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Name:
MATH 215
MIDTERM II
This Exam contains 5 problems. The problems are worth 12 points each. Each part of a problem counts equally. On problems 3, 4 and 5 you can get partial credit. Hence, explain yourself carefully on these problems.
NO CALCULATOR. 2 TWO-SIDED 3in. BY 5in. NOTECARD OK. CHECK YOUR SECTION IN THE TABLE
Sec. Time Exam rm. Professor GSI ME
20 9-10 1210 Chem Angela KUBENA Harlan KADISH
30 10-11 1400 and 1800 Chem Harry D’SOUZA Robin LASSONDE
40 11-12 1400 and 1800 Chem Harry D’SOUZA Giwan KIM
50 12-1 1400 and 1800 Chem Harry D’SOUZA Holly CHUNG
60 1-2 170 Denn Zuoqin WANG Timothy FERGUSON
70 2-3 182 Denn Zuoqin WANG Crystal ZEAGER
X X 269 Denn Extended time 5-11pm X
1
Problem 1. TRUE/FALSE QUESTIONS. NO PARTIAL CREDIT. CIRCLE TRUE OR FALSE. IF YOU THINK A STATEMENT DOESN’T MAKE SENSE, CIRCLE FALSE.
(a) (^) ∫ 1
0
∫ (^) π/ 4
0
y 1 + (xy)^2
dxdy < π TRUE / FALSE
(b) ∫ (^1)
0
0
x + y dxdy =
0
0
x +
y dxdy = 2
0
0
x dxdy =
TRUE / FALSE
(c) The double integral of the function f (x, y) = x^4 y^2 over the disc x^2 + y^2 ≤ 1 is equal to
4
0
0
x^4 y^2 dxdy TRUE / FALSE
(d) Let F~ =< P, Q, R > be a vector field on R^3. Suppose that P, Q, R have continuous first-order derivatives. Suppose also that F~ is conservative. Then there is a function f so that
F^ ~ (x, y, z) = ∇f (x, y, z) TRUE / FALSE
(e) The integral of the function f (x, y) = x + y over the triangle {(x, y); 0 ≤ x ≤ 1 , 0 ≤ y ≤ x} is equal to (^) ∫ x
0
0
(x + y) dx
dy TRUE / FALSE
(f) The area between the two graphs y = ±
1 − x^2 , − 1 ≤ x ≤ 1 is equal to the integral
C x dy where C is the circle x^2 + y^2 = 1 oriented counterclockwise. TRUE / FALSE
WORKSPACE:
Problem 2. NO PARTIAL CREDIT
In this problem you have to simplify your answers. For example an expression like 2 sin( π 4 ) should
be simplified to something like
(a) The point (x, y) = (− 1 ,
3). Find the corresponding coordinates (r, θ).
(b) The point (r, θ) = (3, − π 4 ). Find the corresponding coordinates (x, y).
(c) The point (ρ, θ, φ) = (3, π 4 , 34 π ). Find the corresponding coordinates (x, y, z).
(d) The point (x, y, z) = (1, − 1 , −
2). Find the corresponding coordinates (ρ, θ, φ).
(e) The point (r, θ, z) = (2, 23 π , 1). Find the corresponding coordinates (x, y, z).
(f) The point (x, y, z) = (−
3 , 1 , −1). Find the corresponding coordinates (r, θ, z).
ANSWERS: a:
b:
c:
d:
e:
f:
WORKSPACE:
Problem 4. YOU MUST SHOW YOUR WORK TO GET ANY CREDIT. YOU CAN GET PARTIAL CREDIT IF YOU EXPLAIN YOURSELF CAREFULLY.
(a) Evaluate (^) ∫ ∫ ∫
E
(x^2 + y^2 + z^2 ) dV
where E is that part of the unit ball centered at the origin, inside the cone x^2 + z^2 = y^2 , 0 ≤ y.
(b) Find the mass of the ball x^2 + y^2 + z^2 ≤ 1 with density function ρ(x, y, z) = (x^2 + y^2 + z^2 )^2.
(c) Find the gradient vector of the function f (x, y, z) = xy + 5 + (^) x (^2) +xz 2 at the point (1, 2 , 3).
(d) Evaluate the line integral (^) ∫
C
F^ ~ · d~r
where F~ (x, y, z) =< z, x, y > and ~r(t) =< t, t^2 , t^3 >, 0 ≤ t ≤ 1.
ANSWERS: a:
b:
c:
d:
WORKSPACE:
Problem 5. YOU MUST SHOW YOUR WORK TO GET ANY CREDIT. YOU CAN GET PARTIAL CREDIT IF YOU EXPLAIN YOURSELF CAREFULLY.
(a) Evaluate the line integral (^) ∫
C
F^ ~ · d~r
where F~ =< yz, xz, xy > and ~r(t) =< tet
(^3) − 1 , t^2 cos(2πt^3 ), t^3 sin πt
5 2 >,^0 ≤^ t^ ≤^1.
(b) Evaluate the integral (^) ∫
C
tan(1 + x^2 ) dx + (
x^3 3
where C is the unit circle x^2 + y^2 = 1 oriented clockwise.
ANSWERS: a:
b:
WORKSPACE:
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
- WORKSPACE PROBLEM
