Calculus Quiz 3

Calculus Quiz 3

Id #:

1. (10 points) Let f (x, y) = x + y, with the constraint function x²+ y²= 2. Use Lagrange multipliers to find all local extrema.

Solution: Let g(x, y) = x²+ y². The candidate point on g = 2 at which f achieves an extremum satisfies the equations∇f =λ∇g. Solving (1, 1) =λ(2x, 2y), we get (x, y) = ( 1

2λ^, 1

2λ). By plugging into the equation g = 2, we getλ ^{= 1}

2 or −1

2 , which implies that (x, y) = (1, 1), or (−1, −1). Hence f (1, 1) = 2 is the maximum value of f on the curve g = 2, while f (−1, −1) = −2 is the minimum value of f on the curve g = 2.

2. (a) (3 points) Find the gradient of f (x, y) = x²+ 4y². Solution: ∇f = (2x, 8y)

(b) (3 points) In what direction does f (x, y) = x²+ 4y²increase most rapidly at (1, 1)?

Solution: At (1, 1), f increases most rapidly in the direction ∇f

k∇f k|_(1,1)= (2, 8)

√4 + 64= 1

√17(1, 4).

(c) (4 points) Sketch the gradient vector field∇f on the xy−plane.

Solution: Step1: Sketch some level curves of f .

For example, we draw the curves (ellipses) f = 0.4, 0.8, 1, 1.4, 1.6.

Step2: Use the facts that the gradient vector is perpendicular to the level curve and is pointing to the increasing direction on each level curve to draw∇f

y 1

0.5 0.5

0 -0.5

-1.5 1.5

-0.5

0 1

-1

x -1