name:
section 23
Quiz 5
Math 133
Oct 14, 2014
1. Find the Arc Length of the function f (x) = ln(sec x) from x = 0 to x =
For f (x) = ln (sec x), we have f 0 (x) =
1
sec x
π
.
4
(sec x tan x) = tan x. So,
π
4
Z
Arc Length =
q
1 + f 0 (x)2 dx
0
π
4
Z
=
q
1 + tan2 (x)dx
0
π
4
Z
=
√
sec2 xdx
0
π
4
Z
sec xdx
=
0
π
= ln |sec x + tan x||04
√
2+1 .
= ln
n ∞
. Does it converge or diverge? If it converges,
2. Consider the sequence 1 + n1
n=1
what does it converge to?
n
ln(1+ n1 )
Let an = 1 + n1 . Then ln an = n ln 1 + n1 =
. We have
1
n
lim ln an =
n→∞
=
lim
ln 1 +
lim
n
(form
1
n
n→∞
1
1
1+ n
− n12
− n12
1
= lim
n→∞ 1 + 1
n
= 1.
n→∞
1
0
)
0
(L’Hopital’s Rule)
Therefore, limn→∞ an = e1 = e, and the sequence converges to e.