Главная Учебники - Разные Лекции (разные) - часть 26
Задача 4
. Вычислить приближенно с помощью дифференциала. 4.1. x0
= 8 ∆x= 7,76-8= -0,24 f(x0
)= 3
√8=2 f'= 1/(33
√х2
) f'(x0
)= 1/(3*4)=1/12 f(x)= 2-0,24/12= 1,98 4.2. x0
= 1 ∆x= 1,012-1= 0,012 f(x0
)= 3
√(1+7)= 2 f'= (3х2
+7)/(33
√(х3
+7х)) f'(x0
)= (3+7)/(3*2)= 5/3 f(x)= 2+0,06/3= 2,02 4.3. x0
= 1 ∆x= 0,98-1= -0,02 f(x0
)= 1/2*(1+√4)=3/2 f'= 1/2-х/(2√(5-х2
)) f'(x0
)= 1/2-1/4=1/4 f(x)= 3/2-0,02/4= 1,495 4.4. x0
= 27 ∆x= 27,54-27= 0,54 f(x0
)= 3 f'= 1/(33
√х2
) f'(x0
)= 1/27 f(x)= 3+0,54/27= 3,02 4.5. x0
= 0 ∆x= 0,08 f(x0
)= arcsin0=0 f'= 1/√(1-х2
) f'(x0
)= 1 f(x)= 0+0,08=0,08 4.6. x0
= 1 ∆x= 0,97-1= -0,03 f(x0
)= 3
√8=2 f'= (2х+2)/33
√(х2
+2х+5) f'(x0
)= 4/(3*2)=2/3 f(x)= 2-0,06/3= 1,98 4.7. x0
= 27 ∆x= 26,46-27= -0,54 f(x0
)= 3
√27=3 f'= 1/(33
√х2
) f'(x0
)= 1/27 f(x)= 3-0,54/27= 2,98 4.8. x0
= 2 ∆x= 1,97-2= -0,03 f(x0
)= √(4+2+3) f'= (2х+1)/2√(х2
+х+3) f'(x0
)= 5/(2*3)= 5/6 f(x)= 3-0,15/6= 2,975 4.9. x0
= 1 ∆x= 1,021-1= 0,021 f(x0
)= 1 f'= 11х10
f'(x0
)= 11 f(x)= 1+11*0,021= 1,231 4.10. x0
= 1 ∆x= 1,21-1= 0,21 f(x0
)= 1 f'= 1/(33
√х2
) f'(x0
)= 1/3 f(x)= 1+0,21/3= 1,07 4.11. x0
= 1 ∆x= 0,998-1= -0,002 f(x0
)= 1 f'= 21х20
f'(x0
)= 21 f(x)= 1-0,002/21= 0,9999 4.12. x0
= 1 ∆x= 1,03-1= 0,03 f(x0
)= 1 f'= 2/(33
√х) f'(x0
)= 2/3 f(x)= 1+0,03*2/3= 1,02 4.13. x0
= 2 ∆x= 2,01-2= 0,01 f(x0
)= 26
=64 f'= 6х5
f'(x0
)= 6*25
=192 f(x)= 64+0,01*192= 65,92 4.14. x0
= 8 ∆x= 8,24-8= 0,24 f(x0
)= 3
√8= 2 f'= 1/(33
√х2
) f'(x0
)= 1/12 f(x)= 2+0,24/12= 2,02 4.15. x0
= 2 ∆x= 1,996-2= -0,004 f(x0
)= 27
=128 f'= 7х6
f'(x0
)= 7*26
= 448 f(x)= 128-0,004*448= 126,208 4.16. x0
= 8 ∆x= 7,64-8= -0,36 f(x0
)= 3
√8= 2 f'= 1/(33
√х2
) f'(x0
)= 1/12 f(x)= 2-0,36/12= 1,97 4.17. x0
= 2,5 ∆x= 2,56-2,5= 0,06 f(x0
)= √(10-1)= 3 f'= 1/√(4х-1) f'(x0
)= 1/√9= 1/3 f(x)= 3+0,06/3= 3,02 4.18. x0
= 1 ∆x= 1,016-1= 0,016 f(x0
)= 1/√(2+1+1)= 1/2 f'= -(4х+1)/2√(2х2
+х+1)3
f'(x0
)= (-4-1)/2√(2+1+1)3
= -5/16 f(x)= 0,5-0,08/16= 0,495 4.19. x0
= 8 ∆x= 8,36-8= 0,36 f(x0
)= 3
√8= 2 f'= 1/(33
√х2
) f'(x0
)= 1/12 f(x)= 2+0,36/12= 2,03 4.20. x0
= 4 ∆x= 4,16-4= 0,16 f(x0
)= 1/2 f'= -1/(2√х3
) f'(x0
)= -1/16 f(x)= 0,5-0,16/16= 0,499 4.21. x0
= 2 ∆x= 2,002-2= 0,002 f(x0
)= 27
=128 f'= 7х6
f'(x0
)= 7*26
= 448 f(x)= 128+0,002*448= 128,896 4.22. x0
= 1 ∆x= 1,78-1= 0,78 f(x0
)= √(4-3)= 1 f'= 2/√(4х-3) f'(x0
)= 2 f(x)= 1+0,78*2= 2,56 4.23. x0
= 1 ∆x= 0,98-1= -0,02 f(x0
)= 1 f'= 3/(2√х) f'(x0
)= 3/2 f(x)= 1-3*0,02/2= 0,97 4.24. x0
= 3 ∆x= 2,997-3= -0,003 f(x0
)= 243 f'= 5х4
f'(x0
)= 5*81= 405 f(x)= 243-405*0,003= 241,785 4.25. x0
= 1 ∆x= 1,03-1= 0,03 f(x0
)= 1 f'= 2/(55
√х3
) f'(x0
)= 2/5 f(x)= 1+2*0,03/5= 1,012 4.26. x0
= 4 ∆x= 3,998-4= -0,002 f(x0
)= 256 f'= 4х3
f'(x0
)= 4*64= 256 f(x)= 256-256*0,002= 255,488 4.27. x0
= 0 ∆x= 0,01-0=0,01 f(x0
)= √(1+0+sin0)=1 f'= (1+cosx)/(2√(1+х+sinx)) f'(x0
)= (1+1)/2= 1 f(x)= 1+0,01= 1,01 4.28. x0
= 0 ∆x= 0,01-0= 0,01 f(x0
)= 1 f'= (3-sinx)/(33
√3x+cosx) f'(x0
)= 3/3= 1 f(x)= 1+0,01= 1,01 4.29. x0
= 1 ∆x= 1,02-1= 0,02 f(x0
)= 4
√(2-1)= 1 f'= (2-π/2*cos(πx/2))/(44
√(2x-sin(πx/2))) f'(x0
)= 2/4√1= 0.5 f(x)= 1+0,02*05= 1,01 4.30. x0
= 2 ∆x= 1,97-2= -0,03 f(x0
)= √9=3 f'= х/√(х2
+5) f'(x0
)= 2/3 f(x)= 3-0,03*2/3= 2,98 4.31. x0
= 1,5 ∆x= 1,58-1,5= 0,08 f(x0
)= 1/2 f'= -1/(√(2х+1)) f'(x0
)= -1/2
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