Find the directional derivative of f(x,y) = 2x2 + y2 at P0(−1, 1) in the direction of u = 3i − 4j.

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To find the directional derivative of f(x,y) at P₀ in the direction of u, we use the formula Duf(P₀) = ∇ f(P₀) · u, where u is the unit vector in the direction of u. Step 1: Find the partial derivatives of f(x,y) with respect to x and y. Given f(x,y) = 2x² + y². (∂ f)/(∂ x) = (∂)/(∂ x)(2x² + y²) = 4x; (∂ f)/(∂ y) = (∂)/(∂ y)(2x² + y²) = 2y Step 2: Form the gradient vector ∇ f(x,y). ∇ f(x,y) = (∂ f)/(∂ x), (∂ f)/(∂ y) = 4x, 2y Step 3: Evaluate the gradient at the given point P₀(−1, 1). ∇ f(−1, 1) = 4(−1), 2(1) = −4, 2; Step 4: Find the unit vector u in the direction of u = 3i − 4j. First, find the magnitude of u: ||u|| = √(3² + (−4)²) = √(9 + 16) = √(25) = 5 Now, find the unit vector: u = u||u|| = 1/5(3i − 4j) = 3/5, −4/5 Step 5: Calculate the directional derivative by taking the dot product of ∇ f(−1, 1) and u. D_uf(−1, 1) = ∇ f(−1, 1) · u; D_uf(−1, 1) = −4, 2 · 3/5, −4/5; D_uf(−1, 1) = (−4)(3/5) + (2)(−4/5); D_uf(−1, 1) = −12/5 − 8/5; D_uf(−1, 1) = −20/5; D_uf(−1, 1) = −4 The directional derivative is −4.

Find the directional derivative of f(x,y) = 2x2 + y2 at P0(-1, 1) in the direction of u = 3i − 4j.

Answer

Step-by-step solution

Find the partial derivatives of $f(x,y)$ with respect to $x$ and $y$ .

Form the gradient vector $\nabla f(x,y)$ .

Evaluate the gradient at the given point $P_0(-1, 1)$ .

Find the unit vector $\hat{\mathbf{u}}$ in the direction of $\mathbf{u} = 3\mathbf{i} - 4\mathbf{j}$ .

Calculate the directional derivative by taking the dot product of $\nabla f(-1, 1)$ and $\hat{\mathbf{u}}$ .

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More Mathematics Questions

Find the directional derivative of f(x,y) = 2x2 + y2 at P0(-1, 1) in the direction of u = 3i − 4j.

Answer

Step-by-step solution

Find the partial derivatives of f(x,y)f(x,y)f(x,y) with respect to xxx and yyy.

Form the gradient vector ∇f(x,y)\nabla f(x,y)∇f(x,y).

Evaluate the gradient at the given point P0(−1,1)P_0(-1, 1)P0​(−1,1).

Find the unit vector u^\hat{\mathbf{u}}u^ in the direction of u=3i−4j\mathbf{u} = 3\mathbf{i} - 4\mathbf{j}u=3i−4j.

Calculate the directional derivative by taking the dot product of ∇f(−1,1)\nabla f(-1, 1)∇f(−1,1) and u^\hat{\mathbf{u}}u^.