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(ii) Use Horner's method to evaluate the polynomial T(r) at a = 132/475 in the FPA7. Present the results of your calculations in a standard output table for the method of the form ak an an-1 an-2 ... a1 ao abn bn-1 abn-1 bn-2 ab2 abı bo ... bk bn b1 ... or in the transposed version of a standard output table, that is, in a table of the form ak an bn an-1 abn bn-1 : a1 ab2 b1 ao abi bo if you'll have trouble fitting the standard version in your work. (iii) Finally, find the relative error and the number of significant digits in the approximation of In(25/19), as produced by your calculator/software, by the result of evaluation of T(132/475) in the FPA7, which you have obtained in (ii). (If you have worked with a scientific calculator, then, in order to provide more accurate results, exit (Sci 7) mode beforehand).

(ii) Use Horner's method to evaluate the polynomial T(r) at a = 132/475 in the FPA7. Present the results of your calculations in a standard output table for the method of the form ak an an-1 an-2 ... a1 ao abn bn-1 abn-1 bn-2 ab2 abı bo ... bk bn b1 ... or in the transposed version of a standard output table, that is, in a table of the form ak an bn an-1 abn bn-1 : a1 ab2 b1 ao abi bo if you'll have trouble fitting the standard version in your work. (iii) Finally, find the relative error and the number of significant digits in the approximation of In(25/19), as produced by your calculator/software, by the result of evaluation of T(132/475) in the FPA7, which you have obtained in (ii). (If you have worked with a scientific calculator, then, in order to provide more accurate results, exit (Sci 7) mode beforehand).

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Polynomial And Rational Functions
Section9.4: Graphing Polynomial Functions
Problem 47PS
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Numerical analysis Q1 part i is answered I only need part ii and iii thanks.
a) b) (i) Part 2) If and only if part, 25 = 19 A x = 775 132 We have to show that x + I+x² 132 25 If x then x + VI+ x² = 475 25 132 Part 1) If part, If x + /1+ x² = then x = 5 132 At x= 75 » 25 x + V1+ x? = 1 132 132 19(a + vI+ x²) =19(#) 25 x + /1+ x²)=19( 19 x + /1+æ? 475 + 1+ 475 19x + 19/T+ x²=25 17424 225625 132 = 75 + 1 19/T+x²=25 – 19x 243049 225625 132 191+x?=25 – 19x = 775 + Squaring on both sides ( 493) 361 (1+ x2) =625 + 361x² – 950x 132 = 775 + V (475) 361 + 361x² =625 + 361x² – 950x 132 493 361=625 – 950x 475 † 475 950x=264 625 475 264 x= 950 25x 25 = 25×19 132 x=175 æ + /1+x²= Hence proved that x + /1+ æ² = 132 475 A x = ||
Transcribed Image Text:a) b) (i) Part 2) If and only if part, 25 = 19 A x = 775 132 We have to show that x + I+x² 132 25 If x then x + VI+ x² = 475 25 132 Part 1) If part, If x + /1+ x² = then x = 5 132 At x= 75 » 25 x + V1+ x? = 1 132 132 19(a + vI+ x²) =19(#) 25 x + /1+ x²)=19( 19 x + /1+æ? 475 + 1+ 475 19x + 19/T+ x²=25 17424 225625 132 = 75 + 1 19/T+x²=25 – 19x 243049 225625 132 191+x?=25 – 19x = 775 + Squaring on both sides ( 493) 361 (1+ x2) =625 + 361x² – 950x 132 = 775 + V (475) 361 + 361x² =625 + 361x² – 950x 132 493 361=625 – 950x 475 † 475 950x=264 625 475 264 x= 950 25x 25 = 25×19 132 x=175 æ + /1+x²= Hence proved that x + /1+ æ² = 132 475 A x = ||
1. (Floating-Point Arithmetic, Taylor Polynomials, Horner's Method; hyperbolic arcsine function It is known that the ninth Taylor polynomial T(x) of the f(x) = arcsinh(x)= ln(x + 1+ x²) centered at zero is 5 -x' + 112 35 1 T(x) = x - 3 40 1152 In particular, one can approximate the values of the hyperbolic arcsine about zero by the corresponding values of this Taylor polynomial, In(x + V1+ x²) = T(x), (1.1) provided that |x| is small enough. Based on this fact, realize the following plan of obtaining an approximation of In(25/19). (i) Show first that x+ V1+x2 = 25/19 + I = 132/475. Thus in view of (1.1) we can approximate In(25/19) by T(132/475): In(25/19) T(132/475). (ii) Use Horner's method to evaluate the polynomial T(x) at a = 132/475 in the FPA7. Present the results of your calculations in a standard output table for the method of the form ak an an-1 an-2 ... a1 ao abn bn-1 bn ab2 abı bo b1 abn-1 ... bn-2 ... or in the transposed version of a standard output table, that is, in a table of the form ak be an bn an-1 abn bn-1 ab2 abi bo a1 b1 ao if you'll have trouble fitting the standard version in your work. (iii) Finally, find the relative error and the number of significant digits in the approximation of In(25/19), as produced by your calculator/software, by the result of evaluation of T(132/475) in the FPA7, which you have obtained in (ii). (If you have worked with a scientific calculator, then, in order to provide more accurate results, exit (Sci 7) mode beforehand).
Transcribed Image Text:1. (Floating-Point Arithmetic, Taylor Polynomials, Horner's Method; hyperbolic arcsine function It is known that the ninth Taylor polynomial T(x) of the f(x) = arcsinh(x)= ln(x + 1+ x²) centered at zero is 5 -x' + 112 35 1 T(x) = x - 3 40 1152 In particular, one can approximate the values of the hyperbolic arcsine about zero by the corresponding values of this Taylor polynomial, In(x + V1+ x²) = T(x), (1.1) provided that |x| is small enough. Based on this fact, realize the following plan of obtaining an approximation of In(25/19). (i) Show first that x+ V1+x2 = 25/19 + I = 132/475. Thus in view of (1.1) we can approximate In(25/19) by T(132/475): In(25/19) T(132/475). (ii) Use Horner's method to evaluate the polynomial T(x) at a = 132/475 in the FPA7. Present the results of your calculations in a standard output table for the method of the form ak an an-1 an-2 ... a1 ao abn bn-1 bn ab2 abı bo b1 abn-1 ... bn-2 ... or in the transposed version of a standard output table, that is, in a table of the form ak be an bn an-1 abn bn-1 ab2 abi bo a1 b1 ao if you'll have trouble fitting the standard version in your work. (iii) Finally, find the relative error and the number of significant digits in the approximation of In(25/19), as produced by your calculator/software, by the result of evaluation of T(132/475) in the FPA7, which you have obtained in (ii). (If you have worked with a scientific calculator, then, in order to provide more accurate results, exit (Sci 7) mode beforehand).
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