Math 3210-1
HW 15
Due Friday, July 20
L’Hospital’s Rule
1. Evaluate the following limits.
tan x − x
x3
sin x − x
lim
x→0 ex − 1
x2
lim x
x→∞ e
log sin x
lim
x→0+ log x
(log x)2
lim
x→∞
x
2x
lim+ x
(a) lim
x→0
(b)
(c)
(d)
(e)
(f)
x→0
2. Indicate what is wrong with the following result:
2x2 − x − 1
4x − 1
4
2
= lim
= lim =
x→1 3x2 − 5x + 2
x→1 6x − 5
x→1 6
3
lim
Taylor’s Theorem
3. √
Use the example from class to approximate the following: (Note: The example dealt with f (x) =
1 + x.)
√
(a) 3
√
(b) 1.2
4. 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]?
5. 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
6. 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 95. Specify the sets S and T
that you use.
1 2
n−1
3
7. Let f (x) = x for x ∈ [0, 1]. Find L(f ) and U (f ) using the partitions Pn = 0, , , . . . ,
,1 .
n n
n
n2 (n + 1)2
Hint: You may use the formula 13 + 23 + · · · + n3 =
.
4
8. 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.
9. Show that if f is integrable on [a, b], then f is integrable on every interval [c, d] ⊆ [a, b].