University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2020
Views:

475

Pages:

1
Author:

InterstellarInnovator

Exercise 1. Find the point p(x; y; z ) on the plane 2x + y (Hass Sec 14.8 Ex 1) z Exercise 2. Find the points on the hyperbolic cylinder x2 origin. (Hass Sec 14.8 Ex 2) 5 = 0 that is closest to the origin. z2 1 = 0 that are closest to the Theorem: Suppose f and g are functions of two variables that have continuous rst partial derivatives, and that r g = 6 0 throughout a region of the xy-plane. If f has an extremum f (x0 ; y0 ) subject to the constraint g (x; y ) = 0, then there is a real number such that r f (x0 ; y0 ) = r g(x0 ; y0 ). De nition: The number is called a Lagrange multiplier. Method of Lagrange Multipliers: To nd the maximum and minimum values of f (x; y; z ) subject to the constraint g (x; y; z ) = k: (a) Find all values of x, y , z , and such that r f (x; y; z) = r g(x; y; z) and g(x; y; z) = k. (b) Evaluate f at all the points (x; y; z ) that result from step (a). The largest of these values is the maximum value of f ; the smallest is the minimum value of f . Exercise 3. Find the extrema of f (x; y) = xy if (x; y) is restricted to the ellipse 4x2 + y2 = 4. (Swok Sec 16.9 Ex 1) Exercise 4. Find the maximum and minimum values of the function f (x; y) = 3x + 4y on the circle x2 + y 2 = 1. (Hass Sec 14.8 Ex 4) Class Exercise 1. Use Lagrange multipliers to nd the maximum and minimum values of the function subject to the given constraint. (#4,6) (a) f (x; y ) = 3x + y ; x2 + y 2 = 10 (b) f (x; y ) = exy ; x3 + y 3 = 16 Exercise 5. Find the volume of the largest rectangular box with faces parallel to the coordinate planes that can be inscribed in the ellipsoid 16x2 + 4y 2 + 9z 2 = 144. (Swok Sec 16.9 Ex 2) Exercise 6. If f (x; y; z ) = 4x2 + y2 + 5z 2 , nd the point on the plane 2x + 3y + 4z = 12 at which f (x; y; z ) has its least value. (Swok Sec 16.9 Ex 3) Class Exercise 2. Use Lagrange multipliers to nd the maximum and minimum values of the function subject to the given constraint. (#8; 10; 12) (a) f (x; y; z ) = x2 + y 2 + z 2 ; x + y + z = 12 (b) f (x; y; z ) = x2 y 2 z 2 ; x2 + y 2 + z 2 = 1 (c) f(x;y; z) = x4 + y4 + z4; x2 + y2 + z2 = 1