ก F กF ก F F ก n x

การประมาณคา 1

2301116 ก II ก ก !""ก

ก 2

ก ก !""ก

#$$

%" $$ &ก n & x₀

ก ก !"

ก "ก'&()!*+,ก"+,ก

ก "ก'&()!*& (

ก "ก'&()!*ก$ก- $(

ก "ก'&()!*ก$ก- ./"

#$$

● 012 8.1.1 3* f $4"ก !".5 f "%" f', f'', ..., f⁽ⁿ⁾ , #

$"6"!, [a, b] f⁽ⁿ⁺¹⁾ ก&"!, (a, b) ( c ∈ (a, b)

$ $(ก#7&*, #$$

● 012 8.1.2 )* f $4"ก !".5"%" "& n+1 7&*ก&)"

!,$8& (a, b) f⁽ⁿ⁾ , #$"6"!,8& [a, b] 3* x x₀ $4"&

&)" (a, b) #ก# ก" 7&*, c .5(, x x₀ )*

fb=faf 'a

1 ! baf ' 'a

2 ! ba²⋯f^na

n ! baⁿ f^n1c

n1! ba^n1

fx=fx0f 'x₀

1! xx0f ' 'x₀

2! xx₀²⋯f^nx₀

n ! xx0ⁿf^n1c

n1! xx₀^n1

%" $$ &ก n & x

₀

● ก#$$ $ ก "&)*

$ $(ก#&ก ,, %" $$ &ก n f & x₀

)* R_n $4", #ก# , f(x) ก T_n(x) 7&*

$ $(ก#&ก ,, $91$6%" $$ '&( c (, x ก x₀ f(x) = T_n(x) + R_n(x)

T_nx=fx₀

∑

k=1

n f^kx₀

k ! xx₀^k

R_nx= f^n1c

n1! xx₀^n1

ก 5

'( %" $$

1. ก "&)* f(x) = ln x

1.1 %" $$ &ก 7 f & 1

1.2 ln(3/2) &*,(%" $$ &ก 7 )!*9"( 5

# " ก$#, &$6")"ก 2. ก "&)* f(x) = e^x

2.1 T_n(x) f & 0 2.2 | x | < 1/2 | R_n(x) |

ก 6

#,( ก

3. ก "&)* f(x) = e^x e^0.2 '&(%" $$ )*, &

$6""*(ก, 0.0001

4. ก "&)* f(x) = ln(1 + x) &*,(%" $$

&ก 5 f & 0

%*$#, &$6"ก n #ก#

7$ก" 0.0001

∫

0 0.5

fxdx

∫

0 0.5

fxdx

∫

0 0.5

T_nxdx

ก "ก

กก f(x) &*,( T_n(x) $ (" %" $$ !,()"ก "ก ก&$#'&(

&":"

3* "," M .5 |f⁽ⁿ⁺¹⁾(x)| < M 7&*,

| R_n(x) | < M |x - x₀|ⁿ⁺¹/(n+1)!

&":"

fx=T_nxR_nx

∫

a b

f xdx=

∫

a b

T_nxdx

∫

a b

R_nxdx

∣ ∫ab fxdx∫ab Tnxdx∣n1!M ∫ab∣xx0∣n1dx

∣

∫

a b

f xdx

∫

a b

T_nxdx∣=∣

∫

a b

R_nxdx∣

∫

a b

∣R_nx∣dx

ก "ก

$ , "กก !" f(x) > 0 "!, [a, b] 6%6:"(

)#*$*"'* f $"6ก" X ก x = a 35 x = b

&":" $ 3)!*+,ก%6:"$(+6"+* !,(($;ก%

)"ก %6:"#*ก

ก "ก&*,(+,ก"+,ก

ก "ก&*,(ก$ก- $!$(

ก "ก&*,(ก$ก- ./"

ก 9

ก "ก

➔ )* f:D → R $6$4"ก !""!,8& [a, b]

