Math 2210-1
Review 2
1. Calculate the two first and four second partial derivatives of the following functions:
(a) f (x, y) = ex−y − ey−x
(b) g(x, y) = x2 y sec xy
2. Evaluate
lim
(x,y)→(0,0)
xy
as (x, y) approaches the origin along:
x2 + y 2
(a) The x-axis
(b) The y-axis
(c) The line y = mx
3. Find the gradient of the following functions:
(a) f (x, y) = x2 y −2
(b) f (x, y, z) = xy + yz + zx
(c) f (x, y, z) = e−x sin(z + 2y) at (0, π4 , π4 )
4. Find a function f whose gradient is the vector valued function F~ (x, y) = 2xy~ı + (1 + x2 )~.
5. Find the directional derivative at the given point in the direction indicated:
ax + by
at (1,1) in the direction of ~ı − ~.
x+y
(b) f (x, y, z) = (x + y 2 + z 3 )2 at (1,-1,1) in the direction of ~ı + ~.
(a)
6. Determine the path of steepest descent along the surface z = a2 x2 +b2 y 2 from the point (a2 , b2 , a4 +b4 ).
7. Find the rate of change of f (x, y) = tan−1 (y 2 −x2 ) with respect to t along the curve ~r(t) = sin t~ı+cos t~.
8. Find
du
dt
if u = ex sin y, x = t2 , and y = πt using the chain rule.
9. If the lengths of two sides of a triangle are x and y, and θ is the angle between the two sides, then the
area A of the triangle is given by A = 12 xy sin θ. If the sides are each increasing at the rate of 3 inches
per second and θ is decreasing at the rate of 0.10 radian per second, how fast is the area changing at
the instant x = 1.5 feet, y = 2 feet, and θ = 1 radian?
∂u
2
2
10. Find ∂u
∂s and ∂t if u = x − xy + z , where x = s cot t, y = sin(t − 1), and z = t sin s. (Draw the tree
diagram too.)
√
11. Find the equation of the tangent plane to the surface x2 − y 2 + z 2 + 1 = 0 at the point (1, 3, 7).
12. Find the critical points for the following functions, and classify them.
(a) f (x, y) = x2 + xy + y 2 − 6x + 2
(b) f (x, y) = ex cos y
13. Let f (x, y) = x2 + kxy + y 2 , where k is a constant.
(a) Show that f has a critical point at (0,0) independant of the value of k.
(b) For what values of k will f have a saddle point at (0,0)?
(c) For what values of k will f have a local minimum at (0,0)?
(d) For what values of k is the second-derivative test inconclusive?
14. Find the maximum or f (x, y) = 4x2 − 4xy + y 2 subject to the constraint x2 + y 2 = 1.
Z
(x1/2 − y 2 ) dA where R is the region enclosed by y = x1/4 , and y = x2 .
15. Evaluate
R
√
√
√
16. Calculate the area of the region R which lies between x + y = a and x + y = a, a > 0.
Z
x2 dA where R is the region for which 0 ≤ x ≤ 1, 0 ≤ y ≤ 3.
17. Evaluate
R
18. Evaluate
Z
R
√
xy dA where R is the region 0 ≤ y ≤ 1, y 2 ≤ x ≤ y.
p
19. Using polar coordinates, calculate the volume of the solid bounded above by the cone z = 2 − x2 + y 2
and bounded below by the disc (x − 1)2 + y 2 ≤ 1.
Z
1
20. Evaluate
dx dy where R is the triangle with verticies at (0,0), (1,0), and (1,1).
2 + y 2 )3/2
(1
+
x
R
21. Integrate f (x, y) = cos(x2 + y 2 ) over
(a) the closed unit disc
(b) the annular region 1 ≤ x2 + y 2 ≤ 4.
1
Z
22. Calculate
1/2
Z
√
1−x2
(x2 + y 2 )3/2 dy dx.
0
23. Find the mass and center of mass of 0 ≤ x ≤ 8, 0 ≤ y ≤ x1/3 where δ(x, y) = y 2 .
24. Find the surface area of the part of the conical surface x2 + y 2 = z 2 that is directly over the triangle
in the xy-plane with vertives (0,0), (4,0), and (0,4).
25. Evaluate the following:
Z 1 Z 2y Z x
(a)
(x + 2z) dz dx dy.
0
(b)
Z
1
1
2
Z
0
y2
y
Z
ln x
yez dz dx dy.
0
26. Find the volume of the solid bounded above by the parabolic cylinder z = 1 − y 2 , below by the plane
2x + 3y + z + 10 = 0, and on the sides by the circular cylinder x2 + y 2 − x = 0.
27. Evaluate the following:
Z π/4 Z 1 Z r cos θ
(a)
r sec3 θ dz dr dθ.
0
(b)
Z
0
0
π/4
Z
0
0
π
Z
2 cos φ
ρ2 sin φ dρ dθ dφ.
0
Z
28. Evaluate using cylindrical coordinates
1
0
29. Evaluate using spherical coordinates
Z
0
3
Z
0
√
1−x2
Z √4−x2 −y2
dz dy dx.
0
Z √9−y2 Z √9−x2 −y2 p
z x2 + y 2 + z 2 dz dx dy.
0
0