Math 241
Name:
Final Exam
1. [6 points] Consider the function
f (x, y) =
1 4
3
x − 2xy + y 2 − y.
2
2
Determine whether the critical point (1, 1) for f is a local maximum, a local minimum, or neither.
2. [6 points] Evaluate
Z 1Z 1 y
e
0
x2
√ dy dx by changing the order of integration.
y
3. [6 points] Find the equation of the tangent line to the curve
x(t) = t 3 − 2t, t 3 + 1
at the point (−1, 2). (Your answer should be an equation involving x and y.)
4. [6 points] Find the equation of the tangent plane to the surface X(s,t) = 2t cos s, t sin s, t 3 at the
point (4, 0, 8).
5. [24 points (4 pts each)] The following figure shows a contour plot for a function f : R2 → R.
5
0
1
2
3
4
6
7
5
8
20
4
5
4
10
18
16
3
12
14
14
3
16
2
2
6
4
2
12 14
8 10
18
1
0
1
0
1
2
3
4
5
6
7
0
(a) The function f has three critical points in the rectangle 0 ≤ x ≤ 7, 0 ≤ y ≤ 5. Estimate the
coordinates of each of these points.
(b) For each of the three critical points in part (a), indicate whether the Hessian H f at the point is
positive definite, negative definite, or neither.
(c) Estimate the maximum value of the function f (x, y) on the rectangle 0 ≤ x ≤ 7, 0 ≤ y ≤ 5.
For the problems on this page, your answer must be correct to within 10% to receive full credit.
5
0
1
2
3
4
6
7
5
8
20
4
5
4
10
18
16
3
12
14
14
3
16
2
2
6
4
2
12 14
8 10
18
1
0
1
0
1
2
3
4
5
6
7
0
Z 2Z 2
f (x, y) dy dx.
(d) Estimate
1
0
ZZ
f (x, y) dA, where D is the disk of radius 1 centered at the point (3, 3).
(e) Estimate
D
Z
f (x, y) ds, where L is the line segment from the point (0, 2) to the point (3, 2).
(f) Estimate
L
ZZ
6. [10 points] Evaluate
S
and (3, 1).
y2 dA, where S is the square region in R2 with vertices (1, 0), (0, 2), (2, 3),
Z
7. [10 points] Evaluate
terclockwise.
(−y, x) · ds, where C is the top half of the ellipse x2 + 4y2 = 4, oriented coun-
C
8. [4 points] Find the coordinates of the point at which the helix x(t) = cos(πt), sin(πt),t intersects the
paraboloid z = 4 − x2 − y2 .
9. [16 points)] Let S be the portion of the surface z = ln(r) in the range 0 ≤ z ≤ 1, with upward-pointing
normal vectors.
(a) Evaluate
ZZ p
x2 + y 2 dS.
S
ZZ
(b) Evaluate
S
e z k · dS.
10. [12 points] Evaluate
Z 1 Z √1−x2Z x2 +y2p
−1 0
0
x2 + y2 dz dy dx by changing to cylindrical coordinates.