해법
cosh(4x)=16sinh(x)+1
해법
x=0,x=ln(2.41421…)
+1
도
x=0∘,x=50.49898…∘솔루션 단계
cosh(4x)=16sinh(x)+1
삼각성을 사용하여 다시 쓰기
cosh(4x)=16sinh(x)+1
하이퍼볼라식별사용: sinh(x)=2ex−e−xcosh(4x)=16⋅2ex−e−x+1
하이퍼볼라식별사용: cosh(x)=2ex+e−x2e4x+e−4x=16⋅2ex−e−x+1
2e4x+e−4x=16⋅2ex−e−x+1
2e4x+e−4x=16⋅2ex−e−x+1:x=0,x=ln(2.41421…)
2e4x+e−4x=16⋅2ex−e−x+1
양쪽을 곱한 값 22e4x+e−4x⋅2=16⋅2ex−e−x⋅2+1⋅2
단순화e4x+e−4x=16(ex−e−x)+2
지수 규칙 적용
e4x+e−4x=16(ex−e−x)+2
지수 규칙 적용: abc=(ab)ce4x=(ex)4,e−4x=(ex)−4,e−x=(ex)−1(ex)4+(ex)−4=16(ex−(ex)−1)+2
(ex)4+(ex)−4=16(ex−(ex)−1)+2
다음으로 방정식 다시 쓰기 ex=u(u)4+(u)−4=16(u−(u)−1)+2
u4+u−4=16(u−u−1)+2해결 :u=−1,u=1,u≈−0.41421…,u≈2.41421…
u4+u−4=16(u−u−1)+2
다듬다u4+u41=16(u−u1)+2
양쪽을 곱한 값 u4
u4+u41=16(u−u1)+2
양쪽을 곱한 값 u4u4u4+u41u4=16(u−u1)u4+2u4
단순화
u4u4+u41u4=16(u−u1)u4+2u4
u4u4간소화하다 :u8
u4u4
지수 규칙 적용: ab⋅ac=ab+cu4u4=u4+4=u4+4
숫자 추가: 4+4=8=u8
u41u4간소화하다 :1
u41u4
다중 분수: a⋅cb=ca⋅b=u41⋅u4
공통 요인 취소: u4=1
u8+1=16(u−u1)u4+2u4
u8+1=16(u−u1)u4+2u4
u8+1=16(u−u1)u4+2u4
16(u−u1)u4+2u4 확장 :16u5−16u3+2u4
16(u−u1)u4+2u4
=16u4(u−u1)+2u4
16u4(u−u1)확대한다:16u5−16u3
16u4(u−u1)
분배 법칙 적용: a(b−c)=ab−aca=16u4,b=u,c=u1=16u4u−16u4u1
=16u4u−16⋅u1u4
16u4u−16⋅u1u4단순화하세요:16u5−16u3
16u4u−16⋅u1u4
16u4u=16u5
16u4u
지수 규칙 적용: ab⋅ac=ab+cu4u=u4+1=16u4+1
숫자 추가: 4+1=5=16u5
16⋅u1u4=16u3
16⋅u1u4
다중 분수: a⋅cb=ca⋅b=u1⋅16u4
숫자를 곱하시오: 1⋅16=16=u16u4
공통 요인 취소: u=16u3
=16u5−16u3
=16u5−16u3
=16u5−16u3+2u4
u8+1=16u5−16u3+2u4
u8+1=16u5−16u3+2u4해결 :u=−1,u=1,u≈−0.41421…,u≈2.41421…
u8+1=16u5−16u3+2u4
2u4를 왼쪽으로 이동
u8+1=16u5−16u3+2u4
빼다 2u4 양쪽에서u8+1−2u4=16u5−16u3+2u4−2u4
단순화u8+1−2u4=16u5−16u3
u8+1−2u4=16u5−16u3
16u3를 왼쪽으로 이동
u8+1−2u4=16u5−16u3
더하다 16u3 양쪽으로u8+1−2u4+16u3=16u5−16u3+16u3
단순화u8+1−2u4+16u3=16u5
u8+1−2u4+16u3=16u5
16u5를 왼쪽으로 이동
u8+1−2u4+16u3=16u5
빼다 16u5 양쪽에서u8+1−2u4+16u3−16u5=16u5−16u5
단순화u8+1−2u4+16u3−16u5=0
u8+1−2u4+16u3−16u5=0
표준 양식으로 작성 anxn+…+a1x+a0=0u8−16u5−2u4+16u3+1=0
u8−16u5−2u4+16u3+1인수 :(u+1)(u−1)(u6+u4−16u3−u2−1)
u8−16u5−2u4+16u3+1
합리적인 근정리를 사용하라
a0=1,an=1
의 나눗셈 a0:1,의 나눗셈 an:1
따라서 다음 합리적인 숫자를 확인하십시오:±11
−11이 표현의 어근입니다, 그러니 잘 생각해보세요 u+1
=(u+1)u+1u8−16u5−2u4+16u3+1u+1u8−16u5−2u4+16u3+1=u7−u6+u5−17u4+15u3+u2−u+1
u+1u8−16u5−2u4+16u3+1
u+1u8−16u5−2u4+16u3+1나누다:u+1u8−16u5−2u4+16u3+1=u7+u+1−u7−16u5−2u4+16u3+1
분자의 선행 계수를 나눕니다 u8−16u5−2u4+16u3+1
그리고 나눗셈 u+1:uu8=u7
몫=u7u+1 에 u7로 곱하시오 u8+u7 새 나머지를 얻으려면 u8−16u5−2u4+16u3+1 에서 u8+u7빼십시오 나머지=−u7−16u5−2u4+16u3+1
그러므로u+1u8−16u5−2u4+16u3+1=u7+u+1−u7−16u5−2u4+16u3+1
=u7+u+1−u7−16u5−2u4+16u3+1
u+1−u7−16u5−2u4+16u3+1나누다:u+1−u7−16u5−2u4+16u3+1=−u6+u+1u6−16u5−2u4+16u3+1
분자의 선행 계수를 나눕니다 −u7−16u5−2u4+16u3+1
