HP 40gs hp 40gs_user's guide_English_E_HDPMSG40E07A.pdf - Page 296
Exercise 8
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hp40g+.book Page 22 Friday, December 9, 2005 1:03 AM Exercise 8 Part 1 For this exercise, make sure that the calculator is in exact real mode with X as the current variable. For an integer, n, define the following: ∫ un = 2 2----x----+-----3- e -xn dx 0 x+2 Define g over [0,2] where: g(x) = 2----x----+-----3x+2 1. Find the variations of g over [0,2]. Show that for every real x in [0,2]: 3-- ≤ g(x) ≤ 7-- 2 4 2. Show that for every real x in [0,2]: 3-- -x- en ≤ -x- g(x)en ≤ 7-- e -xn 2 4 3. After integration, show that: 3-- ⎛ ⎜ n e 2-n 2⎝ - ⎞ n⎟ ⎠ ≤ un ≤ 7-- ⎛ ⎜ 2-- nen 4⎝ - ⎞ n⎟ ⎠ 4. Using: lim e---x---------1-- = 1 x→0 x show that if un has a limit L as n approaches infinity, then: 3 ≤ L ≤ 7-2 16-22 Step-by-Step Examples
16-22
Step-by-Step Examples
Exercise 8
For this exercise, make sure that the calculator is in exact
real mode with
X
as the current variable.
Part 1
For an integer,
n
, define the following:
Define g over [0,2] where:
1.
Find the variations of g over [0,2]. Show that for
every real x in [0,2]:
2.
Show that for every real
x
in [0,2]:
3.
After integration, show that:
4. Using:
show that if
has a limit
L
as
n
approaches infinity,
then:
u
n
2
x
3
+
x
2
+
--------------
e
x
n
--
x
d
0
2
∫
=
gx
()
2
x
3
+
x
2
+
--------------
=
3
2
--
gx
()
7
4
--
≤
≤
3
2
--
e
x
n
--
gx
()
e
x
n
--
7
4
--
e
x
n
--
≤
≤
3
2
--
ne
2
n
--
n
–
⎝
⎠
⎜
⎟
⎛
⎞
u
n
7
4
--
ne
2
n
--
n
–
⎝
⎠
⎜
⎟
⎛
⎞
≤
≤
e
x
1
–
x
-------------
x
0
→
lim
1
=
u
n
3
L
7
2
--
≤
≤
hp40g+.book
Page 22
Friday, December 9, 2005
1:03 AM