MA2255-004/Calculus III/Spring 2012
Test 3 (1 hr 40 minutes)
Name (UPPER CASE):
(Sections 14.5, 14.6, 14.7, 15.1, 15.2, 15.3, 15.4, 15.5)
Show your work in detail. No credit will be given if detailed computational, algebraic and/or
graphic arguments are not provided.
1. (5+5=10 points) Let f (x, y) = ln(xy) + ln(yz) + ln(zx).
(a) Find the directions in which f (x, y) increases and decreases most rapidly at P0 (1, 1, 1).
(b) Find the directional derivatives of f at P0 (1, 1, 1) in the directions obtained in part(a).
2. (8 points) Find an equation for the plane that is tangent to the surface
x2 + 2xy − y 2 + z 2 = 7
at (1, −1, 3).
1
3. (8 points) Find parametric equations for the line tangent to the curve of intersection of
the surfaces
x + y 2 + 2z = 4 and x = 1
at the point (1, 1, 1).
4. (5+5=10 points)
(a) Find the linearization L(x, y) of the function f (x, y) = x2 − 3xy + 5 at P0 (2, 1).
(b) Find an upper bound for the magnitude of the error in the approximation
f (x, y) ≈ L(x, y) over the rectangle R ∶ ∣x − 2∣ ≤ 0.1, ∣y − 1∣ ≤ 0.1.
2
5. (10 points) Find all local minima, maxima and saddle points of the function
f (x, y) = 6x2 − 2x3 + 3y 2 + 6xy.
6. (2+4+4=10 points) Consider the iterated integral
π
∫ ∫
0
π
x
(a) Sketch the region of integration.
(b) Reverse the order of integration.
(c) Evaluate the integral.
3
sin y
dy dx.
y
7. (2+6=8 points)
(a) Sketch the region bounded by the curve y = ex and the lines x = 0, y = 0, and x = ln 2.
(b) Express the area of the bounded region in part (a) as an iterated double integral and
evaluate the integral.
8. (8 points)
√ Find the centroid of the semicircular region bounded by the x-axis and the
curve y = 1 − x2 .
4
9. (5+5=10 points) Consider the integral
0
0
√
−1 − 1−x2
∫ ∫
2
√
dy dx.
1 + x2 + y 2
(a) Change the Cartesian integral into an equivalent polar integral.
(b) Evaluate the polar integral obtained in part (a).
10. (8 points) Find the volume of the solid region in the first octant bounded by the coordinate
planes and the plane y + z = 2 and the cylinder x = 4 − y 2 .
5
11. (10 points) Find the center of mass of the solid of constant density bounded below by
the paraboloid z = x2 + y 2 and above by the plane z = 4. [Hint: The integrals involved are
triple iterated integrals.]
Extra-credit Problem (5 points)
√
Find the average height of the hemisphere z = a2 − x2 − y 2 over the circular disk
x2 + y 2 ≤ a2
in the xy-plane.
6