1o.meros μαθηματικα γενικης Κ.Ε
Transcript of 1o.meros μαθηματικα γενικης Κ.Ε
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0.* A1#.!210"f, g !2../*1.1*-
2)20.1#!21S = f + g !02.12 /
1*
1. * $1#.!210"f, g !2../*1.1*-
2)20.1#!21h =J
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2.* .1#!211&")20.2201-
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3. * 0##!210.1#0$"
4.* 1#.!210"f (x) = x .g (x) = 1#x0.1#0-
$0"
5. * 1#!21f (x) =[
, x > 00.1#0$"
6. * 1#!21f (x) =[
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10.x1, x20x1 < x21$*0f (x1) < f (x2).
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1. * .!&"f(x0) .".!.&1"1#!21"f
1.10x02#0/#!1*2"0.!..2
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2.* 1#20012"/0*#1"2"03.2"2".*-
"#0.!.3.!12.1.".!.&1"
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&"2"f12x0.
3. * .!&".".!.&1"1#!21"f1.
10x02#0/#!1*2"03!02!#)0-
2."2"y = f (x), &"!"x)2.x = x0.
4. * .!&"f(x0)."1#!21"f1.10
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5. * .1#!21f0..!.&11.10x0
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6. *1#!21f (x) = [ 0..!.&11210x0 = 0.
7. * 1#!21f (x) = [ 0.1#0$"1210x0 = 0.
8. * . 1#!21 f 0. .!.&1 12 10
x0 2# 0/# !1* 2" 2)20 2 )!
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9. * .!&" 2" 1#!21" f (x) = [ 0.
fx) =[
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0. * .!&"2"1#!21"g (q)# q Q,0.gqq-1.
1. * .!& 2& 1#.!210& f (x) = x .
g (x) = 1#x 0..212$.fx) = x)1#x .
gx) = 1#x)x. 2. * .!&2&1#.!210&x) = e
x.L (x) = lnx
0..212$.(x) = (ex)= e
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[
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3. * !+2.!&"."1#!21"g 0.12.-
0! 1#!21 l, 2)20 g 0. 2" !3"
g (x) = cx, c R - {1}.
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g 0.#.*2)20g0.#.*
5. * /0*20!.!&"."#"1#!21"g 0.12.0!2)20g 0.2*2#.*
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8. * .!&".#!$02"1#!21"g02.
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9. * .!&"2"1#!21"f (x) = 5 0.f(x) = 5x.
0. * .!&"2"1#!21"s (t) = t 0.s(t) = 1.
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6. * 0/!1*2"1#!21"f (x) = [ 0.
[- 1, 1] [- 1, ) + (- 1, 1) (-, 1] (-, +)
7. * 0/!1*2"1#!21"f (x) =
[
0.
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8. * /!..#.!#102.12 /.) 1$. 0. !.3
.!12.12"1#!21"
f (x) = - x f (x) = x
+ f (x) =[
f (x) = -
[
f (x) = - 2x
9. * /!..#.!#102.12 /.) 1$. 0. !.3
.!12.12"1#!21"
f (x) = x2
f (x) = - x2
+ f (x) = -[
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21h =J
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2.x A: f (x) = 0, g (x) 0 2.x A: f (x) = g (x) = 0
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11. * .1#!21f0.1#0$"1210x02#0/#!1*2".
.).
1$*0f (x0) = 0
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+#!$02
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f (x)
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12. *
x 0
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2#1$.2" 2#1$.2"
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0/#!1*2"03!0
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2!#)02."2"f (x) &"!"x - x0
...)2..!.&
15. * .!&f(x0).".!.&1"1#!21"f1.10x02#
0/#!1*2"#0
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2K
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+2122"02$#1"12$!)/12.[t0, t0 + h]
212..22"02$#1" 2$!12t = t0
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17. * S (t) 0.10)"2*2$!12t#02.0#*-
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+2122"02$#1"12$!)/12.[t0, t0 + h]
212..22"02$#1" 2$!12t = t0
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t0$!2$!12t0 + h
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18. ** 1#!21 f0..!.&110/12. R .1&"3#1.122)20f0..!2
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B.10).2.01&20!10.2#
+1210/
)12.10.#/0#2f
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19. * .1#!21f.!.&11./12.1$*#f(x0) = 0
.f(x0) < 0, 0x001&20!)102#2)201#!21f
.!#102)0$12.x = x0
B.0.1&"3#1.10)2/12.
+.!#102)12.x = x0
/0.!#10.!)2.2.x = x0
E.0.12.0!1#!21
20. * .1#!21f, .!.&11./12.1$*#f(x0) = 0
.fx0!0x001&20!)102#2)201#!21f
.!#102)0$12.x = x0
B.0.1&".*#1.10)2/12.
+.!#102)12.x = x0
/0.!#10.!)2.2.x = x0
E.0.12.0!1#!21
21. * H 1#!21f, .!.&11..2)/12.0.1&"
.*#1.12.1$*0
f(x) = 0, .010x 2#
B. f (x) = 0, .010x 2#
+ f (x) > 0, .010x 2#
f (x) < 0, .010x 2#
E....)2..!.&
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22. * 1#!21f, .!.&11..2)/12.0.1&"
3#1.12.1$*0
f(x) = 0.010x 2#
B. f (x) = 0.010x 2#
+ f (x) > 0.010x 2#
f(x) < 0.010x 2#
E....)2..!.&
23. ** 12&1#!21 f, .!.&11./12..x001&20!)
