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Name: Perm number:

Midterm – in-class part

Time: 75 minutes

(1) Find an equation of the tangent plane to the surface z = y ln x at the point (1, 4, 0).

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(2)

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(2) Find approximate values of x(0.02) and y(0.05), if x(0) = 1, y(0) = 2, x0= xy − 12x2−12y2+12, and y0= x2+ y2− 5.

(3)

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(3) Use the Chain Rule to find ∂R∂x, if R = ln(u2+ v2+ w2, u = x + 2y, v = 2x − y, w = 2xy, and x = y = 1

(4)

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(4) Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 6.

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