æ
1. Let z=x2ey, where x=u2v−1 and y=uv −2. The partial derivative ∂ z
∂u at
(u, v) = (1,2) is
(1) 2 (2) 8 (3) 10 (4) 16
2. For the function f(x, y) = x3−12x+y2,
(1) there are no critical points.
(2) fhas a saddle point at the critical point (−2,0).
(3) fhas a local minimum at the critical point (0,0).
(4) fhas a local maximum at the critical point (−2,0).
3. Let f(x, y) = (1 + sin(2x))y3. Using the total differential (or the linearization or
standard linear approximation) of f(x, y) at the point P(0,1), one can approximate the
value of (1 + sin(0.02))(1.01)3by
(1) 1.02 (2) 1.03 (3) 1.05 (4) 1.07
4. The tangent plane to the surface z=xy2at the point P(1,1,1) contains the point
P(0,0, k) when kis
(1) −3 (2) 0 (3) −2 (4) 5
5. At the point P(1,1,1) the function
f(x, y, z) = 2x2y+z4
y2
increases fastest in the direction of the vector
(1) i+k(2) i+j+k(3) 4i(4) j−k
6. The value of
Z6
0Z3
y/2
ex2dx dy
is
(1) e9−1 (2) e36 −1 (3) 5e9−1 (4) e18 −1
A-1