➔ )* x₀, x₁, x₂, ..., x_n $4"& [a, b] ( ,$ ก" h = (b - a)/n a = x₀ < x₁ < x₂ < ... < x_n = b

x₀ x₁ x₂ x₃ x₄ y

x

ก 10

ก '&(+,ก""

➔ $6ก& x_i^* ก [x_i-1, x_i] $ )!* ก !" f(x_i^*)

➔ &":"

∫

a b

fxdx=

∫

x₀ x₁

fxdx

∫

x₁ x₂

fxdx

∫

xn1

x_n

fxdx

≈

∫

x₀ x₁

fx₁^∗dx

∫

x₁ x₂

fx₂^∗dx

∫

x_n1 x_n

fx_n^∗dx

=fx₁^∗x₁x₀fx₂^∗x₂x₁fx_n^∗x_nx_n1

=

∑

i=1 n

fx_i^∗x_ix_i1=h

∑

i=1 n

f x_i^∗

∫

a b

f xdx≈h



^{fx1∗fx2∗fxn∗}



ก '&()!*+,ก"+,ก

● 3* $ $6ก& x_i^* ก#!,(( [x_i-1, x_i] 6& ก !""*(& $ 7&*, +% 7&*6ก "ก ก&$#'&(+,ก

● )" "$&(,ก" 3* $ $6ก& x_i^* ก#!,(( [x_i-1, x_i] 6& ก !"

ก& $ 7&*, +% 7&*6ก "ก ก&$#'&(+,ก"

➔ #ก $6ก& x_i^*)* ก !"# &6& $4"$6( ก &":"$ ก"(

$6ก x_i^* & (!, [x_i-1, x_i] 3* $6ก x_i^* = x_i-1

➔ $4"ก "ก ก&$#&*,(+,ก"" '&()!*& ( .* (

∫

a b

f xdx≈h^fx0f x₁f x_n1

ก '&()!*& ( ,

3* $6ก x_i^* = x_i

$4"ก "ก ก&$#&*,(+,ก"" '&()!*& ( ,

● )"กก !"$4"ก !"$%

ก "ก ก&$#&*,(+,ก"" '&()!*& ( .* (

$ กก "ก ก&$#'&(+,ก

ก "ก ก&$#&*,(+,ก"" '&()!*& ( ,

$ กก "ก ก&$#'&(+,ก"

∫

a b

f xdx≈h^fx1fx₂fx_n

ก 13

x₀ x₁ x₂ x₃ x₄ y

x

ก

x

^*_i3ก$6ก ก& .* (!,

ก '&()!*& ( .* (

ก 14

ก

x

^*_i3ก$6ก$4"& (( ,

x₀ x₁ x₂ x₃ x₄ y

x

ก &*,(& ( ,

012 8.2.1

1. '&()!*+,ก +,ก"

● )* I $4" '&()!*+,ก""

I = h(f(x₁^*) + f(x₂^*) + ... + f(x_n^*))

➔ '&( x₀, x₁, ..., x_n $4"& [a, b] h=(b – a)/n, x_i^*()" [x_i-1, x_i]

➔ 3* f $4"ก !""%" "&"5 M $4" ",".5

| f '(x) | < M ก x )"!, (a, b)

➔ 7&*,

2. )"ก * #*" 3* #*ก )*$#, & $6"7$ก"