그리고 나눗셈 u+1:u−u7=−u6
몫=−u6u+1 에 −u6로 곱하시오 −u7−u6 새 나머지를 얻으려면 −u7−16u5−2u4+16u3+1 에서 −u7−u6빼십시오 나머지=u6−16u5−2u4+16u3+1
그러므로u+1−u7−16u5−2u4+16u3+1=−u6+u+1u6−16u5−2u4+16u3+1
=u7−u6+u+1u6−16u5−2u4+16u3+1
u+1u6−16u5−2u4+16u3+1나누다:u+1u6−16u5−2u4+16u3+1=u5+u+1−17u5−2u4+16u3+1
분자의 선행 계수를 나눕니다 u6−16u5−2u4+16u3+1
그리고 나눗셈 u+1:uu6=u5
몫=u5u+1 에 u5로 곱하시오 u6+u5 새 나머지를 얻으려면 u6−16u5−2u4+16u3+1 에서 u6+u5빼십시오 나머지=−17u5−2u4+16u3+1
그러므로u+1u6−16u5−2u4+16u3+1=u5+u+1−17u5−2u4+16u3+1
=u7−u6+u5+u+1−17u5−2u4+16u3+1
u+1−17u5−2u4+16u3+1나누다:u+1−17u5−2u4+16u3+1=−17u4+u+115u4+16u3+1
분자의 선행 계수를 나눕니다 −17u5−2u4+16u3+1
그리고 나눗셈 u+1:u−17u5=−17u4
몫=−17u4u+1 에 −17u4로 곱하시오 −17u5−17u4 새 나머지를 얻으려면 −17u5−2u4+16u3+1 에서 −17u5−17u4빼십시오 나머지=15u4+16u3+1
그러므로u+1−17u5−2u4+16u3+1=−17u4+u+115u4+16u3+1
=u7−u6+u5−17u4+u+115u4+16u3+1
u+115u4+16u3+1나누다:u+115u4+16u3+1=15u3+u+1u3+1
분자의 선행 계수를 나눕니다 15u4+16u3+1
그리고 나눗셈 u+1:u15u4=15u3
몫=15u3u+1 에 15u3로 곱하시오 15u4+15u3 새 나머지를 얻으려면 15u4+16u3+1 에서 15u4+15u3빼십시오 나머지=u3+1
그러므로u+115u4+16u3+1=15u3+u+1u3+1
=u7−u6+u5−17u4+15u3+u+1u3+1
u+1u3+1나누다:u+1u3+1=u2+u+1−u2+1
분자의 선행 계수를 나눕니다 u3+1
그리고 나눗셈 u+1:uu3=u2
몫=u2u+1 에 u2로 곱하시오 u3+u2 새 나머지를 얻으려면 u3+1 에서 u3+u2빼십시오 나머지=−u2+1
그러므로u+1u3+1=u2+u+1−u2+1
=u7−u6+u5−17u4+15u3+u2+u+1−u2+1
u+1−u2+1나누다:u+1−u2+1=−u+u+1u+1
분자의 선행 계수를 나눕니다 −u2+1
그리고 나눗셈 u+1:u−u2=−u
몫=−uu+1 에 −u로 곱하시오 −u2−u 새 나머지를 얻으려면 −u2+1 에서 −u2−u빼십시오 나머지=u+1
그러므로u+1−u2+1=−u+u+1u+1
=u7−u6+u5−17u4+15u3+u2−u+u+1u+1
u+1u+1나누다:u+1u+1=1
분자의 선행 계수를 나눕니다 u+1
그리고 나눗셈 u+1:uu=1
몫=1u+1 에 1로 곱하시오 u+1 새 나머지를 얻으려면 u+1 에서 u+1빼십시오 나머지=0
그러므로u+1u+1=1
=u7−u6+u5−17u4+15u3+u2−u+1
=u7−u6+u5−17u4+15u3+u2−u+1
u7−u6+u5−17u4+15u3+u2−u+1요인:(u−1)(u6+u4−16u3−u2−1)
u7−u6+u5−17u4+15u3+u2−u+1
합리적인 근정리를 사용하라
a0=1,an=1
의 나눗셈 a0:1,의 나눗셈 an:1
따라서 다음 합리적인 숫자를 확인하십시오:±11
11이 표현의 어근입니다, 그러니 잘 생각해보세요 u−1
=(u−1)u−1u7−u6+u5−17u4+15u3+u2−u+1u−1u7−u6+u5−17u4+15u3+u2−u+1=u6+u4−16u3−u2−1
u−1u7−u6+u5−17u4+15u3+u2−u+1
u−1u7−u6+u5−17u4+15u3+u2−u+1나누다:u−1u7−u6+u5−17u4+15u3+u2−u+1=u6+u−1u5−17u4+15u3+u2−u+1
분자의 선행 계수를 나눕니다 u7−u6+u5−17u4+15u3+u2−u+1
그리고 나눗셈 u−1:uu7=u6
몫=u6u−1 에 u6로 곱하시오 u7−u6 새 나머지를 얻으려면 u7−u6+u5−17u4+15u3+u2−u+1 에서 u7−u6빼십시오 나머지=u5−17u4+15u3+u2−u+1
그러므로u−1u7−u6+u5−17u4+15u3+u2−u+1=u6+u−1u5−17u4+15u3+u2−u+1
=u6+u−1u5−17u4+15u3+u2−u+1
u−1u5−17u4+15u3+u2−u+1나누다:u−1u5−17u4+15u3+u2−u+1=u4+u−1−16u4+15u3+u2−u+1
분자의 선행 계수를 나눕니다 u5−17u4+15u3+u2−u+1
그리고 나눗셈 u−1:uu5=u4
몫=u4u−1 에 u4로 곱하시오 u5−u4 새 나머지를 얻으려면 u5−17u4+15u3+u2−u+1 에서 u5−u4빼십시오 나머지=−16u4+15u3+u2−u+1
그러므로u−1u5−17u4+15u3+u2−u+1=u4+u−1−16u4+15u3+u2−u+1
=u6+u4+u−1−16u4+15u3+u2−u+1
