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Name (Print Clearly):
Student Number:
MTH132 Section 5 & 18, Quiz 6
Nov 17, 2008
Instructor: Dr. W. Wu
Instructions: Answer the following questions in the space provided. There is more than adequate space provided to answer each
question. The total time allowed for this quiz is 15 minutes.
1. [2 pts each].
2
(a)
Convert the sigma notations to addition (summation) notation and evaluate the sums.
6k
 k 1
k 1
6 1 6  2

 3 4  7
11 2 1
4
(b)
 cos k
k 1
cos  cos 2  cos 3  cos 4  1  1  1  1  0
2. Evaluate the sums by formulas.
(a)[2 pts]
(b)[3 pts]
7  (7  1)
 1)  2  (28  1)  2  27  54
2
7
7
k 2
k 1
5
5
5
5
k 1
k 1
k 1
k 1
 (2k )  2( k  1)  2(
 k (3k  5)   (3k 2  5k )  3 k 2  5 k  3
2
5
5
1
1
1
5  6  11
5 6
5
 165  75  240
6
2
3. [2 pts each]. Suppose  f ( x)dx  4 ,  f ( x)dx  6 ,  g ( x) dx  8 . Use the rules to find
4
(a)
 g ( x)dx  0
4
2
(b)

5
(c)
5
5
2
2
1
1
f ( x)dx   f ( x)dx  (  f ( x)dx   f ( x)dx)  (6  (4))  10
5
5
5
1
1
1
 [5 f ( x)  3g ( x)]dx  5 f ( x)dx  3 g ( x)dx  5  6  3  8  30  24  6
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4. Let f ( x)  2 x  1 defined over [0,3] . Consider the Riemann sum:
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n
 f (c )  x
k 1
k
k
. Usually, we choose
equal length xk  x  (b  a) / n . (a) [2 pts]. Partition [0,3] into 4 subintervals of equal length, then
choose the right-hand endpoint of each subinterval ( ck  xk  a  kx ) to evaluate the Riemann sum.
x  (b  a) / n  (3  0) / 4  3 / 4
ck  xk  0  k (3 / 4)  3k / 4
4
 f (ck )  x 
k 1
3 4
3k
9 4
3 4
9 45 3
45
(
2

1
)

k

1 
 4 
3



4 k 1 4
8 k 1
4 k 1
8 2
4
4
(b) [2 pts]. Find a formula for the Riemann sum obtained by dividing the interval into n equal subintervals.
x  (b  a) / n  (3  0) / n  3 / n
ck  xk  0  k (3 / n)  3k / n
n
 f (ck )  x 
k 1
3 4
3k
18 n
3 n
18 n(n  1) 3
9(n  1)
9
(
2

1
)

k

1 2 
 n 
 3  12 


2 
n k 1 n
n k 1
n k 1
n
2
n
n
n
(c) [1 pt]. Find a definite integral to express the limit of the Riemann sum as n approaches infinity.
the limit of the Riemann sum as n approaches infinity is the definite integral of this function over [0,3]
3
 (2 x  1)dx
0
(d) [bonus 2 pts]. Evaluate this definite integral. (Take the limit of this sum or use the area of a certain region)
n
lim
n
9
 f (c )  x  lim (12  n )  12  0  12
k 1
k
n 