Answers to Math 261 SP15 Final Exam
The following are just the final numerical (or otherwise) answers for the Math 261 SP15 Final
Exam. Your solutions on the exam should include full details to allow for partial credit.
If you are having trouble coming up with these answers, please go to a review session or office
hours.
As usual, if you spot any errors, please let Dan know ASAP!
1. (a) True (b) True (c) False (d) False
2. (a) v(t) = h2t3 − 1, 1t + 2, et−1 i.
4
(b) r(t) = h t2 − t + 23 , ln(t) + 2t, et−1 + 2i
3. (a) 12 cos( π2 ) + 1e3·1 = e3 : Yep.
(b) 4e3 (y − 1) − (z − π2 ) = 0.
(c) r(t) = h1, 1 + 4e3 t, π2 − ti.
4. (a) Local max (144 > 0 so min or max, fxx = −12 < 0).
(b) (4, 0), (2, 2), (2, −2).
Z 2π Z 2+cos(θ)
7π
5.
r drdθ =
.
2
0
1
6. (a) No sketching for points.
Z √2 Z 2 Z 2−x
(b)
dzdxdy.
√
y2
− 2
Z
2Z
(c)
0
Z
π
Z
0
2−x Z
√
x
√
− x
0
dydzdx.
2
14
.
3
0
1
8. (a) No sketching for points.
7.
r sin(θ) r drdθ =
(b) hr cos(θ), r, r sin(θ)i, 1 ≤ r ≤ 3, 0 ≤ θ ≤ 2π.
Z 2π Z 3
√
(c)
r sin(θ)( 2r)drdθ.
0
1
9. (a) hx, 1 − z, zi .
(b) h0, −1, −1i.
(c) No sketching for points.
Z π Z sin(θ)
(d)
(−r cos(θ) − 1)rdrdθ.
0
0