Maths 4200-Fall 2014
Review and practice exercises for Midterm 2
1. Let f (x) = x1/3 for x ∈ R.
(i) Prove that f 0 (x) = 13 x−2/3 for x 6= 0.
(ii) Show that f is not differentiable at x = 0.
2. In each of the following cases, determine the intervals in which the function f is
increasing and find the maxima and minima (if any) in the set where each function
is defined
a)f (x) = x2 − 3x + 4,
x ∈ [0, 2].
2
b)f (x) = ln(x − 9),
|x| > 3.
c)f (x) = (sin x)/x if x 6= 0, f (0) = 1, 0 ≤ x ≤ π/2.
3. Define f, g as follows f (0) = g(0) = 0 and if x 6= 0, f (x) = sin(1/x) and g(x) =
x sin(1/x). Show that
a) f 0 (x) = −1/x2 cos(1/x), if x 6= 0;
f 0 (0) does not exist.
b)g 0 (x) = sin(1/x) − 1/x cos(1/x), if x =
6 0;
g 0 (0) does not exist.
4. Suppose that f is differentiable on R and that f (0) = 0, f (1) = 2, and f (2) = 2.
a) Show that there exists c1 ∈ (0, 1) such that f 0 (c1 ) = 2.
b) Show that there exists c2 ∈ (1, 2) such that f 0 (c2 ) = 0.
c) Show that there exists c3 ∈ (0, 2) such that f 0 (c3 ) = 1.
5. Use the mean value theorem to establish the following inequalities
a) sin x ≤ x, for x ≥ 0.
sin(ax) − sin(bx) ≤ |a − b|, for x 6= 0.
b) x
P
6. Show that if
|an − an+1 | < ∞ then (an ) converges, but not conversely.
7. Let (an )n andP(bn )n be sequences P
in R.
P
Show that if
bn converges and
|an − an+1 | < ∞, then
an bn converges.
8. Let (xn )n be a sequence of real numbers and let yn = xn − xn+1 for each n ∈ N.
P∞
(a) Prove that the series
n=1 yn converges if and only if the sequence (xn )n
converges.
P
(b) If ∞
n=1 yn converges, what is the sum?
9. Determine whether each series converges or diverges. Justify your answer.
P 2n
(a)
n!
P sin2 n
(b)
n2
P 1
√
(c)
n n+1
P (−1)n log n
(d)
n
(e)
P cos nπ
(f)
P
√
n
(−1)n
√
n2 +1
√
√
a, and define xn+1 = a + xn for n ≥ 1.
√
(a) Show that xn < 1 + a for all n.
(b) Show that (xn )n≥1 is an increasing sequence.
(c) Show that (xn )n≥1 converges and find its limit.
10. Suppose a > 0. Let x1 =
11. Let fn (x) =
x
x+n
for x ≥ 0.
(a) Show that f (x) = lim fn (x) = 0 for all x ≥ 0.
(b) Show that if t > 0, the convergence is uniform on [0, t].
(c) Show that the convergence is not uniform [0, ∞).
12. Let fn (x) =
nx
1+nx
for x ≥ 0.
(a) Find f (x) = lim fn (x) = 0.
(b) Show that if t > 0, the convergence is uniform on [t, ∞).
(c) Show that the convergence is not uniform [0, ∞).
13. Let fn (x) =
nx
enx
for x ∈ [0, 2].
(a) Show that lim fn (x) = 0 for all x ∈ [0, 2].
(b) Show that the convergence is not uniform on [0, 2].
(c) Let 0 < t < 2. Determine on which interval, [0, t] or [t, 2], the convergence is
uniform. Justify your answer.
14. Let fn (x) = nxn (1 − x) for x ∈ [0, 1]0.
(a) Find f (x) = lim fn (x) = 0.
(b) Show that the convergence is not uniform [0, 1].
R1
R1
(c) Does lim 0 fn (x)dx = 0 f (x)dx?
15. Let fn (x) = x +
1
n
and f (x) = x for x ∈ R.
(a) Show that (fn ) converges uniformly to f on R.
(b) Show that (fn2 ) converges pointwise to f on R but not uniformly.
16. Determine whether or not the given series of functions converges uniformly on the
indicated set. Justify your answers.
P x2n
(a)
,
[0, 1]
(n+x)2
P x2
(b)
,
[5, ∞)
n2
√
P −x
(c)
n ,
( 2, ∞)
P 1 q sin nx
(d)
,
x∈R
n
n
P 1
(e)
,
(0, 1]
1+(nx)2

Solutions for 2, 6, 8, 10

Bid #3048 - Audio Visual Equipment

縦方向の文字位置揃え