102#.2#!$0 f (x0)01&20!)10x00.
10.!22#2" f.1$*0
f (x0) = 0 B. f(x0) 0 + f (x0) = 0
f (x0) = 0 . f (x0) 0
E. f (x0) > 0. f (x0) = 0
24. * .!&"2"1#!21"f (x) = x20..h 0)
KOLP K
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!&210".212$1"
1. * 212$120202*1#!21"2"12"02/12.
&1/.122&2"12"#0.20/!1*2"
2 2
f (x) = 2x
f (x) =[
f (x) =[
f (x) = [
f (x) =[
[
+
R
(0, 1)
(- , 1) (1, + )
(- , - 1) (- 1, + )
(- , 0) (0, + )
(1, )
[1, )
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2. * 212$120202*1#!21"2"12"02/12.
&1/.122&2"12"#0.20/!1*2"
2 2
f (x) = [
f (x) = [+
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f (x) =
[
+
[0, + )
[- 2, + )
(- 2, 0) (0, + )
(- , - 2] [0, + )
(0, + )
(- 2, 0) (0, )
(- 2, +
)
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3. * 212$120202*2"1#!21"2"12"02!.3
2".!12.11212
2 2
1. f (x) = - 3x2 + 2
2. 3(x) =[
3. h (x) = - 2x + 5
x y
y
x
2
y
x
y
x
(A )
2
x
y
y
x
(B )
x
y
y
x
x
y
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+
5
2 ,5 x
y
y
x
5
-2 , 5
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5. * 212.!#12.!.3".!.1210"/*1#.!210&
212$120.0.).#2"02!.3.!12.12"!+2".-
!.+#2"#!102.1212
2 2
(2 )
1
xx
y
y
y = l n x
0
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1
xx
y
y
y = e x
0
(A )
xx
y
y 0
(B )
1
y
y
0 xx
+ y
y
0 xx
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6. * 212.!#12.!.3".!.1210"/*1#.!210&
212$120.0.).#2"02!.3.!12.12"!+2".-
!.+#2"#!102.1212
2 2
1
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y= x
y
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x0
1
1
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\1#[
y
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x0
1
1
y
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0 32 x
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+
0 x
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7. * 212$12002*1#!21"#0.1212022*
2"1#!21"2"!+2".!.+#2"#0.1212
2
f (x)
2
f(x)
3x
2
3x
2 (x
2- 1)
(3x)
2
(3x - 1)
2
3x
2- x
6x2
- 1
6x
3
4x
3x - 1
18x
6 (3x - 1)
6x2
6x - 1
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8.* 212$12002*1#!21"#0.1212022*
2"1#!21"2"!+2".!.+#2"#0.1212
2 2B
f (x) f (x)
.
.x
[.
.x2
x2
.x2 - x
x2
+ .x -
0
.
.x +
.x
2x
x
.x -
.x
x .
.x
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9. * 2122#.!.2&..#!$#2.!+2.122&
0" 03!# 2#" .)0" .!.+1" 2 12 #!$# 2.
/0*20!.2&122&.#2+212$1202.12$0.2"12"
000.2"12"+120.!*%#&12.)0".!.+-
1"
2 2
(c f(x))=
(f(x) + g (x))=
(f(x) g (x))=
[J
[I=
[f (g (x))@=
f(x) g (x) + f (x) g(x)
[J
[J[I[J[I
f(x) + g (x)
c fx)
f(x) g (x)
f(g (x)) g (x)
[J
[I
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!&210"1#!&1"1*2".21"
1. * .1#!+10202.0/.!1*2&1#.!210&
.f (x) =
[ A = ..................................
f (x) =
[
A = ..................................
f (x) =[
+
A = ..................................
/f (x) = [ + A = ..................................
0f (x) =[
+A = ..................................
2. *
[[OLP
f (x) = - 2.!020..1#!+10202.
[[OLP
g (x))2.
.g (x) = 3 f (x) - 1
[[OLP
g (x) = ..............
g (x) = 2 - 4 f (x)[[
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g (x) = ..............
g (x) = (2 f (x))2
[[OLP
g (x) = ..............
/g (x) =[I
[I
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g (x) = ..............
0g (x) = [I +
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g (x) = ..............
3. * .1#!+10202..!.2&)!.
.[
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(2x2
+ 6x - 1) = .....................................
[
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[
[+= ................................................
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/[
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[(3x + 2) (5x - 3)]2
= .............................
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0
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[x + 31#x] = ....................................
12[
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4. * .1#!+10202..!.2&)!.
.[
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[= ................................................
[
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+= .............................................
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= .....................................
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+= ........................................
5.* .1#!+10202"2"2&.!.+&2&.!.2&1#.!210&
12..212$.10.
.f (x) = x2 f (0) =............................
f (x) = x2
+ 1 f (1) =............................
f (x) = 2x2
- 3 f (- 1) = .........................
/f (x) = x f (
) = .........................
0f (x) =[
f (0) =............................
6.* .!0202"01+10"2&03.2&2&!.3+.!.1210&2&
.!.2&1#.!210&12..212$.10.
