3r 4y? 4. Find the directions in which f(r, y) = %3D 6. i) Increases most rapidly at the point (1, 1) ii) Decreases most rapidly at (1, 1) ii) What are the directions of zero change in f(z, y) at (1, 1). 5. Find absolute maxima and minima of the function f(r, y) = 1 + y on the closed triangular plate bounded by the lines z = 0, y = 0, y + 2r = 2 in the first quadrant.

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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4y?
4. Find the directions in which f(x, y)
%3D
6.
i) Increases most rapidly at the point (1, 1)
ii) Decreases most rapidly at (1, 1)
iii) What are the directions of zero change in f(x, y) at (1, 1).
5. Find absolute maxima and minima of the function f(r, y) = x² + y? on the closed triangular plate
bounded by the lines z = 0, y = 0, y + 2x = 2 in the first quadrant.
6. Find the maxium andd minimum values of the following functions by using Lagrange Multiplier.
a) f(x, y) = 81z + y° subject to the constrait 42 + y? = 9
b) f(x, y) = 82? – 2y subject to the constrait r + y? = 1.
7. Find the equation of tangent plane to the surface r? + 2y² + 322 = 21 which is parallel to the plane
2r + 4y + 6z = 3.
8. Find the extremum and saddle points of the function f(x, y) = 2° – 3zy + y if any.
9. Compute each of the following double integrals over the indicated rectangles.
1
dA, R=(0,1) x (1, 2)
(2.r + 3y)2"
R
b) / re""dA, R= (-1,2) × [0, 1)
R
10. i) Sketch the region of given integral
ii) Change integration order
iii) Evaluate the integral
2
1 1
b)
drdy
0 1
V V4-2y
1 3r+2
c) / ydrdy
-2a+4z
--V4-2v
11. Evaluate the following integral by converting it into polar coordinates.
// 2rydA, D is the portion of the region between the circles of radius 2 and radius 5centered at the
origin that lies in the first quadrant.
Transcribed Image Text:4y? 4. Find the directions in which f(x, y) %3D 6. i) Increases most rapidly at the point (1, 1) ii) Decreases most rapidly at (1, 1) iii) What are the directions of zero change in f(x, y) at (1, 1). 5. Find absolute maxima and minima of the function f(r, y) = x² + y? on the closed triangular plate bounded by the lines z = 0, y = 0, y + 2x = 2 in the first quadrant. 6. Find the maxium andd minimum values of the following functions by using Lagrange Multiplier. a) f(x, y) = 81z + y° subject to the constrait 42 + y? = 9 b) f(x, y) = 82? – 2y subject to the constrait r + y? = 1. 7. Find the equation of tangent plane to the surface r? + 2y² + 322 = 21 which is parallel to the plane 2r + 4y + 6z = 3. 8. Find the extremum and saddle points of the function f(x, y) = 2° – 3zy + y if any. 9. Compute each of the following double integrals over the indicated rectangles. 1 dA, R=(0,1) x (1, 2) (2.r + 3y)2" R b) / re""dA, R= (-1,2) × [0, 1) R 10. i) Sketch the region of given integral ii) Change integration order iii) Evaluate the integral 2 1 1 b) drdy 0 1 V V4-2y 1 3r+2 c) / ydrdy -2a+4z --V4-2v 11. Evaluate the following integral by converting it into polar coordinates. // 2rydA, D is the portion of the region between the circles of radius 2 and radius 5centered at the origin that lies in the first quadrant.
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