Calculus II (Math162)
Exam 1 (Fall, 2018)
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1. (12 pts.)
(a) (6 pts.) Find the limit, if it exists, or show that the limit does not exist.
lim
(x,y)→(0,0)
x2y sin(x)
x4+ y2 .
(b) (6 pts.) Find the limit, if it exists, or show that the limit does not exist.
lim (x,y)→(1,−1) xy + 1 x2− y2. 2. (12 pts.) Let f (x, y) = sin(x3+y4) x2+y2 if (x, y) 6= (0, 0); 0 if (x, y) = (0, 0). (a) (6 pts.) Compute ∇f (0, 0).
(b) (6 pts.) Find the directional derivative of f (x, y) at (0, 0) in the direction of the vector ~v = (1, 1).
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3. (12 pts.) Let C be the curve of intersection of the plane y = 34x and the surface z = f (x, y). Suppose f has contin-uous second partial derivatives with
fx(0, 0) = fy(0, 0) = fxx(0, 0) = fyy(0, 0) = 1,
fxy(0, 0) = −1.
(a) (6 pts.) Find parametric equations for the line tangent to C at 0, 0, f (0, 0).
(b) (6 pts.) Determine if the curve C is concave upward or concave downward near 0, 0, f (0, 0).
4. (8 pts.) Suppose we need to know an equation of the tangent plane to a surface S at the point P (2, 1, 3). You don’t have an equation for S but you know that the curves
~r1(t) = h 2 + 3t, 1 − t2, 3 − 4t + t2i
~r2(s) = h 1 + s2, 2s3− 1, 2s + 1 i
both lie on S. Find an equation of the tangent plane at P (2, 1, 3).
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5. (16 pts.) Let C be the curve of intersection of two surfaces f (x, y, z) = x2+ y2− 2 = 0
and
g(x, y, z) = x + z − 4 = 0.
(a) (5 pts.) Find parametric equations for the line tangent to C at (1, 1, 3).
(b) (5 pts.) Find the curvature of C at the point (1, 1, 3). (c) (6 pts.) Find the center of the osculating circle of the
curve C at (1, 1, 3).
6. (12 pts.) Let f (x, y) = 6x2− 2x3+ 3y2+ 6xy.
(a) (8 pts.) Find all the local maxima, local minima, and saddle points of the function f .
(b) (4 pts.) In which direction does f change most rapidly at P (1, 1) ? What is the maximum rate of change at P (1, 1) ?
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7. (16 pts.)
(a) (8 pts.) Find the mininum distance from the cone z =px2+ y2to the point (−6, 4, 0).
(b) (8 pts.) Find extreme values of f (x, y, z) = x(y + z) on the intersection of two surfaces x2+ y2 = 1 and
xz = 1.
8. (12 pts.) Find the absolute maximum and minimum val-ues of the function f (x, y) = 2y2− 2x − 4y e−x on the set D, where D is the closed region bounded by the parabola x − y2= 0 and the line x = 4.
