Math 3210-3
HW 21
Due Tuesday, November 13, 2007
Taylor’s Theorem
1. √
Use the example from class to approximate the following: (Note: The example dealt with f (x) =
1 + x.)
√
(a) 3
√
(b) 1.2
2. Assuming we know the familiar derivative formulas for the trigonometric functions, let f (x) = sin x.
(a) Find p6 for f at x = 0.
(b) How accurate is this on the interval [−1, 1]?
3. ♣ Show that if x ∈ [0, 1], then
x−
x2
x3
x4
x2
x3
+
−
≤ log(1 + x) ≤ x −
+ .
2
3
4
2
3
The Riemann Integral
4. ♣ Theorem 33 revised says: Let S and T be nonempty subsets of R with s ≤ t for all s ∈ S and t ∈ T .
Then sup S ≤ inf T . Use this theorem to give another proof of Theorem 96. Specify the sets S and T
that you use.
n−1
1 2
,1 .
5. ♣ Let f (x) = x3 for x ∈ [0, 1]. Find L(f ) and U (f ) using the partitions Pn = 0, , , . . . ,
n n
n
2
2
n
(n
+
1)
Hint: You may use the formula 13 + 23 + · · · + n3 =
.
4
6. Let f be integrable on [a, b] and suppose that g is a function on [a, b] such that g(x) = f (x) except for
Rb
Rb
finitely many x ∈ [a, b]. Show that g is integrable and that a f = a g.
7. ♣ Show that if f is integrable on [a, b], then f is integrable on every interval [c, d] ⊆ [a, b].