How could I have prepared for this test?

As written in the Term Test 3 instructions, we suggested you complete all lecture worksheets, solve textbook exercises, review the problem sets, review the readings, and memorize definitions. Please read these comments carefully and the detailed remarks in correspond solutions. Use them to reflect on the feedback from your test.

Q1) No part marks were awarded for any part.

(1a) was similar to worksheet F1 Q10.

(1b) was similar to worksheet F2 Q2. It is a standard definition.

(1c) was similar to worksheet F3 Q3. It is a standard definition.

(1d) was similar to worksheet F3 Q2.

(1e) was similar to worksheet F4 Q4. It is a standard definition.

Q2) No part marks were awarded for any part.

(2a) was similar to worksheet F5 Q8.

(2b) was similar to worksheet F8 Q9.

(2c) was similar to worksheet F9 Q1.

(2d) was similar to worksheet G1 Q1.

(2e) is a basic computation.

Q3) Part marks were awarded for (3a) to (3d). No part marks were awarded for (3e).

(3a) was similar to worksheet F5 Q3.

(3b) was similar to worksheet F6 Q1.

(3c) was similar to worksheet F7 Q1.

(3d) was similar to worksheet F9 Q7.

(3e) was similar to worksheet G2 Q4.

Q4) Part marks were awarded if there was one minor error in the bounds or integrand. Multiple minor errors or significant errors received no credit. These are all standard setups of iterated integrals which are similar

to many provided problems. A specific suggestion is provided for each.

(4a) was similar to worksheet G4 Q2.4.

(4b) was similar to worksheet G3 Q5.3 and F8 Q2.

(4c) and (4d) were similar to worksheet G6 Q6.

Q5) This was similar to worksheet G7 Q8 and almost identical to G7 Q11.1, which was discussed in Tutorial 17.

Q6) This was very similar to worksheet G9 Q5, which was discussed in lecture.  Moreover, this was nearly

identical to worksheet G9 11.5, which was discussed in Tutorial 17.  Note this was a proof of the basic comparison test, so you cannot use the basic comparison test in your proof.

Q7) This was similar to many problems in worksheet G5, most notably Q7 which was discussed in the Tutorial

16 worksheet. Writing down a correct integral without an explanation of your process is worth very little credit. The instructions explicitly requested the area of a slice, and to sketch a labelled typical slice.

Do not tear this page off. This page is a formula sheet and will not be graded under any

circumstances. It can be used for rough work only.

Trigonometry

sin = cos = sin = cos = sin = cos =

sin(  e ) =    sine         cos(  e ) = cose         tan(  e ) =    tane             tane = sece = cote = = csce =

sin2 e + cos2 e = 1       1 + tan2 e = sec2 e        1 + cot2 e = csc2 e

cos2e = cos2 e sin2 e = 2cos2 e 1 = 1 2sin2 e         sin2e = 2sine cose         tan2e = cos2 e = sin2 e = 1 2(co)s2e         tan2 e =1(1)

Antiderivatives

l x n dx = + C l dx = ln|x| + C l ax dx = + C l sin xdx = cos x + C l cos xdx = sin x + C l tan xdx =    ln| cos x| + C

l sec xdx = ln| sec x + tan x| + C l dx = arcsin x + C l dx = arctan x + C l udv = uv + l vdu l f (g(x))g\ (x)dx = l f (u)du

Linear algebra

a · b = aT b = a1 b1 + · · · + an bn = ||a||||b|| cose a b = (a2 b3 a3 b2 , a3 b1 a1 b3 , a1 b2 a2 b1 ) e1 e2 = e3

|a · b| ≤ ||a||||b||

Coordinate systems

||a + b|| ≤ ||a|| + ||b||        (AB)T = BTAT

(x , y) = (r cose , r sine )

(AB) 1 = B 1A 1

dA = dxdy = rdrde

(x , y, z) = (r cose , r sine , z) = (ρ cose sinϕ , ρ sine sinϕ , ρ cosϕ)

r 2 = x 2 + y2 , ρ 2 = x 2 + y2 + z2

dV = dxdydz = rdrde dz = ρ 2 sinϕdρe dϕ

lg(Ω) f dV = lΩ (f o g)| det Dg|dV

1.  (5 points) The parts of this question are unrelated. No justification is necessary.

