Practice II.1. Consider the case n = 2.(a) Find the gradients of the following functions from R² to R::= x² + x²-2(ii) ƒ₂ (21) := 4x² +(i) fi(iii) f321I2I1I₂:= 9(x₁ − 1)² + (2+¹)²-2 (iv) f₁(i) fi(.); x =• (-¹1); v = (3)=(-¹)ONLY PART B - PLESAE(b) Find the following directional derivative f'(x; v) for the following functions atthe given x and v. (The functions refer to part (a))(iii) f3(-); x =I1Inv anything1:= = √(x₁ + 1)² + (x₂ - 2)²=4x²+2-1(ii) f2(.); x =(iv) f₁(); x =(3);3(2)- (2¹)V =V =

Given : if1x1 ,x2=x12+x22-2 , x=1 , -1 , x=2 , 3ii f2(x1 , x2 ) =4x12 +x229-1 , x= 3 , 1 , v=1 ,…

Answered: (b) Find the following directional…

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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Related questions

Question

Practice II.1. Consider the case n = 2.
(a) Find the gradients of the following functions from R² to R:
:= x² + x²-2
(ii) ƒ₂ (21) := 4x² +
(i) fi
(iii) f3
21
I2
I1
I₂
:= 9(x₁ − 1)² + (2+¹)²-2 (iv) f₁
(i) fi(.); x =
• (-¹1); v = (3)
=(-¹)
ONLY PART B - PLESAE
(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 =

Expert Solution

Step 1

Given :

$(i) f_{1} (x_{1}, x_{2}) = x_{1}^{2} + x_{2}^{2} - 2, x = (1, - 1), x = (2, 3) (i i) f_{2} (x_{1}, x_{2}) = 4 x_{1}^{2} + \frac{x_{2}^{2}}{9} - 1, x = (3, 1), v = (1, 2) (i i i) f_{3} (x_{1}, x_{2}) = 9 {(x_{1} - 1)}^{2} + \frac{{(x_{2} + 1)}^{2}}{4} - 2, x = (1, - 1)$