ANALIZA MATEMATYCZNA LISTA ZADA 12
(1) Podaj wzór na Cn= Xn
i=1
b − a n f¡
a + ib − a n
¢, a nast¦pnie oblicz
n→∞lim Cn
(a) f(x) = 1, a = 5, b = 8; (b) f(x) = x, a = 0, b = 1;
(c) f(x) = x, a = 1, b = 5; (d) f(x) = x2, a = 0, b = 5; (e) f(x) = x3, a = 0, b = 1; (f) f(x) = 2x+5, a = −3, b = 4;
(g) f(x) = x2+ 1, a = −1, b = 2;
(h) f(x) = x3+x, a = 0, b = 4; (i) f(x) = ex, a = 0, b = 1.
(2) Oblicz nast¦puj¡ce caªki oznaczone poprzez konstrukcj¦ ci¡gu podziaªów przedziaªu, odpowiadaj¡cego mu ci¡gu sum Riemanna, oraz jego granicy
(a) Z 4
2
x10dx, (ti = 2 · 2i/n); (b) Z e
1
log x
x dx, (ti = ei/n);
(c) Z 20
0
x dx; (d) Z 10
1
e2xdx; (e)
Z 1
0
√3
x dx, (ti = ni33); (f) Z 1
−1
|x| dx; (g)
Z 2
1
dx
x dx, (ti = 2i/n); (h) Z 4
0
√x dx, (ti = 4in22).
(3) Oblicz caªki oznaczone (a)
Z π
−π
sin x2007dx; (b) Z 2
0
arctan([x]) dx; (c)
Z 2
0
[cos(x2)] dx; (d) Z 1
0
√1 + x dx;
(e) Z −1
−2
1
(11 + 5x)3 dx; (f) Z 2
−13
1 p5
(3 − x)4 dx; (g)
Z 1
0
x
(x2+ 1)2 dx; (h) Z 3
0 sgn (x3− x) dx; (i)
Z 1
0
x e−xdx; (j) Z π/2
0
x cos x dx;
1
(k) Z e−1
0
log(x + 1) dx; (l) Z π
0
x3 sin x dx; (m)
Z 9
4
√x
√x − 1dx; (n) Z e3
1
1 x√
1 + log xdx; (o)
Z 2
1
1
x + x3 dx; (p) Z 2
0
√ 1
x + 1 +p
(x + 1)3 dx; (q)
Z 5
0
|x2− 5x + 6| dx; (r) Z 1
0
ex
ex− e−x dx; (s)
Z 2
1
x log2x dx; (t) Z √7
0
x3
√3
1 + x2 dx; (u)
Z 6π
0
| sin x| dx; (w) Z π/2
0
cos x sin11x dx; (x)
Z log 5
0
ex√ ex− 1
ex+ 5 dx; (y) Z π
−π
x2007cos x dx; (z)
Z 2π
0
(x − π)2007cos x dx.
(4) Udowodni¢ nast¦puj¡ce oszacowania (a)
Z π/2
0
sin x
x dx < 2; (b) 1 5 <
Z 2
1
1
x2+ 1dx < 1 2; (c) 1
11 <
Z 10
9
1
x + sin xdx < 1
8; (d) Z 2
−1
|x|
x2+ 1 dx < 3 2; (e)
Z 1
0
x(1 − x99+x) dx < 1
2; (f) 2√ 2 <
Z 4
2
x1/xdx; (g) 5 <
Z 3
1
xxdx < 31; (h) Z 2
1
1
xdx < 3 4. (5) Obliczy¢ nast¦puj¡ce granice
(a) lim
n→∞
1
n +n+11 +n+21 +n+31 + · · · + 2n1 ; (b) lim
n→∞
120+220+320+···+n20 n21 ; (c) lim
n→∞
³ 1
n2 +(n+1)1 2 +(n+1)1 2 + (n+3)1 2 + · · · + (2n)1 2
´
· n; (d) lim
n→∞
√ 1 n√
2n +√n√12n+1 + √n√12n+2 +√n√12n+3 + · · · + √n1√3n; (e) lim
n→∞
¡sinn1 + sin2n+ sinn3 + · · · + sinnn¢
·n1; (f) lim
n→∞
¡√4n +√
4n + 1 +√
4n + 2 + · · · +√ 5n¢
· n√1n; (g) lim
n→∞
³ 1
√3
n+ √3 1
n+1 + √3 1
n+2+ · · · + √31
8n
´
· √31
n2; (h) lim
n→∞
√6
n·(√3 n+√3
n+1+√3
n+2+···+√3
√ 2n) n+√
n+1+√
n+2+···+√ 2n ; (i) lim
n→∞
n
n2 + n2n+1 + n2n+4 +n2n+9 + n2n+16+ · · · + n2+nn 2; (j) limn→∞5n4 +5n+34 + 5n+64 +5n+94 + · · · + 26n4 ;
2
(k) limn→∞7n1 +7n+21 + 7n+41 +7n+61 + · · · + 9n1 ; (l) lim
n→∞
1
7n2 + 7n21+1 + 7n21+2 +7n21+3 + · · · + 8n12; (m) limn→∞n1 ³
e√1
n + e√2
n + e√3
n + · · · + e√n
n
´;
(n) limn→∞³
√1
n+√n+31 +√n+61 + √n+91 + · · · + √17n
´ √1 n; (o) lim
n→∞
n2+0
(3n)3 + (3n+1)n2+13 +(3n+2)n2+23 + (3n+3)n2+33 + · · · + n(4n)2+n3; (p) lim
n→∞
n
2n2 + 2(n+1)n 2 +2(n+2)n 2 +2(n+3)n 2 + · · · + 50nn2; (r) limn→∞2nn2 + n2+(n+1)n 2 + n2+(n+2)n 2 +n2+(n+3)n 2 + · · · + 50nn2.
3