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Newton's Method for Roots worksheets

Approximate positive square through sixth roots with the specialized Newton update.

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Use f(x)=x^n-N and Newton's method to approximate the positive indexed radical.

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Learning objective

What this activity builds

Model an indexed radical as a zero of x^n-N and compute successive upper approximations.

Worked example

What function approximates the positive nth root of N?

f(x)=x^n-N.

Its positive zero is the indexed radical, and Newton's update simplifies to ((n-1)x+N/x^(n-1))/n.

Teaching notes

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  • Keep each iterate positive.
  • Carry unrounded values between rows.
  • Connect the specialized update to the common Newton formula.

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Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.1. index 6Approximate693;f(x)=x6-93Start x0 = 2.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:2. index 3Approximate3498;f(x)=x3-498Start x0 = 8.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:3. index 5Approximate5303;f(x)=x5-303Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 1 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.4. index 4Approximate4465;f(x)=x4-465Start x0 = 5.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:5. index 2Approximate381;f(x)=x2-381Start x0 = 20.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:6. index 6Approximate6285;f(x)=x6-285Start x0 = 3.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 2 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.7. index 5Approximate5435;f(x)=x5-435Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:8. index 3Approximate360;f(x)=x3-60Start x0 = 4.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:9. index 4Approximate4431;f(x)=x4-431Start x0 = 5.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 3 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.10. index 2Approximate185;f(x)=x2-185Start x0 = 14.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:11. index 5Approximate5340;f(x)=x5-340Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:12. index 6Approximate671;f(x)=x6-71Start x0 = 2.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 4 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.13. index 2Approximate412;f(x)=x2-412Start x0 = 20.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:14. index 3Approximate322;f(x)=x3-22Start x0 = 3.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:15. index 4Approximate4420;f(x)=x4-420Start x0 = 5.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 5 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.16. index 3Approximate3455;f(x)=x3-455Start x0 = 8.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:17. index 5Approximate5308;f(x)=x5-308Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:18. index 6Approximate6205;f(x)=x6-205Start x0 = 2.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 6 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.19. index 2Approximate438;f(x)=x2-438Start x0 = 21.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:20. index 4Approximate4136;f(x)=x4-136Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:21. index 5Approximate5269;f(x)=x5-269Start x0 = 3.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 7 of 8
Math SheetNameScoreTeacherDateNewton's Method for RootsUse the stated positive starting value and Newton update to approximate each indexed radical. Carry full precision between iterations.22. index 4Approximate4382;f(x)=x4-382Start x0 = 4.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:23. index 2Approximate301;f(x)=x2-301Start x0 = 17.500000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:24. index 3Approximate325;f(x)=x3-25Start x0 = 3.000000. Complete 2 Newton iterations using full precision.Derivative f'(x):ix_if(x_i)f'(x_i)x_(i+1)01Nth iterate:math-sheet.comPage 8 of 8

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