ y - x^2 - 1 = 0 $$

3. Solve for y

   To find the equation of the level curve, solve for $y$:
   $$ y = x^2 + 1 $$

4. Sketch the curve

   The equation $y = x^2 + 1$ is a parabola that opens upwards. It has its vertex at $(0, 1)$.
   To sketch this curve, plot points for different values of $x$ (like -2, -1, 0, 1, 2) and join them smoothly in a parabolic shape.