0.001 $ #*!, [1, 2] $4"ก!,((

∫

1 2 1

xdx , n=5

∫

a b

fxdx

∣

^{∫ab fxdxI}

∣

^{b2 na2M}

กก %6:"$( , ^∆x &* "" "6 y₁

y₂$ 7&*, %6:"6

$ $(กก$ก- )!*6ก$ก- $!$(

y2

y₁

∆x

ก$ก- $!$(

1

2y1y2∆x

∫

a b

f xdx≈1

2

 ∑i=1n fxi1fxi∆x

=∆^x

2 ^{f x}0fx₁f x₁fx₂⋯f x_n1fx_n

=∆^x

2 ^fx02 f x₁2 fx₂⋯2 fx_n1f x_n

ก 17

3*

f

$4"ก !"#$"6

x₀ x₁ x₂ x₃ x₄ y

x

ก$ก- $!$(

∫

a b

f xdx≈T_n=x

2 ^{f x02 f x1⋯2 f xn1f xn}

ก 18

012 8.3.1

● )* I $4" '&(ก$ก- $!$(

I = h/2 (f(x₀) + 2f(x₁) + ... + 2f(x_n-1) + f(x_n))

➔ '&( x₀, x₁, ..., x_n$4"&!, [a, b], h = (b – a)/n

➔ 3* f $4"ก !""%" "& M $4" ",".5

| f''(x) | < M ก x )"!, (a, b)

➔ 7&*,

1. "ก '&()!*ก$ก- $!$( $6 n=8

* #*" 3* #*ก )*$#, &$6"7$ก" ^0.001 $ #*!,

[1, 2] $4"ก!,((

∫

0 1

e^x2dx

∫

a b

fxdx

∣

^∫a b

fxdxI

∣

^{ba12 n23M}

$ #*ก ก !"&*,( y = Ax² + Bx + C

% %6:")#*% ' ก–h7(h7&*,

-h 0 h y

x P₀(-h,y₀) P₁(0,y₁) P₂(h,y₂)

ก$ก- ./"

∫

h h

A x²B xC dx=Ax³ 3Bx²

2C x∣_h^h =2Ah³ 3C h

y₀ = A(-h)² + B(-h) + C = Ah² – Bh + C

y₁ = C

y₂=Ah² + Bh + C

y₀ + 4y₁ + y₂ = 2Ah² + 6C

$ 7&*,

ก &*,(ก$ก- ./"

∫

a b

f xdx≈S_n=h

3y₀4 y₁y₂h

3y₁4 y₂y₃⋯

h

3^{f x}n24 f x_n1f x_n

ก 21

#ก ./"

● ก$ก- ./"

I = h/3 (f(x₀) + 4f(x₁) + 2f(x₂) + ... + 2f(x_n-2) + 4f(x_n-1) + f(x_n))

➔ *$ก# ก$ก- ./"#*)!* ","&$4"$ $% ก )!* 3 &

➔ 012 8.4.1 )* I $4"ก &*,(ก$ก- ./"

➔ '&( x₀, x₁, ..., x_n$4"&!, [a, b] .5 n $4" "," h = (b – a)/n

3* f $4"ก !""%" "& M $4" ",".5

| f⁽⁴⁾(x) | < M ก x )"!, (a, b)

➔ 7&*,

∣

^{∫ab fxdxI}

∣

^{ba180 n45M}

ก 22

'(

1. ก "& f(x) )*&#

x 2 3 4 5 6 7 8 9

f(x) 4.5010 5.0246 5.5865 6.1271 6.9037 7.6814 8.6312 9.4258

➔

➔

➔

2. )"ก &*,(ก$ก- $( $6 n = 8

$#, &$6"$ก&5:" (9"( ⁴ # ")

∫

2

9 1

fxxd x ,∫

2 9

x fxd x

∫

0 1

e^x2dx

'(

3. ก "& f(x) )*&#

x 0 0.5 1.0 1.5 2.0 2.5 3.0

f(x) 0.0625 0.0615 0.0588 0.0548 0.05 0.045 0.004

➔ '&()!*

● ก$ก- $!$( $6 n = 3

● ก$ก- ./" $6 n = 6

4. )"ก &*,(ก$ก- ./"'&()*$#

, &$6"7$ก" 0.0001 #* [1, 2] $4"ก!,

∫

0 3

xfxdx

∫

1 2

ln1xdx