u−1−16u4+15u3+u2−u+1나누다:u−1−16u4+15u3+u2−u+1=−16u3+u−1−u3+u2−u+1
분자의 선행 계수를 나눕니다 −16u4+15u3+u2−u+1
그리고 나눗셈 u−1:u−16u4=−16u3
몫=−16u3u−1 에 −16u3로 곱하시오 −16u4+16u3 새 나머지를 얻으려면 −16u4+15u3+u2−u+1 에서 −16u4+16u3빼십시오 나머지=−u3+u2−u+1
그러므로u−1−16u4+15u3+u2−u+1=−16u3+u−1−u3+u2−u+1
=u6+u4−16u3+u−1−u3+u2−u+1
u−1−u3+u2−u+1나누다:u−1−u3+u2−u+1=−u2+u−1−u+1
분자의 선행 계수를 나눕니다 −u3+u2−u+1
그리고 나눗셈 u−1:u−u3=−u2
몫=−u2u−1 에 −u2로 곱하시오 −u3+u2 새 나머지를 얻으려면 −u3+u2−u+1 에서 −u3+u2빼십시오 나머지=−u+1
그러므로u−1−u3+u2−u+1=−u2+u−1−u+1
=u6+u4−16u3−u2+u−1−u+1
u−1−u+1나누다:u−1−u+1=−1
분자의 선행 계수를 나눕니다 −u+1
그리고 나눗셈 u−1:u−u=−1
몫=−1u−1 에 −1로 곱하시오 −u+1 새 나머지를 얻으려면 −u+1 에서 −u+1빼십시오 나머지=0
그러므로u−1−u+1=−1
=u6+u4−16u3−u2−1
=u6+u4−16u3−u2−1
=(u−1)(u6+u4−16u3−u2−1)
=(u+1)(u−1)(u6+u4−16u3−u2−1)
(u+1)(u−1)(u6+u4−16u3−u2−1)=0
제로 인자 원리 사용:\4각형이면 ab=0그렇다면 a=0or b=0u+1=0oru−1=0oru6+u4−16u3−u2−1=0
u+1=0해결 :u=−1
u+1=0
1를 오른쪽으로 이동
u+1=0
빼다 1 양쪽에서u+1−1=0−1
단순화u=−1
u=−1
u−1=0해결 :u=1
u−1=0
1를 오른쪽으로 이동
u−1=0
더하다 1 양쪽으로u−1+1=0+1
단순화u=1
u=1
u6+u4−16u3−u2−1=0해결 :u≈−0.41421…,u≈2.41421…
u6+u4−16u3−u2−1=0
다음을 위한 하나의 솔루션 찾기 u6+u4−16u3−u2−1=0 뉴턴-랩슨을 이용하여:u≈−0.41421…
u6+u4−16u3−u2−1=0
뉴턴-랩슨 근사 정의
f(u)=u6+u4−16u3−u2−1
f′(u)찾다 :6u5+4u3−48u2−2u
dud(u6+u4−16u3−u2−1)
합계/차이 규칙 적용: (f±g)′=f′±g′=dud(u6)+dud(u4)−dud(16u3)−dud(u2)−dud(1)
dud(u6)=6u5
dud(u6)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=6u6−1
단순화=6u5
dud(u4)=4u3
dud(u4)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=4u4−1
단순화=4u3
dud(16u3)=48u2
dud(16u3)
정수를 빼라: (a⋅f)′=a⋅f′=16dud(u3)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=16⋅3u3−1
단순화=48u2
dud(u2)=2u
dud(u2)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=2u2−1
단순화=2u
dud(1)=0
dud(1)
상수의 도함수: dxd(a)=0=0
=6u5+4u3−48u2−2u−0
단순화=6u5+4u3−48u2−2u
렛 u0=1계산하다 un+1 까지 Δun+1<0.000001
u1=0.6:Δu1=0.4
f(u0)=16+14−16⋅13−12−1=−16f′(u0)=6⋅15+4⋅13−48⋅12−2⋅1=−40u1=0.6
Δu1=∣0.6−1∣=0.4Δu1=0.4
u2=0.32945…:Δu2=0.27054…
f(u1)=0.66+0.64−16⋅0.63−0.62−1=−4.639744f′(u1)=6⋅0.65+4⋅0.63−48⋅0.62−2⋅0.6=−17.14944u2=0.32945…
Δu2=∣0.32945…−0.6∣=0.27054…Δu2=0.27054…
u3=0.03701…:Δu3=0.29243…
f(u2)=0.32945…6+0.32945…4−16⋅0.32945…3−0.32945…2−1=−1.66761…f′(u2)=6⋅0.32945…5+4⋅0.32945…3−48⋅0.32945…2−2⋅0.32945…=−5.70244…u3=0.03701…
Δu3=∣0.03701…−0.32945…∣=0.29243…Δu3=0.29243…
u4=−7.14264…:Δu4=7.17966…
f(u3)=0.03701…6+0.03701…4−16⋅0.03701…3−0.03701…2−1=−1.00217…f′(u3)=6⋅0.03701…5+4⋅0.03701…3−48⋅0.03701…2−2⋅0.03701…=−0.13958…u4=−7.14264…
Δu4=∣−7.14264…−0.03701…∣=7.17966…Δu4=7.17966…
u5=−5.91974…:Δu5=1.22290…
