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(a) Find the gradients of the following functions from R² to R: (1) f₁ (21) := x² + x² − 2 (iii) fs (¹) (ii) ƒ2 (22) := 4x² + ¾ − 1 := 9(x₁ - 1)²+(2+1)²-2 (iv) f₁ I1 1₂ = √√√(x₁ + 1)² + (x₂ − 2)²

(a) Find the gradients of the following functions from R² to R: (1) f₁ (21) := x² + x² − 2 (iii) fs (¹) (ii) ƒ2 (22) := 4x² + ¾ − 1 := 9(x₁ - 1)²+(2+1)²-2 (iv) f₁ I1 1₂ = √√√(x₁ + 1)² + (x₂ − 2)²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Practice II.1. Consider the case n = 2. (a) Find the gradients of the following functions from R² to R: (i) fi := x² + x²-2 (ii) ƒ₂ (21) := 4x² + (iii) f3 21 I2 I1 I₂ := 9(x₁ − 1)² + (2+¹)²-2 (iv) f₁ (i) fi(.); x = • (-¹1); v = (3) =(-¹); (b) Find the following directional derivative f'(x; v) for the following functions at the given x and v. (The functions refer to part (a)) (iii) f3(.); x = I1 In v anything 1:= = √(x₁ + 1)² + (x₂ - 2)² =4x²+2-1 (ii) f2(.); x = (iv) f₁(-); x = (3); 3 (2) - (2¹) V = V =
Transcribed Image Text:Practice II.1. Consider the case n = 2. (a) Find the gradients of the following functions from R² to R: (i) fi := x² + x²-2 (ii) ƒ₂ (21) := 4x² + (iii) f3 21 I2 I1 I₂ := 9(x₁ − 1)² + (2+¹)²-2 (iv) f₁ (i) fi(.); x = • (-¹1); v = (3) =(-¹); (b) Find the following directional derivative f'(x; v) for the following functions at the given x and v. (The functions refer to part (a)) (iii) f3(.); x = I1 In v anything 1:= = √(x₁ + 1)² + (x₂ - 2)² =4x²+2-1 (ii) f2(.); x = (iv) f₁(-); x = (3); 3 (2) - (2¹) V = V =
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