.f (x) = x2
- 1 A (0, f (0)) y = ............................
f (x) = 2x2 - 1 A (1, f (1)) y = ............................
f (x) = 3x2
- 2 A (- 1, f (- 1)) y = ............................
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7.* +.0!.3.!12.12"y = f (x) $.!202.212$!.3
.!12.12"!+2".!.+#2"
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8.* 212/2.2*1#.!210!+12012122#"
.212$#"2*#"2&!+2&.!.+&2#"
2
f (x)
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x - 1
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(x2
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9.* 212/2.2*1#.!210!+12012122#"
.212$#"2*#"2&!+2&.!.+&2#"
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f (x)
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1#[[
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.212$#"2*#"2&!+2&.!.+&2#"
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x e [
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11.* 2.!.2&../2.2*2011!&1#.!210&.1#-
!+102021202.212$0/!1*2#"212+0
2!+2.!&)2#".212.2/0*20!.!&)2#"
2 2
0/!1*
2+
!+2.!-
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2
/0*20!.!-
&"
h (x) =[
3 (x) =[
f (x) =[
[
+
g (x) =
[
[
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!&210".2#"
1. ** 02.1#!21f0f (x) = x2
- 3x + 2. .!020
.20/!1*2"
.0"2"2#x $#0f (x) = 0
20/!1*2"1#!21"g (x) =[[
[ +
2. ** 02.1#!21g 0g (x) = x2 + 2.
.+.0"2"2#x R $#0g (x) = 0;
.!020i) 20/!1*2"1#!21"f (x) =[
[ +
ii) 20/!1*2"1#!21"h (x) = [ +
3. ** 02.1#!21g 0g (x) = x2
- 1.
.+.0"2"2#x R $#0g (x) = 0;
+.0"2"2#x R 1#!21g (x) 0.02
.!020i) 20/!1*2"1#!21"f (x) =[
[
ii) 20/!1*2"1#!21"h (x) = [
iii) 20/!1*+2"1#!21"3(x) =[
[
4. ** 02.1#!21g0g (x) = x - 4.
.+.0"2"2#x R $#0g (x) = 0;.!02020/!1*2"1#!21"f (x) =
[
[
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5. ** 2.1#.!210"f, g 0f (x) = x2 - 4x - 2 .g (x) = 3x - 2, x R..!020
.22*2"1#!21"f (x) + g (x) ..!1/!102020/!-
1*2",
22*2"1#!21"f (x) - 2g (x) ..!1/!102020/-
!1*2",
22*2"1#!21"f (x) g (x) ..!1/!102020/!-1*2",+
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- 3x + 1, g (x) = 5x2
- 1, x R.
.!020
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OLP
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g (x)
2[
OLP
[f (x) + 2g (x)]
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[
+.!020
.20/!1*2"
2[
OLP
3 (x)
2[
OLP
>3 (x)]3
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. .!020.20/!1*2"
2
[
OLP
f (x)
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9. ** 2.1#.!210"f, g 0f (x) = 6x3 + 5x - 1, g (x) = 2x - 1, x R..!020
.2.[
OLP
f (x),[
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2[
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[I
10. **
[[OLP
f (x).!0202
[[OLP
3(x), )2.
.3x) = 3 f (x)3(x) = 3 f (x) - 2
3(x) =[I
[I
/3(x) = [I
11. ** 02.1#!21f0f (x) =[
[
+.!020
.20/!1*2",2
[OLP
f (x)
12. ** 02.1#!21f0f (x) =[
[
+.!020
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2
[
OLP
f (x)
13. ** 02.1#!21f0f (x) =[
[.!020
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2[
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14. ** +.0"2"2#. R 1#!21f (x) =.[
[ +
$00/!1*
21*R 2&!..2+.!+
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[ ++
$00/
!1*21*R 2&!..2+.!+
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[
+. .!020
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2[
OLP
f (x).
.0021020,.f (x) 0.1#0$"121x0 = 1.
17. ** 02.1#!21f0f (x) =
=
+
[.
[[[
.
.+.x 3 0.1#0$"1#!21
+..22#. R 1#!21f (x) 0.1#0$"1210x0 = 3;
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=
[.
[[
[[
.!020
.2[
OLP
f (x)
222#. R+1201#!21f.0.1#0$"1210x0 = 2.
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19. ** 02.1#!21f0f (x) =
=
+
[.
[[
[[
.!020
.20/!1*2",
2[
OLP+
[
[[ +
222#. R+120f.0.1#0$"1210x0 = 1
20. ** 02.1#!21 f0 f (x) =
=
+
[.
[[
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. .!020 2
22#. R, +1201#!21f.0.1#0$"1210x0 = 2.
21. ** /.+"0)"202!.+#0./.03!1020&"1#!212"
/./
.20!02!)2# 20./)2#
22. ** 020"0#!"+0)"!!+#+ )02.2. 21+120 2 0./) 2#..!.012.0!). 10
12 m2. .03!10202"x 2"0#!"&"1#!21 2#-
#"y 2"0#!"+
23. ** ."#)"2.".2."r $00./)cm2. .03!10202
0!02!)2#&"1#!212".2."r.
24. ** 02.1#!21 f0f (x) =[
, x R. .!020
.2f(3)
21#20012/0*#1"2"03.2"2".*"2"1#!2
1"f, 12100x = 3
201&12".!.&03.2"
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25. ** 02.1#!21f0f (x) = .x2, x R. R...!0202f(2).