f(u4)=(−7.14264…)6+(−7.14264…)4−16(−7.14264…)3−(−7.14264…)2−1=141168.16963…f′(u4)=6(−7.14264…)5+4(−7.14264…)3−48(−7.14264…)2−2(−7.14264…)=−115436.50403…u5=−5.91974…
Δu5=∣−5.91974…−(−7.14264…)∣=1.22290…Δu5=1.22290…
u6=−4.88878…:Δu6=1.03095…
f(u5)=(−5.91974…)6+(−5.91974…)4−16(−5.91974…)3−(−5.91974…)2−1=47545.59081…f′(u5)=6(−5.91974…)5+4(−5.91974…)3−48(−5.91974…)2−2(−5.91974…)=−46117.92631…u6=−4.88878…
Δu6=∣−4.88878…−(−5.91974…)∣=1.03095…Δu6=1.03095…
u7=−4.01362…:Δu7=0.87515…
f(u6)=(−4.88878…)6+(−4.88878…)4−16(−4.88878…)3−(−4.88878…)2−1=16068.08422…f′(u6)=6(−4.88878…)5+4(−4.88878…)3−48(−4.88878…)2−2(−4.88878…)=−18360.23113…u7=−4.01362…
Δu7=∣−4.01362…−(−4.88878…)∣=0.87515…Δu7=0.87515…
u8=−3.26329…:Δu8=0.75033…
f(u7)=(−4.01362…)6+(−4.01362…)4−16(−4.01362…)3−(−4.01362…)2−1=5457.33921…f′(u7)=6(−4.01362…)5+4(−4.01362…)3−48(−4.01362…)2−2(−4.01362…)=−7273.21083…u8=−3.26329…
Δu8=∣−3.26329…−(−4.01362…)∣=0.75033…Δu8=0.75033…
u9=−2.61197…:Δu9=0.65132…
f(u8)=(−3.26329…)6+(−3.26329…)4−16(−3.26329…)3−(−3.26329…)2−1=1865.40737…f′(u8)=6(−3.26329…)5+4(−3.26329…)3−48(−3.26329…)2−2(−3.26329…)=−2864.03467…u9=−2.61197…
Δu9=∣−2.61197…−(−3.26329…)∣=0.65132…Δu9=0.65132…
u10=−2.04081…:Δu10=0.57115…
f(u9)=(−2.61197…)6+(−2.61197…)4−16(−2.61197…)3−(−2.61197…)2−1=641.38974…f′(u9)=6(−2.61197…)5+4(−2.61197…)3−48(−2.61197…)2−2(−2.61197…)=−1122.97668…u10=−2.04081…
Δu10=∣−2.04081…−(−2.61197…)∣=0.57115…Δu10=0.57115…
u11=−1.54238…:Δu11=0.49843…
f(u10)=(−2.04081…)6+(−2.04081…)4−16(−2.04081…)3−(−2.04081…)2−1=220.42864…f′(u10)=6(−2.04081…)5+4(−2.04081…)3−48(−2.04081…)2−2(−2.04081…)=−442.24518…u11=−1.54238…
Δu11=∣−1.54238…−(−2.04081…)∣=0.49843…Δu11=0.49843…
u12=−1.12448…:Δu12=0.41790…
f(u11)=(−1.54238…)6+(−1.54238…)4−16(−1.54238…)3−(−1.54238…)2−1=74.45277…f′(u11)=6(−1.54238…)5+4(−1.54238…)3−48(−1.54238…)2−2(−1.54238…)=−178.15724…u12=−1.12448…
Δu12=∣−1.12448…−(−1.54238…)∣=0.41790…Δu12=0.41790…
u13=−0.80272…:Δu13=0.32175…
f(u12)=(−1.12448…)6+(−1.12448…)4−16(−1.12448…)3−(−1.12448…)2−1=24.10604…f′(u12)=6(−1.12448…)5+4(−1.12448…)3−48(−1.12448…)2−2(−1.12448…)=−74.92024…u13=−0.80272…
Δu13=∣−0.80272…−(−1.12448…)∣=0.32175…Δu13=0.32175…
u14=−0.58368…:Δu14=0.21904…
f(u13)=(−0.80272…)6+(−0.80272…)4−16(−0.80272…)3−(−0.80272…)2−1=7.31448…f′(u13)=6(−0.80272…)5+4(−0.80272…)3−48(−0.80272…)2−2(−0.80272…)=−33.39326…u14=−0.58368…
Δu14=∣−0.58368…−(−0.80272…)∣=0.21904…Δu14=0.21904…
u15=−0.46184…:Δu15=0.12183…
f(u14)=(−0.58368…)6+(−0.58368…)4−16(−0.58368…)3−(−0.58368…)2−1=1.99663…f′(u14)=6(−0.58368…)5+4(−0.58368…)3−48(−0.58368…)2−2(−0.58368…)=−16.38770…u15=−0.46184…
Δu15=∣−0.46184…−(−0.58368…)∣=0.12183…Δu15=0.12183…
