MATH 112 QUIZ # 1
(1) Let P = cfw_0, /4, /2, 3/4, be a partition of [0, ], and let f (x) = sin x. Calculate
the upper and lower Riemann sums U (f, P ) and L(f, P ). Solution. Since f is increasing on the intervals [0, /4], [/4, /2] and decreasing on [/2,
MATH 112 QUIZ # 2
1
x x
x
f (t)dt = a .
(1) Suppose that f is continuous and lim f (x) = a. Show that lim
x
0
Solution. Given > 0 there is M > 0 such that |f (x) a| /3 for x M . Choose
N M such that 3|a|M N , and so that if x N , then:
1
x
M
f (t)dt
0
.
MATH 112 QUIZ # 4
(1) Compute the following:
(a) f (x), where f (x) = xx . Solution.
d
d
f (x) = f (x) log f (x) = xx (x log x) = xx (log x + 1) .
dx
dx
1
ax dx, where a > 0. Solution.
(b)
0
1
1
ex log a dx =
ax dx =
0
0
ex log a
log a
1
=
0
elog a 1
a1
=
MATH 112 QUIZ # 5
(1) Compute the following:
(a)
dx
x log x
Solution. Do a u substitution: u = log x. Then the integral is:
du
=
= log u = log(log x) + C ,
u
where C is a constant.
dx
1 + ex
(b)
Solution. Do a u substitution: u = 1 + ex . Then
du
= dx. So
MATH 112 QUIZ # 6
(1) (a) Compute the n-th degree Taylor polynomial of f (x) = log(1 + x) at x = 0.
(b) Prove that if f (a) exists, then:
f (a + h) + f (a h) 2f (a)
f (a) = lim
.
h0
h2
(2) (a) Dene lim supn an for a bounded sequence cfw_an .
(b) Prove tha