.!1/!10202.+1201#20012"/0*#1"2"03.2"
2".*"2"f1210f (2)) .0.
26. ** 02.1#!21f0f (x) = x2
+ 1, x R...!0202f(0).
.!1/!102021#20012/0*#1"2"03.2"2".*
"2"f12100x = 0.
.!020201&12"03.2"2".*"2"f1210
(0, f (0)).
27. ** 02.1#!21f0f (x) = x2
- 5x + 6, x R. .!020.2f(x)
201&12"03.2"2"!.3".!12.1"2"1#!21"
f, #0..!12.xx.
28. ** 02.1#!21f0f (x) = 2x2 - .x, x R, . R...!0202f(2).
.!1/!10202.+12003.22"!.3".!12.1"2"
1#!21"f 1210f.1$.2002. xx&.
45.
29. ** .!00 &.#1$.20 0 2. xx 03.2 2"
.*"#0.!.3.!12.12"1#!21"f (x) = - 2x2
+ x - 3
1210
, f (
)).
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30. ** 1 0)"2*#02. 0#*!../02.1#.!210 2#
$!)#.)22*S (t) = 2t + t2)#2t 02!2.10sec .2S 10
2!..!020
.212.$*22.2#2*12$!)/12.>] sec
212..2.$*22.2#2*)2.t = 1 sec (1 sec0220
12#
31.** 10)"2*#02000#*!.1/02.1#.!210
2#$!)#t 10sec) .)22*S (t) = 3t2 - t. .!020
.212.$*22.2#2*12$!)/12.[2, 4] sec
212..2.$*22.2#2*)2.t = 3 sec (3 sec 0220
12#
32. ** 2.$*22. 0)" 2* # 02. 0#*!.. 1#.!210 2#
$!)#t (10 sec), /02..)22*#(t) = 3t2 - 5.
..03!10202!#)02."2"2.$*22."02$#12#-
2*&"!"t, )2.t = t0.
) .#10202!#)02."2"2.$*22."02$#12#-
2*&"!"t, )2.t = 10 sec (10 sec 022012#.
33. ** ."#1)"!&02.02.1#.!2102#$!)#t 10
+!0"1*3&.022* (t) = 103 - 5 102 (1 + t)-1...!0202.!$).!)!&(t = 0).
.!0202.!)2&!&)2.t = +!0".
.!0202!#)02."2##1*2&!&&"!"2
$!))2.t = 9+!0".
34. ** #1)"."0!$"/02.1#.!2102#$!)#t 102
.)22*t) = 10 e004t10$/0".!0202!#)02."2##1*.#2"2"0!$"&"!"2$!)*120!..)2
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35.** 2.1#.!210"f, g 0f (x) =[
H
[, g (x) = e
xx2. .!020
.!+2.!&i) 2"f .ii) 2"g.
".!.+#"i) f. ii) g
36. ** . !020 #+# (x) 2!2# .* 22 +120
37. ** 02.1#!21f0f (x) = 2x - x2.
..!020i) 2f (x) ii) 2f(x)
../0$0)2(1 - x) f(x) + f(x) = 0, .0x R.
38. ** 02.1#!21f0f (x) = e2x
.
..!020i) 2f (x) ii) 2f(x)
./0020)2f(x) - f(x) = 0, .0x R.
39. ** 02.1#!21f0f (x) = e.x, . R. .!020.f(x)
f(x)
"2"2#.+120.1$*01$1f(x) + 2f(x) = 3f (x), .-
0x R.
40. ** 02.1#!21f0f (x) = (3x - 2) [+ .!020
.f(x)
f
41. ** 02.1#!21f0f (x) =H
H[
[
+.!020
.0/!1*2"
f(x).
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42. ** 02.1#!21f0f (x) =H
[[
.!020
.0/!1*2"
f(x).
43. ** 02.1#!21f0f (x) =1#[
[.!020
.0/!1*2"
f(x).
44. ** 02.1#!21f0f (x) =
x
3+ 2x
2+ 3x + 1, x R. .!020
.f(x)
. 10. 2" .*" 2" 1#!21" 12. . 03.2)00"
1.#20..!0"12.xx.
45. ** 02.1#!21f0f (x) = (x + 1)2, x R. .!020
.f(x)
1#20012 /0*#1" 2" 03.2" 2" .*" 2" f12
100202
46. ** 02.1#!21f0f (x) = - x2 + 3x - 1, x R. .!020 .f (x)
01&12"03.2"2".*"2"f, #1$.2002
.xx&..
47. ** 02.1#!21f0f (x) = .x + 1)2, x R. R.
..!0202f (x).
.!1/!10202.+1201#20012"/0*#1"2"03.2
"2".*"2"f1210f(1)).0.
.!020201&12".!.&03.2"0#0."
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48. ** 02.1#!21f0f (x) = x2 - 4x + 2, x R...!0202f(x)
.!1/!10202102"!.3".!12.1"2"1#!21"
f, 1203.22"1$.20&.02.xx.
49. ** 02.1#!21 f 0 f (x) = 2x2 .x . R . 0#0.
y = 3x - 1, x R. .#10202..+1200#0.y = 3x - 1 .0.03.22"!.3".!12.1"2"f12100202
50. ** 02.1#!21f0f (x) =
x
3+ x
2- 2x + 1, x R. .!020
.f(x).