u16=−0.41933…:Δu16=0.04251…
f(u15)=(−0.46184…)6+(−0.46184…)4−16(−0.46184…)3−(−0.46184…)2−1=0.41814…f′(u15)=6(−0.46184…)5+4(−0.46184…)3−48(−0.46184…)2−2(−0.46184…)=−9.83510…u16=−0.41933…
Δu16=∣−0.41933…−(−0.46184…)∣=0.04251…Δu16=0.04251…
u17=−0.41428…:Δu17=0.00505…
f(u16)=(−0.41933…)6+(−0.41933…)4−16(−0.41933…)3−(−0.41933…)2−1=0.04030…f′(u16)=6(−0.41933…)5+4(−0.41933…)3−48(−0.41933…)2−2(−0.41933…)=−7.97447…u17=−0.41428…
Δu17=∣−0.41428…−(−0.41933…)∣=0.00505…Δu17=0.00505…
u18=−0.41421…:Δu18=0.00006…
f(u17)=(−0.41428…)6+(−0.41428…)4−16(−0.41428…)3−(−0.41428…)2−1=0.00052…f′(u17)=6(−0.41428…)5+4(−0.41428…)3−48(−0.41428…)2−2(−0.41428…)=−7.76725…u18=−0.41421…
Δu18=∣−0.41421…−(−0.41428…)∣=0.00006…Δu18=0.00006…
u19=−0.41421…:Δu19=1.19705E−8
f(u18)=(−0.41421…)6+(−0.41421…)4−16(−0.41421…)3−(−0.41421…)2−1=9.29452E−8f′(u18)=6(−0.41421…)5+4(−0.41421…)3−48(−0.41421…)2−2(−0.41421…)=−7.76450…u19=−0.41421…
Δu19=∣−0.41421…−(−0.41421…)∣=1.19705E−8Δu19=1.19705E−8
u≈−0.41421…
긴 나눗셈 적용:u+0.41421…u6+u4−16u3−u2−1=u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…
u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…≈0
다음을 위한 하나의 솔루션 찾기 u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…=0 뉴턴-랩슨을 이용하여:u≈2.41421…
u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…=0
뉴턴-랩슨 근사 정의
f(u)=u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…
f′(u)찾다 :5u4−1.65685…u3+3.51471…u2−32.97056…u+5.82842…
dud(u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…)
합계/차이 규칙 적용: (f±g)′=f′±g′=dud(u5)−dud(0.41421…u4)+dud(1.17157…u3)−dud(16.48528…u2)+dud(5.82842…u)−dud(2.41421…)
dud(u5)=5u4
dud(u5)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=5u5−1
단순화=5u4
dud(0.41421…u4)=1.65685…u3
dud(0.41421…u4)
정수를 빼라: (a⋅f)′=a⋅f′=0.41421…dud(u4)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=0.41421…⋅4u4−1
단순화=1.65685…u3
dud(1.17157…u3)=3.51471…u2
dud(1.17157…u3)
정수를 빼라: (a⋅f)′=a⋅f′=1.17157…dud(u3)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=1.17157…⋅3u3−1
단순화=3.51471…u2
dud(16.48528…u2)=32.97056…u
dud(16.48528…u2)
정수를 빼라: (a⋅f)′=a⋅f′=16.48528…dud(u2)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=16.48528…⋅2u2−1
단순화=32.97056…u
dud(5.82842…u)=5.82842…
dud(5.82842…u)
정수를 빼라: (a⋅f)′=a⋅f′=5.82842…dudu
공통 도함수 적용: dudu=1=5.82842…⋅1
단순화=5.82842…
dud(2.41421…)=0
dud(2.41421…)
상수의 도함수: dxd(a)=0=0
=5u4−1.65685…u3+3.51471…u2−32.97056…u+5.82842…−0
단순화=5u4−1.65685…u3+3.51471…u2−32.97056…u+5.82842…
렛 u0=0계산하다 un+1 까지 Δun+1<0.000001
u1=0.41421…:Δu1=0.41421…