When necessary, fill in EXACTLY ONE circle.                                                                 (unfilled filled )

(1a) Let R be a rectangle in Rn . Let P, P\ , P\\ be partitions of the rectangle R.

Which statement is necessarily TRUE?

If R1 and R2 are distinct subrectangles of P , then R1 U R2 = ?.

The set P\ n P\\  is a partition of R.

If P\  is a refinement of P , then P\  contains P .

If P is the common refinement of P\  and P\\ , then P contains P\  and P\\ .

None of the above statements are true.

(1b){(L)o(e)f(t)Let(ion) f(o)y(=)x(2)2] y([).C(1),om(3])put(de)e(fi)t(n)h(e)e(d) low(by t)er(he)su(p)f(n)ri(1)e(2)y(})ou(of)r[fi(0)a(2)l(]) a(a)n(n)swe(d p)r(a)o(rt)n(it)l(i)y(o)n

(1c) Let R be a rectangle in Rn . Let f : R 一 R be bounded.

Which statement is necessarily TRUE?

For every partition P of R, IRf ≤ LP (f ).

There exists a partition P of R such that IRf = LP (f ).

If f is continuous on R, then there exists a partition P of R such that lR f dV = LP (f ).

There exists a partition P of R such that IRf 0.01 ≤ LP (f ).

None of the above statements are true.

(1d) Let R be a rectangle in Rn . Let f : R 一 R be bounded. Here is an attempted proof that IRf ≤ IRf .

Select the most accurate assessment of this argument.

The proof is essentially correct, but missing some minor details and justifications. Line 1 is flawed since it is possible that LP (f ) UP (f ).

Line 1 is flawed since LP (f ) or UP (f ) are not necessarily defined.

Line 2 is flawed since it assumes an invalid property of suprema and infima.

Line 3 is flawed since IRf = infP LP (f ) and IRf = supP UP (f ).

(1e) Let D Rn be a set and let f : D 一 Rm be a function.

Which statement is EQUIVALENT to "f is uniformly continuous on D"?

Aa = D, Aϵ > 0,3δ > 0 s.t. Ax = D, ||x a|| < δ =>  ||f (x) f (a)|| < ϵ

Aϵ > 0,Aa = D, 3δ > 0 s.t.  Ax = D, ||x a|| < δ =>   ||f (x) f (a)|| < ϵ

Aϵ > 0,3δ > 0 s.t. Ax = D, Aa = D, ||x a|| < δ =>   ||f (x) f (a)|| < ϵ

Aϵ > 0,Aa = D, 3δ > 0 s.t. f (Bδ (a) U D) Bϵ (f (a))

2.  (5 points) The parts of this question are unrelated. No justification is necessary. Fill in EXACTLY ONE circle. (unfilled filled

TRUE?

If Sc = Rn / S does not have zero Jordan measure, then S has zero Jordan measure. If So is empty, then S has zero Jordan measure.

If ∂ S has zero Jordan measure, then S is bounded.

If S has zero Jordan measure, then S has zero Jordan measure.

None of the above statements are true.

(2b) For a cheese slice S R2 with mass density δ, approximate the position of its centre of mass.

Which statement is TRUE?

Σ is the set of all subsets of Ω .

0 ≤ ϕ(x) ≤ 1 for all x 2 Ω .

P(n1 An ) =z1 P(An ) for any collection of disjoint events {An }1 in Σ .

If A 2 Σ, then A 2 Σ and P(A) = P(A).

None of the above statements are true.

(2d) Let f : [0,1] [2,3] ! R be bounded. Define the three quantities

A = l 1 l 3 f (x , y)dydx ,

B = l 3 l 1 f (x , y)dxdy,

C = f dA.

