Exam 2 MTH 231 Summer 2006 Total Pts:100 8/3/2006
Name: Total Received:
Show all work for full credit. Do not write anything on question papers except your name.
1. Find and sketch the domain of the function f(x, y) = √x+y+1
x+1 . Find f(2,3). (6 Pts)
2. Sketch the level curves of the function f(x, y) = xy for k= 1,2. (5 Pts)
3. For the limit lim(x,y)→(0,0) x2
x2+y2, compute the following and determine whether the
above limit exists.
(a) lim(x,y)→(0,0) x2
x2+y2along the line y= 0.
(b) lim(x,y)→(0,0) x2
x2+y2along the line y=x.
(c) lim(x,y)→(0,0) x2
x2+y2along the curve x=y2. (7 Pts)
4. Determine the set of points at which the function f(x, y) = ln(x2+y2−1) is
continuous and sketch. (5 Pts)
5. For the function f(x, y) = x2+2xy+y3−x2y3, find fx(x, y ), fy(x, y), fxx(0,1), fyy (1,0),
and fxy(0,0). (7 Pts)
6. If f(x, y) = xex2y, find ∂ f
∂x and ∂f
∂y . (7 Pts)
7. Find an equation of the tangent plane to the surface z= 4x2−y2+ 2yat (1,-1,1).
(7 Pts)
8. Find ∂z
∂x and ∂z
∂y if x3+y3+z3+ 6xyz = 1 using the formula ∂z
∂x =−Fx
Fzand ∂z
∂y =−Fy
Fz.
(6 Pts)
9. Use the chain rule to find ∂z
∂s and ∂z
∂t where z=x2+xy +y2,x=s+t, y =st.
(7 Pts)
10. Find the directional derivative of the function f(x, y, z) = y
x+zat (4,1,1) in the
direction of the vector −→
v=<1,2,−2>. What is the maximum value of this
derivative at the point (4,1,1)? (8 Pts)
11. Find the equation of the tangent plane to the surface x2+ 2y2+ 3z2= 21 at the
point (4,-1,1). (6 Pts)
(Hint: Fx(x0, y0, z0)(x−x0) + Fy(x0, y0, z0)(y−y0) + Fz(x0, y0, z0)(z−z0) = 0)
12. Find the local maximum and minimum values and saddle points of the function
f(x, y) = 9 −2x+ 4y−x2−2y2using the Second Derivative Test. (7 Pts)
13. Calculate the iterated integral R2
0Rπ
2
0xsin y dy dx. (6 Pts)
14. Find the volume of the solid that lies under the plane 3x+ 2y+z= 12 and above
the rectangle R={(x, y)|0≤x≤1,−2≤y≤3}. (7 Pts)
