Math 171-501
Quiz 12
Fall 2007
Calculus
Instructions Please write your name in the upper right-hand corner of the
page. Write complete sentences to explain your solutions.
1. Suppose f (x) = 4 cos x. Show that the Riemann sum for the function f on the interval [0, π/2] using left-hand√endpoints
for the parti√
tion {0, π/6, π/4, π/3, π/2} is equal to π(6 + 3 + 2 )/6.
[This is exercise 8 on page 377 of the textbook.]
Solution. Writing out
4 cos(0)
P4
i=1
f (xi−1 )∆xi gives the sum
π π − 0 + 4 cos(π/6)
−
6
4 6
π π π π
+ 4 cos(π/3)
,
−
−
+ 4 cos(π/4)
3
4
2
3
π
which simplifies to
√
√
π
1 π
3 π
2 π
4· +4·
·
+4·
·
+4· ·
6
2 12
2 12
2 6
√
√
or π6 (6 + 3 + 2 ).
2
n
X
4i
2
sin
as a definite integral.
2. Express lim
2
n→∞
n
n
i=1
Solution. The solution is not unique, but perhaps the most natural
interpretation of the expression is the limit of Riemann sums for regular
partitions of the interval [0, 2] into n subintervals each of width 2/n,
where right-hand endpoints are used, and the function is sin(x2 ). The
R2
limit is then equal to the definite integral 0 sin(x2 ) dx.
Another interpretation is that the interval is [0, 1], the n subintervals
of the partition each have width 1/n, Rthe function is 2 sin(4x2 ), and the
1
limit is equal to the definite integral 0 2 sin(4x2 ) dx.
November 15, 2007
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Dr. Boas
Math 171-501
Quiz 12
Fall 2007
Calculus
3. Let p1 = 2, p2 = 3, p3 = 5, and in general let pn be the nth prime
k
X
1
> 1?
number. What is the smallest value of k for which
p
i=1 i
Solution. When k = 1, the sum equals 21 . When k = 2, the sum
equals 12 + 31 , or 56 . When k = 3, the sum equals 12 + 31 + 15 , or 31
. Thus
30
the sum first exceeds 1 when k = 3.
4. RThe figure shows the graph of a function f . Estimate the value of
4
f (x) dx by using a regular partition with four subintervals and choos0
ing right-hand endpoints.
y
4
3
2
1
0
1
2
3
4
x
Solution. The estimate is the shaded area below: namely, 10.
y
4
3
2
1
0
November 15, 2007
1
2
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3
4
x
Dr. Boas