) "01+10"2&03.2&2"!.3".!12.1"2"f, #0-
..!0"120#0.y = x + 3.
51. ** 02.1#!21f0f (x) =[
, x R, x 0.
../0020)2f.
.
.0. R. 0.
. !1/!1020 2 01&1 2" 0#0." # 03202. 12 10
.
.
2"!.3".!12.1"2"f.
52. ** 02.1#!21f 0 f (x) = x3 - 9x2 + 15x - 3, x R...!0202f(x).
.002102022.2".!1/!10202..!)2.22".#!$#
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53. ** 2. 1#.!210" f, g 0 2*#" f (x) = 2x2
- 4x - 1 .
g (x) = 4x - x2
+ 2, x R. .!020.i) 2f(x) .ii) 2g (x).
"10".2"0"1#.!210".!#1#.!)2.2
"2"2&.!22&.#2+
54. ** 02.1#!21f0f (x) =
x
3- 2x
2- 5x - 2, x R. .!020
.f(x)
+.0"2"2#x $#0f(x) = 0
0".)2".!.&2"2&x 0.10".!22&.2f
/"2"2&.!22&
55. ** 02.1#!21f0f (x) = x2x + 3, x R R...!0202.+120f.$0121x = 1 2).!)2.210
0/#".!)2.2.!#101#!21121x = 1;
56. ** 02.1#!21f0f (x) = x3
- 3x, x R. .!0*2./.12.2.#f0.
.*#1.
-#1.
57. ** 02.1#!21f0f (x) = x2e-x...!0*f (x), f (x).
.00201#!21f&"!"22.2"
.!1/!12*2..!)2.22".#!$#
58. ** 02.1#!21f0f (x) = (2x - x2) e
x, x R.
..!0*i) 20/!1*2"ii) f(x).f (x).
.0020f&"!"i) 22.2",
ii) 2. .!)2.2 2" .. 0212* .#2
.#!$#.
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59. ** 02.1#!21f0f (x) = x3x2 + 3x - 1, x R R...!0202f (x).
.!1/!10202.+120f.$02.!)2.2.12.10.
02020"x1 = 2, x2 = - 2.
.!0202"2"2&.!22&
60. ** )). 2. !+..!.)!..0 2 /.0!02!
0.00#$02120./)
61. ** )).2.!+..!.)!..00./)m2, .!020
2"/.1210"00##$02!)20!0!02!
62. ** ../0020)2.)).2.1102!&.#0.00!..
10*.2."R, 21)0#!$00.*20!0./)
63. ** .!0*/*.!x, y 012.0!)!1.#.$#2
0.*20!)0
64. ** 2+1"0)"$.*0.!2.2"0./!$)12"
2# 1#.!210 2# $!)# .2.10#" 10+!0" !1002. .) 2
2*2"1#!21"
(t) = t2
+ 250t-1
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65. ** 0!0.#.2..+0."!!.1)"#02.1.12
..0)".10*"02.$*22.#!1002..)22*2"1#!-
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66. ** 0!0.W (t), # .//02. .) . 02.02. 0 2
$!)t1*3&.022*2"1#!21"
W (t) = 6t2
- t4
.02!2.10Joules.
..03!10202!#)02."2"0!0."&"!"2$!)2
1$*2##2$!12t = t0.
0.$!122$0121$*
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67. ** 2012!#2&.12+0&3!0&0.12.0!2.200#2..
$!).12"/!$)12"02.3!".0212/!0.2&
$!)&!1002..)22*2"1#!21"
(t) = t2
+W
)#t @$!)"..!1/!120$!12.22.!..202
12!/"
)10..#2)2!/"
68. ** 02.2/!.10)"!.1*1.3!.0!!302./02.
.)22*2"1#!21"f (x) = x2.x.!12.0!.x 0!-
1./)12#3.!#10mg..0.0/0/01)22./)1"2#
3.!#+120.$#020.*20!02.2/!.12#!.1*
69. ** .0!121.$.!.12".!.10#002.*&2.%.
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1200!#[
+ 25x/!$2+1"2#2.%*0.
(1000 -
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$/
,
.1#!21f0..!.&11./12.
.)20f0.1&".*#1..)201&"3#1.12
x0.)20f.!#1012/12..12.
x = x0.)200$12.x = x0;
2 /.) 1$. .!#102.
!.3.!12.1."1#0$*"1#-
!21"f. .$.!.2!10200
2"!210"#.#* 2 1
1 2 3 4 5 6 7
x
y
0
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0/!1*2"f0.>@ 0/!1*2"f0.@ 2/12.1#!21f.!#102-)12.x = 3.
1$*0)2f 0. 1$*0)2fx) > 0 .x (2, 3) .
fx) > 0 .x (3, 4).
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,
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.210"12"0!&210"2#2*#&12)"
1. 20. 39.
2. 21. 40.
3. 22. 41.
4. 23. 42. 5. 24. 43.
6. 25. 44.
7. 26. 45.
8. 27. 46.
9. 28. 47. i)
10. 29. ii)
11. 30. iii)
12. 31. iv)
13. 32. v)
14. 33. vi)
15. 34. vii)
16. 35. viii)
17. 36. ix)
18. 37. x)
19. 38.
