Write down the equation of the tangent line to the level

Question

Consider an elliptic paraboloid

f(x, y) = 4 - x² - y²/4.

a. Find the xy-, yz- and xz-traces of the graph of f(x, y) and sketch the graph.

b. Write down the equation of the level curve which passes through a point (2, 4). Sketch the level curve.

c. Find the unit vectors in the directions of the steepest descent and the steepest ascent at (2, 4). Mark these directions on the graph in (b). Verify that the gradient at (2, 4) is orthogonal to the level curve through this point.

d. Write down the equation of the tangent line to the level curve at (2, 4).