f(u0)=05−0.41421…⋅04+1.17157…⋅03−16.48528…⋅02+5.82842…⋅0−2.41421…=−2.41421…f′(u0)=5⋅04−1.65685…⋅03+3.51471…⋅02−32.97056…⋅0+5.82842…=5.82842…u1=0.41421…
Δu1=∣0.41421…−0∣=0.41421…Δu1=0.41421…
u2=0.03272…:Δu2=0.38148…
f(u1)=0.41421…5−0.41421…⋅0.41421…4+1.17157…⋅0.41421…3−16.48528…⋅0.41421…2+5.82842…⋅0.41421…−2.41421…=−2.74516…f′(u1)=5⋅0.41421…4−1.65685…⋅0.41421…3+3.51471…⋅0.41421…2−32.97056…⋅0.41421…+5.82842…=−7.19595…u2=0.03272…
Δu2=∣0.03272…−0.41421…∣=0.38148…Δu2=0.38148…
u3=0.50422…:Δu3=0.47149…
f(u2)=0.03272…5−0.41421…⋅0.03272…4+1.17157…⋅0.03272…3−16.48528…⋅0.03272…2+5.82842…⋅0.03272…−2.41421…=−2.24108…f′(u2)=5⋅0.03272…4−1.65685…⋅0.03272…3+3.51471…⋅0.03272…2−32.97056…⋅0.03272…+5.82842…=4.75313…u3=0.50422…
Δu3=∣0.50422…−0.03272…∣=0.47149…Δu3=0.47149…
u4=0.14569…:Δu4=0.35852…
f(u3)=0.50422…5−0.41421…⋅0.50422…4+1.17157…⋅0.50422…3−16.48528…⋅0.50422…2+5.82842…⋅0.50422…−2.41421…=−3.51061…f′(u3)=5⋅0.50422…4−1.65685…⋅0.50422…3+3.51471…⋅0.50422…2−32.97056…⋅0.50422…+5.82842…=−9.79171…u4=0.14569…
Δu4=∣0.14569…−0.50422…∣=0.35852…Δu4=0.35852…
u5=1.88887…:Δu5=1.74318…
f(u4)=0.14569…5−0.41421…⋅0.14569…4+1.17157…⋅0.14569…3−16.48528…⋅0.14569…2+5.82842…⋅0.14569…−2.41421…=−1.91147…f′(u4)=5⋅0.14569…4−1.65685…⋅0.14569…3+3.51471…⋅0.14569…2−32.97056…⋅0.14569…+5.82842…=1.09654…u5=1.88887…
Δu5=∣1.88887…−0.14569…∣=1.74318…Δu5=1.74318…
u6=4.63637…:Δu6=2.74749…
f(u5)=1.88887…5−0.41421…⋅1.88887…4+1.17157…⋅1.88887…3−16.48528…⋅1.88887…2+5.82842…⋅1.88887…−2.41421…=−23.55475…f′(u5)=5⋅1.88887…4−1.65685…⋅1.88887…3+3.51471…⋅1.88887…2−32.97056…⋅1.88887…+5.82842…=8.57317…u6=4.63637…
Δu6=∣4.63637…−1.88887…∣=2.74749…Δu6=2.74749…
u7=3.79830…:Δu7=0.83806…
f(u6)=4.63637…5−0.41421…⋅4.63637…4+1.17157…⋅4.63637…3−16.48528…⋅4.63637…2+5.82842…⋅4.63637…−2.41421…=1737.96023…f′(u6)=5⋅4.63637…4−1.65685…⋅4.63637…3+3.51471…⋅4.63637…2−32.97056…⋅4.63637…+5.82842…=2073.76688…u7=3.79830…
Δu7=∣3.79830…−4.63637…∣=0.83806…Δu7=0.83806…
u8=3.17364…:Δu8=0.62465…
f(u7)=3.79830…5−0.41421…⋅3.79830…4+1.17157…⋅3.79830…3−16.48528…⋅3.79830…2+5.82842…⋅3.79830…−2.41421…=550.45789…f′(u7)=5⋅3.79830…4−1.65685…⋅3.79830…3+3.51471…⋅3.79830…2−32.97056…⋅3.79830…+5.82842…=881.21646…u8=3.17364…
Δu8=∣3.17364…−3.79830…∣=0.62465…Δu8=0.62465…
u9=2.74529…:Δu9=0.42835…
f(u8)=3.17364…5−0.41421…⋅3.17364…4+1.17157…⋅3.17364…3−16.48528…⋅3.17364…2+5.82842…⋅3.17364…−2.41421…=167.42483…f′(u8)=5⋅3.17364…4−1.65685…⋅3.17364…3+3.51471…⋅3.17364…2−32.97056…⋅3.17364…+5.82842…=390.85896…u9=2.74529…
Δu9=∣2.74529…−3.17364…∣=0.42835…Δu9=0.42835…
u10=2.50516…:Δu10=0.24012…