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.210"12"0!&210"."0"
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.210"12"0!&210".212$1"
1. 2 2
f (x) = 2x R
f (x) =[
(- , 1) (1, + )
f (x) = [
(- , 0) (0, + )
f (x) = [ [1, )
f (x) =[
[
+ (- , - 1) (- 1, + )
2. 2 2
f (x) = [ [0, + )
f (x) = [+ [- 2, + )
f (x) =[
(0, + )
f (x) =[
+(- 2, + )
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7.2
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2
I[
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6x
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2 (x2
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.210"12"0!&210"1#!&1"1*2".21"
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+ A = R
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g (x) = 10
J[I[2
[[OLP
g (x) = 16
/J[[I[I
[[OLP
g (x) = -
0J[ [I +
[[OLP
g (x) = 3
3..[
OLP
(2x2
+ 6x - 1) = 7
[
OLP
[
[+=
[
OLP
(5 [ ) = 5
/[
OLP
[(3x + 2) (5x - 3)]2
= 64
0
[
OLP
>[1#[@
12[
OLP
>[1#[@
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4..[
OLP
[
[=
[
OLP
[
[
+= -
[
OLP
[
[[
+++
=
/[
OLP
[[
[
+= -
5..I
I
I
/I
) = 0
0I
6..\
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8.
2
f (x)
2
I[
x - 1 1
(x - 1)2
2 (x - 1)
(x2
- 1)2
4x (x2
- 1)
(x - 1)
(x - 1)
[
[
[
-
(x - 1)
[
-
(x - 1)
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9.
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f (x)
2
I[
[[
1#[
1#[[ 1#[[
[[1#
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[1#2
x
[
[
[
[1#[[
[
[
[
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10.
2
f (x)
2
I[
x - lnx 1 -[
x e [
e [
(1 -[
)
e-2x
3+1
- 6x2 e-2x
3+1
ln [
[
[
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11.
22
0/!1*
2+
..!&"
2
.!&"
h (x) =[
R - {0} - [
[
3[[
R - {- 1, 1} - [
[
[
[ +
f (x) =[
[
+ R [
[
+
[[ +
g (x) =[
[R - {0} [
[[
[
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1. .5 x = 1, x = 2 B = R - {1, 2}
2. .0#!$#[ R : g (x) = 0
A = R B = R
3. . x = 1 J[!)2.[ (- , - 1) (1, + )LA = R - {- 1, 1}
ii) B = (- , - 1] [1, + )
iii) +, - 1) (1, + )
4. . x = 4 A = R - {4}
5. .f (x) + g (x) = x2
- x - 4 5
3f (x) - 2g (x) = 3x2
- 18x - 2 5
f (x) g (x) = 3x3
- 14x
2
+ 2x + 4 += R
/[J
[I=
[
[[
5^
}
6. .[
OLP
f (x) = 1,[
OLP
g (x) = - 1 [
OLP
[f (x) + 2g (x)] = - 1
7. .5^
}
[OLP
3[
[OLP
>3[@3
=
8. ., -
] [
, + )
[
OLP
f (x) = 0
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9. .[
OLP
f (x) = - 12,[
OLP
g (x) = - 3 [
OLP
[J
[I= 4
.
[[OLP
3[
[[OLP
3[
[[OLP
3[ /
[[OLP
3[
.5^` [
OLP
f (x) = - 4
.5^
}
[
OLP
f (x) = - 2
.5^` [
OLP
f (x) =
14. . (0, + )
15. .!
.5^` [
OLP
f (x) = -
I[0.1#0$"121[0 = 1
.+.[1#!21I[0.1#0$"&"#
..
[OLP
f (x) = 3 .
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.5 [
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[
[[ += 1 .
20. .
./ / /
/
22. x (y) =\
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.I. .
.I 1#20012"/0*#1" y = 1
27. .I[[ y = -
.I. .
29.
& = 0
.# = 6 m/sec #PVHF0WVHF
.# = 17 m/sec #PVHF0WVHF
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2.2"H 2,7).
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LIH
LLJH
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x
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ii) I[H2x
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[ ..
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H
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)
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.I[.[ . y = 4x
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, -
)
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. y = x + 10
y = -
.
x +
.
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(- @I.*#1. >@I3#1. [5, + I.*#1.+.[IPD[ +.[IPLQ
.LI[[ LLJ[[
I.[.!#10PLQ J.[.!#10PD[
f (1) = - 3 min g (2) = 6 max
.I[[2
- 4x - 5 x1 = 5, x2 = - 1
+.[1I.!#10PLQ +.[2I.!#10PD[
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/f (5) = -
min f (- 1) =
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57. .I[[H-x - x2e-x = e-x (2x - x2) I[H-x (x2 - 4x + 2)
1#!21I0.1&"3#1.12, 0], [2, .1&".*#1.12/12.>@
+.[.!#100$1210I
+.[.!#101210I
.LA = R ii) I[Hx
(- x2I[H
x(- x
2- 2x + 2)
LI0.1&"3#1.12., - ], [ , + )
. 1&".*#1.12> , ]ii) +.[ $#01202012#2I ) = 3, 4
+.[ $#00$12020.$12#2I ) = - 1,174
.I[[2[
+.[I.!#10PD[I
+.[I.!#10PLQI
60. .2202!&0[\
)#0!02!"
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63. x = y = 6.
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xy =364, x2 = 524, y2 = 256.
