Н.W(1) In Exercises 1-4, find the gradient of the function at the given point. Then sketch thegradient together with the level curve that passes through the point.1. fa., у) 3 у - х, (2. 1)2. f(x, y) = In (x + y), (1, 1)3. g(x, у) — ху?, (2, - 1)4. g(x, y) =y?(V2, 1)(2) In Exercises 1-3, find the derivative of the function at PO in the direction of u.1. f(x, y) = 2xy - 3y, P(5, 5), u2. f(x, y) = 2x + y, Po(-1, 1), u =x - y4i + 3j3i - 4j3. g(x, y) =Po(1, -1), u = 12i + 5jxy + 2'(3) In Exercises 1-4, find the directions in which the functions increase and decrease mostrapidly at PO. Then find the derivatives of the functions in these directions.1. f(x, y) = x + xy + y, Po(-1, 1)2. f(x, y) = x'y + e" sin y, P(1, 0)3. f(x, y, z) = (x/y) - yz, P(4, 1, 1)(4) In Exercises 1-3, find equations for the(a) tangent plane and(b) normal line at the point PO on the given surface..1. x + y? + ? = 3, P(1, 1, 1)2. x? + y? - = 18, P(3, 5,-4)3. 2z - x = 0, P(2, 0, 2)%3D(5) In Exercises I and 2, find an equation for the plane that is tangent to the given surface atthe given point.1. 2 In (x + y), (1,0,0)2. z = Vy - x, (1, 2, 1)%3D

Since you have asked multiple questions, we will solve the first question for you. If you want any…

)In Exercises 1-4, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. 1. f(x, y) = y - x, (2, 1) 2. f(x, y) = In (x + y²). (1, 1) 3. g(x, y) xy, (2, -1) 4. g(x, y) = 21 2' (V2. 1)

)In Exercises 1-4, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. 1. f(x, y) = y - x, (2, 1) 2. f(x, y) = In (x + y²). (1, 1) 3. g(x, y) xy, (2, -1) 4. g(x, y) = 21 2' (V2. 1)

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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Question

Н.W
(1) In Exercises 1-4, find the gradient of the function at the given point. Then sketch the
gradient together with the level curve that passes through the point.
1. fa., у) 3 у - х, (2. 1)
2. f(x, y) = In (x + y), (1, 1)
3. g(x, у) — ху?, (2, - 1)
4. g(x, y) =
y?
(V2, 1)
(2) In Exercises 1-3, find the derivative of the function at PO in the direction of u.
1. f(x, y) = 2xy - 3y, P(5, 5), u
2. f(x, y) = 2x + y, Po(-1, 1), u =
x - y
4i + 3j
3i - 4j
3. g(x, y) =
Po(1, -1), u = 12i + 5j
xy + 2'
(3) In Exercises 1-4, find the directions in which the functions increase and decrease most
rapidly at PO. Then find the derivatives of the functions in these directions.
1. f(x, y) = x + xy + y, Po(-1, 1)
2. f(x, y) = x'y + e" sin y, P(1, 0)
3. f(x, y, z) = (x/y) - yz, P(4, 1, 1)
(4) In Exercises 1-3, find equations for the
(a) tangent plane and
(b) normal line at the point PO on the given surface..
1. x + y? + ? = 3, P(1, 1, 1)
2. x? + y? - = 18, P(3, 5,-4)
3. 2z - x = 0, P(2, 0, 2)
%3D
(5) In Exercises I and 2, find an equation for the plane that is tangent to the given surface at
the given point.
1. 2 In (x + y), (1,0,0)
2. z = Vy - x, (1, 2, 1)
%3D

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