f(u9)=2.74529…5−0.41421…⋅2.74529…4+1.17157…⋅2.74529…3−16.48528…⋅2.74529…2+5.82842…⋅2.74529…−2.41421…=45.99071…f′(u9)=5⋅2.74529…4−1.65685…⋅2.74529…3+3.51471…⋅2.74529…2−32.97056…⋅2.74529…+5.82842…=191.52771…u10=2.50516…
Δu10=∣2.50516…−2.74529…∣=0.24012…Δu10=0.24012…
u11=2.42337…:Δu11=0.08179…
f(u10)=2.50516…5−0.41421…⋅2.50516…4+1.17157…⋅2.50516…3−16.48528…⋅2.50516…2+5.82842…⋅2.50516…−2.41421…=9.50260…f′(u10)=5⋅2.50516…4−1.65685…⋅2.50516…3+3.51471…⋅2.50516…2−32.97056…⋅2.50516…+5.82842…=116.17302…u11=2.42337…
Δu11=∣2.42337…−2.50516…∣=0.08179…Δu11=0.08179…
u12=2.41431…:Δu12=0.00905…
f(u11)=2.42337…5−0.41421…⋅2.42337…4+1.17157…⋅2.42337…3−16.48528…⋅2.42337…2+5.82842…⋅2.42337…−2.41421…=0.86400…f′(u11)=5⋅2.42337…4−1.65685…⋅2.42337…3+3.51471…⋅2.42337…2−32.97056…⋅2.42337…+5.82842…=95.43428…u12=2.41431…
Δu12=∣2.41431…−2.42337…∣=0.00905…Δu12=0.00905…
u13=2.41421…:Δu13=0.00010…
f(u12)=2.41431…5−0.41421…⋅2.41431…4+1.17157…⋅2.41431…3−16.48528…⋅2.41431…2+5.82842…⋅2.41431…−2.41421…=0.00977…f′(u12)=5⋅2.41431…4−1.65685…⋅2.41431…3+3.51471…⋅2.41431…2−32.97056…⋅2.41431…+5.82842…=93.27961…u13=2.41421…
Δu13=∣2.41421…−2.41431…∣=0.00010…Δu13=0.00010…
u14=2.41421…:Δu14=1.39211E−8
f(u13)=2.41421…5−0.41421…⋅2.41421…4+1.17157…⋅2.41421…3−16.48528…⋅2.41421…2+5.82842…⋅2.41421…−2.41421…=1.29821E−6f′(u13)=5⋅2.41421…4−1.65685…⋅2.41421…3+3.51471…⋅2.41421…2−32.97056…⋅2.41421…+5.82842…=93.25483…u14=2.41421…
Δu14=∣2.41421…−2.41421…∣=1.39211E−8Δu14=1.39211E−8
u≈2.41421…
긴 나눗셈 적용:u−2.41421…u5−0.41421…u4+1.17157…u3−16.48528…u2+5.82842…u−2.41421…=u4+2u3+6u2−2u+1
u4+2u3+6u2−2u+1≈0
다음을 위한 하나의 솔루션 찾기 u4+2u3+6u2−2u+1=0 뉴턴-랩슨을 이용하여:솔루션 없음 u∈R
u4+2u3+6u2−2u+1=0
뉴턴-랩슨 근사 정의
f(u)=u4+2u3+6u2−2u+1
f′(u)찾다 :4u3+6u2+12u−2
dud(u4+2u3+6u2−2u+1)
합계/차이 규칙 적용: (f±g)′=f′±g′=dud(u4)+dud(2u3)+dud(6u2)−dud(2u)+dud(1)
dud(u4)=4u3
dud(u4)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=4u4−1
단순화=4u3
dud(2u3)=6u2
dud(2u3)
정수를 빼라: (a⋅f)′=a⋅f′=2dud(u3)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=2⋅3u3−1
단순화=6u2
dud(6u2)=12u
dud(6u2)
정수를 빼라: (a⋅f)′=a⋅f′=6dud(u2)
전원 규칙을 적용합니다: dxd(xa)=a⋅xa−1=6⋅2u2−1
단순화=12u
dud(2u)=2
dud(2u)
정수를 빼라: (a⋅f)′=a⋅f′=2dudu
공통 도함수 적용: dudu=1=2⋅1
단순화=2
dud(1)=0
dud(1)
상수의 도함수: dxd(a)=0=0
=4u3+6u2+12u−2+0
단순화=4u3+6u2+12u−2
렛 u0=1계산하다 un+1 까지 Δun+1<0.000001
u1=0.6:Δu1=0.4
f(u0)=14+2⋅13+6⋅12−2⋅1+1=8f′(u0)=4⋅13+6⋅12+12⋅1−2=20u1=0.6
Δu1=∣0.6−1∣=0.4Δu1=0.4
u2=0.29338…:Δu2=0.30661…
f(u1)=0.64+2⋅0.63+6⋅0.62−2⋅0.6+1=2.5216f′(u1)=4⋅0.63+6⋅0.62+12⋅0.6−2=8.224u2=0.29338…
Δu2=∣0.29338…−0.6∣=0.30661…Δu2=0.30661…
u3=−0.16852…:Δu3=0.46190…
