Sketch the level curve through p showing level value and

Question

Let f(x,y) = 3x² - y + 5 (perhaps the function represents temperature distribution on the plane), and let P be the point with coordinates (1,4).

a. Sketch the level curve through P showing level value, and find a vector normal to this curve at P.

b. Find the direction (i.e. a unit vector) along which f increases most rapidly at the point P.

c. Find the rate of change of f at P along the above direction.

d. Starting from the above direction at P, if you now turn 30degrees clockwise, what will be the rate of change of f at P along this new direction?

e. Find the rate of change of f at P toward the point (2,3).

f. If y is a vector which is tangent to the level curve through P at P, what is v. Δf(1,4)? And why?