[0,1] [2,3]

Which statement is necessarily TRUE?

If A and B exist, then C exists.

If A and B exist and A = B, then C exists.

If C exists, then both A and B exists.

If C exists, then at least one of A or B exists.

None of the above statements are true.

(2e)  Evaluate I = l01 l0x (x + 2y)dydx . Write your final numerical answer only.

3.  (5 points) The parts of this question are unrelated. No justification is necessary.

Fill in ALL boxes that apply. If none apply, leave it blank.

(3a) Which of the following sets have zero Jordan measure?

铡 B1 (0)

铡 {(x , y) 2 R2  : y = 2x}

1  {1,2} ([0,1] \ Q)

1  {(x , y, z) 2 R3  : x 2 + y2 = 4,2 ≤ z ≤ 37}

(3b) Which of the following sets are Jordan measurable?

铡 Rn

1 ∂ B1 (0)

铡 [3,4]n \ Qn

1  {(x , y, z) 2 R3  : x 2 + y2 < 4,2 ≤ z < 37}

(3c) Which of the following functions f are integrable on the sets S ?

1 S = {(x , y) 2 R2  : 0 < x 2 + y2 ≤ 1} and f (x , y) = x 2 + y2 .

铡 S = {(x , y) 2 R2  : 0 < x 2 + y2 ≤ 1} and f (x , y) = 1/(x2 + y2 ).

1 S = {(x , y) 2 R2  : x 2 + y2 = 1} and f (x , y) = 237 if (x , y) 2 Q2 and f (x , y) = 0 otherwise.

铡 S = {(x , y) 2 R2  : x 2 + y2 ≤ 1} and f (x , y) = 237 if (x , y) 2 Q2 and f (x , y) = 0 otherwise.

(3d) Let (Ω, Σ, P) be a continuous probability space in R2  for selecting a vector uniformly inside the square Ω = [一8,8]2 . Which statements are events occurring with probability zero?

铡 The vector lies inside the first quadrant.

1 The vector has magnitude 1.

铡 The vector points downward but not directly down.

铡 The vector has two rational components.

(3e)  Let R = [a, b] [c, d] [e, f ] be a rectangle in R3 and let φ : R ! R be bounded. According to Fubini’s theorem, the identity

lR φdV = lab lef lc d φ(x , y, z)dydzdx

holds and both quantities exist provided which assumption(s) hold? Select as few as possible.

1 φ is integrable on [a, b] [c, d] [e, f ].

1 For every x 2 [a, b], the x -slice φx is integrable on [c , d] [e, f ]

铡 For every y 2 [c, d], the y -slice φy is integrable on [a, b] [e, f ]

铡 For every z 2 [e, f ], the z -slice φz is integrable on [a, b] [c, d]

铡 For every x 2 [a, b], y 2 [c , d], the (x , y)-slice φx ,y is integrable on [e, f ]

1 For every x 2 [a, b], z 2 [e, f ], the (x , z)-slice φx ,z is integrable on [c , d]

铡 For every y 2 [c, d], z 2 [e, f ], the (y, z)-slice φy,z is integrable on [a, b].

4.  (8 points) The parts below are unrelated. No justification is necessary. Do not evaluate any integral(s)

(4a) Let W = {(x , y) 2 R2  : 1 ≤ x 2 + y2 ≤ 4, x ≥ 0, y ≥ 0} be a plate with continuous density f : R2 ! R.

Express the mass of W as an iterated double integral using polar coordinates with order drde .

(4b) Let H = {x2 + y2 + z2 ≤ 4, z ≥ 0} be the solid upper hemisphere of radius 2.

Express vol(H ) as an iterated double integral with order dydx .

Express vol(H ) as an iterated triple integral using cylindrical coordinates with order drde dz .

Express vol(H ) as an iterated triple integral using spherical coordinates with order dρde dϕ .