Problem 3 (Like Class Text 3.8). (a) Evaluate the polynomial function f (x) = x3

Question

Problem 3 (Like Class Text 3.8). (a) Evaluate the polynomial function f (x) = x3 - 5x2 + 6x + 0.55 at x = 2.727 using 4-digit arithmetic with chopping. Note that 3/7 = 0.42857143 rounded to eight significant digits, but is 0.4285 with 4-digit chopping. Also, use only the , , +, and keys and chop after each operation. This means that x3 involves chopping after x*x and again after multiplying that result by x. (b) Repeat part (a) with an equivalent expression f (x) = [(x ? 5)x + 6]x + 0.55 and compare relative error to part (a). You may assume that your calculator result for either function form rounded to 6 significant digits is the true result. Problem 4 (Like Class Text 3.8). Determine the number of terms of the Mclaurin series for cos x needed to approximate cos a to eight significant digit accuracy using approximate relative error (the true value at 2x is exact at the first term, but that is trivial, so do not stop there). The series is the following:

Explanation / Answer

Ans 4:

4.49E-12

We can see that 15 terms of the Mclaurian are needed to approximate to the eight digit for the particular value (2*pi()) provided
Obviously, the computations are too long to be performed by hand. The following piece of VBA code did it. We have kept it as simple in features as possible to distinguish jargon from proceedure. Using this, the calculation of the remainder in excel is straightforward.

Function mclaurian(ByRef term As Double, angle As Double) As Double
Dim value As Double, remainder As Double, i As Integer
value = 1
For i = 2 To term
value = value - (-1) ^ i * angle ^ (2 * (i - 1)) / WorksheetFunction.Fact(2 * (i - 1))
Next i
mclaurian = value
End Function

A. x 2.727 x*x 7.436529 chopped 7.4365 x*x*x 20.27934 chopped 20.2793 5*x*x (since5*x is 3-digit) 37.18265 chopped 37.8126 6*x 16.362 f(x) by chopping -0.6213 f(x) 0.00877 Rel error -71.8438 B. x-5 -2.273 (x-5)x -6.19847 chopped -6.1984 (x-5)x+6 -0.1984 [(x-5)x+6]*x -0.54104 chopped -0.541 f(X) IN B 0.009 REL ERROR = [f(x) in B -f(x)]/f(x) 0.026226 Rel error in B<< Rel errror in A

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