Gradient and directional derivatives practice problems
∇f, and the rate of change in any direction: D_u f = ∇f · u. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 0 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (-1, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x y^{2} + 2 x \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle 3, 4 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 5, 12 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} + 3 x y + y^{2} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (1, 0) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (3, 4) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x^{2} y - y^{3} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (2, 1) \) in the direction of \( \displaystyle \langle 1, 1 \rangle \).
Find the directional derivative of \( \displaystyle f(x, y) = x e^{y} \) at \( \displaystyle (1, 2) \) in the direction of \( \displaystyle \langle -1, 2 \rangle \).