I. Draw a contour map of z = Vx2 + (y – 2)2 using k = 0, 1, 2, 3, 4, then sketch its graph in R³. II. Let f(r, y, z) = 23 sin r+ Determine 2 dydzdrəz?` III. Consider the function g defined by 1 g(r, y) = sin log3(r – y) 1. Calculate the instantaneous rate of change of g at the point (4, 1, 1) in the direction of the vector v = (1,2). 2. In what direction does g have the maximum directional derivative at (r, y) = (4, 1)? What is the maximum directional derivative? az s'. Solve for set IV. Let w = r In a + 2ryz + 5z°y, r = tan s+r, y =,z = V. Let z be a function of r and y, and tan Vy2 +r2 = z*e®w. Solve for az az and dy VI. Determine all the relative minimum and maximum values, and saddle points of the function h defined by h(r, y) = r – 3y +3ry?. VII. Use Lagrange Multipliers to solve the following: Maximize f(r, y, z) = 4x + 2y + z subject to r? + y? + z? = 1.
I. Draw a contour map of z = Vx2 + (y – 2)2 using k = 0, 1, 2, 3, 4, then sketch its graph in R³. II. Let f(r, y, z) = 23 sin r+ Determine 2 dydzdrəz?` III. Consider the function g defined by 1 g(r, y) = sin log3(r – y) 1. Calculate the instantaneous rate of change of g at the point (4, 1, 1) in the direction of the vector v = (1,2). 2. In what direction does g have the maximum directional derivative at (r, y) = (4, 1)? What is the maximum directional derivative? az s'. Solve for set IV. Let w = r In a + 2ryz + 5z°y, r = tan s+r, y =,z = V. Let z be a function of r and y, and tan Vy2 +r2 = z*e®w. Solve for az az and dy VI. Determine all the relative minimum and maximum values, and saddle points of the function h defined by h(r, y) = r – 3y +3ry?. VII. Use Lagrange Multipliers to solve the following: Maximize f(r, y, z) = 4x + 2y + z subject to r? + y? + z? = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:(y – z)²
|
I. Draw a contour map of z = r² +
R3.
using k = 0, 1, 2, 3, 4, then sketch its graph in
II. Let f(x, y, 2) = 2 sin a+
Determine
dydzdrdz?
III. Consider the function g defined by
1
g(r, y) = sin
log3(r – y)"
1. Calculate the instantaneous rate of change of g at the point (4, 1, 1) in the direction
of the vector v = (1,2).
2. In what direction does g have the maximum directional derivative at (x, y) = (4, 1)?
What is the maximum directional derivative?
dz
IV. Let w = x lna³ + 2xyz + 52°y, r = tan s +r, y = z = s'. Solve for
set
dz
dz
V. Let z be a function of r and y, and tan Vy2 + r2 = zew. Solve for
and
%3D
dr
dy
VI. Determine all the relative minimum and maximum values, and saddle points of the function
h defined by
h(x, y) = x – 3y + 3ry?.
VII. Use Lagrange Multipliers to solve the following:
Maximize f(r, y, z) = 4x + 2y + z
subject to a2 + y² + z? = 1.
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