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$/
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fi %
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10.050 50
+ 5.040 25,08 2.010 10
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10$/0"/!$i fi %
[25, 35) 10 12,5
[35, 40) 16 20
[40, 45) 20 25
[45, 50) 14 17,5
[50, 55) 10 12,5
[55, 65) 5 6,25
[65, 90) 5 6,25
80 100
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24.
!)".0!." i fi % i
[0, 3) 95 19 95
[3, 6) 193 38,6 288
[6, 12) 122 24,4 410
[12, 24) 68 13,6 478
[24, 36) 22 4,4 500
500 100
25.
xi i fi Ni Fi fi % F i %
1 8 0,4 8 0,4 40 40
2 2 0,1 10 0,5 10 50
3 5 0,25 15 0,75 25 75
4 3 0,15 18 0,9 15 90
5 2 0,1 20 1 10 100
* 20 1 100
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/02.20.2!"2&&2&.21*2+2"0.
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5.
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1 5 5
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6.
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22 5 9
23 6 15
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25 3 22
26 6 28
27 2 30
30
L!0" LL!0" LLL!0"
7.
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27 3 19 19
28 3 19 38
29 5 31 69
30 2 12 81
31 3 19 100
16 100
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8..
xi . i i
0 1 1
2 1 2
3 1 3
4 3 6
5 4 10
6 2 12
7 3 15
8 3 18
9 2 20
10 3 23
13 1 24
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26
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10..212$".."0.
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02.3!",..11&" /!/!+" /+" 0!!+"
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0.2/!$ 48.500 13.500 28.000 7.000 97.000
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.
0!" 2.1 fi %
0! 20,8 18
1. 44 38
3! 30,5 27
#!+ 10,5 9
0.. 9 8
114,8 100
+.2#)/!.. 0! 0!#1.: 1383! 96#!+ 330.. 28
12. ,..11&" 180/!/!+" 38,8
/+" 116,5
0!!+" 24,7
13.
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0 0 0
1 10 3
2 20 6
3 20 6
4 30 9
5 40 13
6 50 16
7 50 16
8 40 13
9 30 9
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320 100
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0 135 0,338 34 0
1 220 0,55 55 220
2 8 0,02 2 16
3 15 0,037 4 45
4 12 0,03 3 48
5 10 0,025 2 50
400 1 100 379
L [ 0,947 LL/
18.0./)2#!+2#!. 1 = 10.
0./)2#2!2#!. 3 = 30.
0./)2#22.!2#!. 4 = 20.
!.20./)2#/0*20!#!. 2 = 40.
0/22"2#0.2*%"2#.0.
19../.2#02...12.2..2.*#1.2!1#0)2/
01"0..!.2!1.0./FP
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12.
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[150 - 155) 1 1
[155 - 160) 2 3
[160 - 165) 3 6
[165 - 170) 8 14
[170 - 175) 10 24
[175 - 180) 7 31[180 - 185) 6 37
[185 - 190) 3 40
[190 - 195) 1 41
41
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12+ 1#$22& 2." ./1 $#0 . .)1FP
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[30 - 35) 12
[35 - 40) 9
[40 - 45) 5
[45 - 50) 3
[50 - 55) 2
[55 - 60) 1
50
L# LL#
-
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21.
!0.&"10
+!0"i fi % i Fi %
[400, 500) 15 3,75 15 3,75
[500, 600) 45 11,25 60 15
[600, 700) 60 15 120 30
[700, 800) 75 18,75 195 48,75
[800, 900) 70 17,5 265 66,25
[900, 1000) 60 15 325 81,25[1000, 1100) 50 12,5 375 93,75
[1100, 1200) 25 6,25 400 100
400 100
22. [ .2"
23. [ $!).
24..!0..
25. .L [ = 184,25 LL/ iii) R = 23
0!2&1i) [ = 186 LL/ iii) R = 20
0!2&1i) [ = 185,66 LL/ iii) R = 25
0!2&1i) [ = 184,62 LL/ iii) R = 26
26. [ w = 59,25
28. . [ = 11 V2 = 26,2 V 5,11// 041 = 7,5 Q2 = 11,5 Q3 = 14,5
12 5 &9
-
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144
29.
10"2!
1"[i)i xii xi - [ (xi - x)
2 i (xi - [ )2
[4, 6) 5 7 35 - 6 36 252
[6, 8) 7 13 91 - 4 16 208
[8, 10) 9 17 153 - 2 4 68
[10, 12) 11 18 198 0 0 0
[12, 14) 13 29 377 2 4 116
[14, 16) 15 11 165 4 16 176
[16, 18) 17 5 85 6 36 180
100 1104 1000
L [ 11 ii) s2 = 10 iii) s 3,16 iv) CV 29%
30.
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02!"
2"[i
i xi2
xii xi2i
[30, 35) 32,5 8 1056,25 260 8450
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[45, 50) 47,5 15 2256,25 712,5 33843,75
[50, 55) 52,5 10 2756,25 525 27562,5
[55, 60) 57,5 8 3306,25 460 26450
[60, 65) 62,5 3 3906,25 187,5 11718,75
70 3200 150987,5
L [ $ ii) s2 = 67 iii) s 8,5 iv) CV = 17,39%
-
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31.
!)".2$"
10+!0"!)"1#10#+
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100
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22.