f(u2)=0.29338…4+2⋅0.29338…3+6⋅0.29338…2−2⋅0.29338…+1=0.98759…f′(u2)=4⋅0.29338…3+6⋅0.29338…2+12⋅0.29338…−2=2.13808…u3=−0.16852…
Δu3=∣−0.16852…−0.29338…∣=0.46190…Δu3=0.46190…
u4=0.21863…:Δu4=0.38715…
f(u3)=(−0.16852…)4+2(−0.16852…)3+6(−0.16852…)2−2(−0.16852…)+1=1.49867…f′(u3)=4(−0.16852…)3+6(−0.16852…)2+12(−0.16852…)−2=−3.87099…u4=0.21863…
Δu4=∣0.21863…−(−0.16852…)∣=0.38715…Δu4=0.38715…
u5=−0.69789…:Δu5=0.91652…
f(u4)=0.21863…4+2⋅0.21863…3+6⋅0.21863…2−2⋅0.21863…+1=0.87272…f′(u4)=4⋅0.21863…3+6⋅0.21863…2+12⋅0.21863…−2=0.95220…u5=−0.69789…
Δu5=∣−0.69789…−0.21863…∣=0.91652…Δu5=0.91652…
u6=−0.14461…:Δu6=0.55328…
f(u5)=(−0.69789…)4+2(−0.69789…)3+6(−0.69789…)2−2(−0.69789…)+1=4.87554…f′(u5)=4(−0.69789…)3+6(−0.69789…)2+12(−0.69789…)−2=−8.81206…u6=−0.14461…
Δu6=∣−0.14461…−(−0.69789…)∣=0.55328…Δu6=0.55328…
u7=0.24442…:Δu7=0.38903…
f(u6)=(−0.14461…)4+2(−0.14461…)3+6(−0.14461…)2−2(−0.14461…)+1=1.40910…f′(u6)=4(−0.14461…)3+6(−0.14461…)2+12(−0.14461…)−2=−3.62201…u7=0.24442…
Δu7=∣0.24442…−(−0.14461…)∣=0.38903…Δu7=0.38903…
u8=−0.42403…:Δu8=0.66846…
f(u7)=0.24442…4+2⋅0.24442…3+6⋅0.24442…2−2⋅0.24442…+1=0.90238…f′(u7)=4⋅0.24442…3+6⋅0.24442…2+12⋅0.24442…−2=1.34994…u8=−0.42403…
Δu8=∣−0.42403…−0.24442…∣=0.66846…Δu8=0.66846…
u9=0.02045…:Δu9=0.44448…
f(u8)=(−0.42403…)4+2(−0.42403…)3+6(−0.42403…)2−2(−0.42403…)+1=2.80677…f′(u8)=4(−0.42403…)3+6(−0.42403…)2+12(−0.42403…)−2=−6.31459…u9=0.02045…
Δu9=∣0.02045…−(−0.42403…)∣=0.44448…Δu9=0.44448…
u10=0.56930…:Δu10=0.54885…
f(u9)=0.02045…4+2⋅0.02045…3+6⋅0.02045…2−2⋅0.02045…+1=0.96162…f′(u9)=4⋅0.02045…3+6⋅0.02045…2+12⋅0.02045…−2=−1.75205…u10=0.56930…
Δu10=∣0.56930…−0.02045…∣=0.54885…Δu10=0.54885…
해결 방법을 찾을 수 없습니다
해결책은u≈−0.41421…,u≈2.41421…
해결책은u=−1,u=1,u≈−0.41421…,u≈2.41421…
u=−1,u=1,u≈−0.41421…,u≈2.41421…
솔루션 확인
정의되지 않은 (특이점) 점 찾기:u=0
의 분모를 취하라 u4+u−4 그리고 0과 비교한다
u4=0해결 :u=0
u4=0
규칙 적용 xn=0⇒x=0
u=0
의 분모를 취하라 16(u−u−1)+2 그리고 0과 비교한다
u=0
다음 지점은 정의되지 않았습니다u=0
정의되지 않은 점을 솔루션과 결합:
u=−1,u=1,u≈−0.41421…,u≈2.41421…
u=−1,u=1,u≈−0.41421…,u≈2.41421…
다시 대체 u=ex,을 해결하다 x
ex=−1해결 :솔루션 없음 x∈R
ex=−1
af(x) 에 대해 0 또는 음수일 수 없습니다 x∈R솔루션없음x∈R
ex=1해결 :x=0
ex=1
지수 규칙 적용
ex=1
만약에 f(x)=g(x), 그렇다면 ln(f(x))=ln(g(x))ln(ex)=ln(1)
로그 규칙 적용: ln(ea)=aln(ex)=xx=ln(1)
ln(1)간소화하다 :0
ln(1)
로그 규칙 적용: loga(1)=0=0
x=0
x=0
ex=−0.41421…해결 :솔루션 없음 x∈R
ex=−0.41421…
af(x) 에 대해 0 또는 음수일 수 없습니다 x∈R솔루션없음x∈R
ex=2.41421…해결 :x=ln(2.41421…)
ex=2.41421…
지수 규칙 적용
ex=2.41421…
만약에 f(x)=g(x), 그렇다면 ln(f(x))=ln(g(x))ln(ex)=ln(2.41421…)
로그 규칙 적용: ln(ea)=aln(ex)=xx=ln(2.41421…)
x=ln(2.41421…)
x=0,x=ln(2.41421…)
x=0,x=ln(2.41421…)