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16s MATH 2451 thq 09
Multivariable Calculus
2016-04-12 (Tue)
McCary
T.C. Use
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ID
Exam
Course
Due Date
Instructor
(First Name)
(Last Name)
2016-04-07 17:50
1. (10 points) Be sure to clearly label every cube.
(a) For each shaded cube C~k,N determine its “address” (the values of ~k and N ).
(b) Shade the following cubes: C

2
,1
−3
and C


10
,2
1
and C

3
4
 ,0
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
4
3
2
1
−1
−2
−3
−4
−4
−3
−2
−1
1
2
3
4
2. (10 points) Recall the definition of the n-volume of a set:
Z
voln (A) = 1A |dn x|
Rn
A set A ⊂ R2 is shaded on the axis below. The 2-volume of A is defined as the integral above (with n = 2).
(a) Compute L1 (1A )
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(b) Compute U1 (1A )
4
3
2
1
−1
−2
−3
−4
−4
−3
−2
−1
1
2
3
4
3. (10 points) Recall that a set A ⊂ Rn is said to have n-volume zero if for every ε > 0, there
is N > 0 such
that the
x sum of the volume of all of the cubes C ∈ DN needed to cover A is less than ε. Let A =
x ∈ (0, 1) ⊂ R2 . In
x this exercise, you’ll show that A has 2-volume zero.
(a) For the specific cases N = 0, 1 and 2, determine which cubes C ∈ DN are needed to cover A, and determine the
total 2-volume of the cubes in each cases. Provide a sketch.
(b) For a general N , determine which cubes C ∈ DN are needed to cover A, and determine the total 2-volume of
these cubes.
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(c) Use the above to show that A has 2-volume zero.
4. (10 points)
(a) Find two functions f and g which are not integrable, yet f + g is integrable.
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(b) Find two functions f and g which are not integrable, yet f · g is integrable.
(c) Find a function f which is not integrable, yet |f | is integrable.
5. (10 points) Our definition of integral defines LN (f ), UN (f ). One way to visualize/understand these definitions is that
they are defining piecewise constant functions. That is, LN (f ) is the lower sum on paving DN , but it can also be
understood as defining and N th lower function which is constant on each cube C ∈ DN (and similarly for UN (f )).
Then the integral lower and upper functions is trivial because they are piecewise constant: the integral is a sum.1
y
y
y
y
L1 (f )
L2 (f )
L3 (f )
L4 (f )
x
x
x
x
y
y
y
U2 (f )
U3 (f )
U4 (f )
ID
U1 (f )
y
x
x
x
x
(
x2 + 1 0 ≤ x < 1
For f (x) =
, draw LN (f ) and UN (f ) for N = 0, 1, 2 on the provided axes, and caculate the value
0
otherwise
of each lower and upper sum.2
y
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y
x
x
x
L0 (f ) =
L1 (f ) =
L2 (f ) =
y
y
y
x
U0 (f ) =
1
y
x
U1 (f ) =
x
U2 (f ) =
Soon (in lecture) we’ll define a piecewise constant function which is exactly the integral of f : the lower and upper integrals are defined with
inf and sup, how will we define this exact variant?.
2
Just the value, showing the calculation is not necessary.
6. (10 points)
(a) In a similar fashion
to the previous exercise, draw3 and compute4 LN (f ), UN (f ) for the given f for N = 0, 1, 2.
( 2
x + y 2 0 < x, y < 1
x
Let f
=
.
y
0
otherwise
L0 (f ) =
L1 (f ) =
L2 (f ) =
2
2
2
1
1
1
1
0.2
0.4
0
0
0.5
0.6
0.8
0.2
0.4
1 0
0.8
2
1
1
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2
0.4
0
0
0.5
0.6
(b) Now compute
R
0.8
1 0
0.2
0.4
0.6
1 0
1
1
0.5
0.8
1
0
0
0.2
0.4
1 0
Z
f |dx dy| ≤ U2 (f ) ≤ U1 (f ) ≤ U0 (f )
R2
4
0.8
2
f |dx dy| using your prior knowledge of iterated integrals, and show that:
L0 (f ) ≤ L1 (f ) ≤ L2 (f ) ≤
0.5
0.6
U2 (f ) =
R2
3
0.4
1 0
1
0.2
0.2
U1 (f ) =
U0 (f ) =
0
0
1
0
0
0.5
0.6
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0
0
1
Make it look nice! You may need to practice.
Just the value, showing the calculation is not necessary. By all means, write a program or use a spreadsheet.
0.5
0.6
0.8
1 0
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This page won’t be graded.