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10 70 140
20 5 5
20 5 15
80
33. [ 0.122"2)20V2 =
=
L
(ti - [ )2.
VV2 =
L
(ti - [ )2
(t1 - [ )2
+ (t2 - [ )2
+ + (t - [ )2
= 0 t1 = t2 = = t = [ .
34.. ) 173
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146
35. . \ =
[
\ L
LL
L
+=
== = [ = 2, s
\ = s
[ .
\ = [ + c, s\ = s
[ .
2"."02.2".#*.2F/0"2)20/.*.1
0.022.12.#02..2F
36. . [+ =
W
W
W ++++++ = = [
[ = [ [ [
/
[
=
[ 0) 0 [ + [
37. .
!+2
02.1[
0*20!
02.1\x
2xy y
2
6 8 36 48 645 7 25 35 49
8 7 64 56 49
8 10 64 80 100
7 5 49 35 25
6 8 36 48 64
10 10 100 100 100
4 6 16 24 36
9 8 81 72 64
7 6 49 42 36[ \ [2 = 520 [\ \2 = 587
\ = 4 + 0,5x
-
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147
38. .
x y x2
xy y2
1 1 1 1 1
3 2 9 6 4
4 4 16 16 16
6 4 36 24 16
8 5 64 40 25
9 7 81 63 49
11 8 121 88 64
14 9 196 126 81
[ \ [2
= 524 [\ \2
= 256
\ [$01#20012/0*#1"./!$02..)
210
\ = 8,18
39. .
0!.
(x)
+0&02!.
(y)x
2xy y
2
75 82 5625 6150 6724
80 78 64000 6240 6084
93 86 8649 7998 7396
65 72 4225 4680 5184
87 91 7569 7917 8281
71 80 5041 5680 6400
98 95 9604 9310 9025
68 72 4624 4896 5184
84 89 7056 7476 7921
77 74 5929 5698 5476
[ [\ [2
= 64722 [\ \2
= 67675
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\ = 29,23 + 0,66x
[2)2012+0&02!..!0
40. .U [\0.02!.1#1$0210"
41.
x y x - [ y - \ (x - [ )2
(y - \ )2
(x - [ ) (y - \ )
35 102 - 9 5 81 25 - 45
39 100 - 5 3 25 9 - 15
41 97 - 3 0 9 0 0
43 97 - 1 0 1 0 0
40 98 - 4 1 16 1 - 4
53 100 9 3 81 9 27
50 97 6 0 36 0 0
49 91 5 - 6 25 36 - 30
40 94 - 4 - 3 16 9 12
50 94 6 - 3 36 9 - 18
*. 440 970 326 98 - 73
.U
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0/#!$0 .!2 1#1$21 02.* 2" 2" 2& $!0!3&
.2"2"2&+1.0)2)2.22&$!0!3&32022&+.0.0
-
8/7/2019 1o.meros .
148/151
149
42..
x y x2
xy y2
2 18 4 36 324
4 12 16 48 144
5 10 25 50 100
6 8 36 48 64
8 7 64 56 49
11 5 121 55 25
[ [\ [2 = 266 [\ \2 = 706
r = - 0,92.
[[.\\2)20
[ \ [2 [\ \2
10 39 100 390 1521
14 21 196 294 441
16 15 256 240 225
18 9 324 162 81
22 6 484 132 36
28 0 784 0 0
[ [\ [2 = 2144 [\ \2 = 2304
r = - 0,92.
-
8/7/2019 1o.meros .
149/151
150
43.
.
(x)
01
..2"\x
2xy y
2
56 17 3136 952 289
42 12 1764 504 144
72 14 5184 1008 196
36 10 1296 360 100
63 13 3969 819 169
47 09 2209 423 81
55 11 3025 605 121
49 08 2401 392 64
38 11 1444 418 121
60 15 3600 900 225
[ [\ [2
= 28028 [\ \2
= 1510
.U
\ = 4,852 + 0,138x
+01#.."."02+0.0!#
44. .
x y x - [ y - \ (x - [ )2
(y - \ )2
(x - [ ) (y - \ )
1 2,5 - 2 - 1,5 4 2,25 3
1 4,0 - 2 0 4 0 0
2 3,0 - 1 - 1 1 1 1
3 4,5 0 0,5 0 0,25 0
3 4,0 0 0 0 0 0
4 3,5 1 - 0,5 1 0,25 - 0,5
7 6,5 4 2,5 16 6,25 10
*. 21 28 0 0 26 10 13,5
U 0,83
-
8/7/2019 1o.meros .
150/151
151
45. .
!+2
02.1[
0*20!
02.1\x
2xy y
2
6 8 36 48 64
5 7 25 35 49
8 7 64 56 49
8 10 64 80 100
7 5 49 35 25
6 8 36 48 6410 10 100 100 100
4 6 16 24 36
9 8 81 72 64
7 6 49 42 36
[ \ [2
= 520 [\ \2
= 587
U.0.02!.1#1$0210".2
-
8/7/2019 1o.meros .
151/151
,
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2)1#2!&1" /.1& ..2+ (Klein - Hilbert - Minkowski 02.
#32"12.0122#Gttingen02.2.2)".2"12#20$02#)0!# 0+ 2 02. .2" 12#20$0 2" Breslaw. 02.
2.2)" .2" 12 .012 2# Gttingen . /./$02. 2 Felix Klein 12
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