Maximum rate of change practice problems
The gradient points uphill: the maximum rate of change is ‖∇f‖, and D_u f = ∇f · u. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} + x y + y^{2} \) at \( \displaystyle (-1, 2) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} y + y^{3} \) at \( \displaystyle (1, 1) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} y + y^{3} \) at \( \displaystyle (2, 1) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = e^{x} \sin{\left(y \right)} \) at \( \displaystyle (3, 4) \), and the direction in which it occurs.
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 y^{2} \) at \( \displaystyle P(1, 1) \) in the direction toward \( \displaystyle Q(4, 2) \).
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} y - y^{2} \) at \( \displaystyle (2, 1) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = x^{2} y - y^{2} \) at \( \displaystyle (1, 1) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \), and the direction in which it occurs.
Find the maximum rate of change of \( \displaystyle f(x, y) = e^{x} \sin{\left(y \right)} \) at \( \displaystyle (-1, 2) \), and the direction in which it occurs.
Find the directional derivative of \( \displaystyle f(x, y) = x e^{- y} \) at \( \displaystyle P(2, 1) \) in the direction toward \( \displaystyle Q(1, 2) \).