This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back — missed you this week. To find the directional derivative of f(x,y) at P_0 in the direction of u, we use the formula D_uf(P_0) = f(P_0) · 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) = 2x^2 + y^2. ( f)/( x) = ()/( x)(2x^2 + y^2) = 4x ( f)/( y) = ()/( y)(2x^2 + y^2) = 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 P_0(-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|| = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(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. Send me the next one 📸