Question
Near certain values of ? each of the following functions cannot hbe accurately computed using the formula given due to cancellation error Identify the values of € which are involved (e.g nCar = 0 large positive z) and prO- pose reformulation of the function (e.g using Taylor serics_ rationalization trigonometric identities; etc:) to remedy the problem: This is problem # 12 from the texthook Please also see pages 48-49 for examples and more details_ f()=1+ COS f(r) = IT In(/1) f (s) =1 2 sin? (
Near certain values of ? each of the following functions cannot hbe accurately computed using the formula given due to cancellation error Identify the values of € which are involved (e.g nCar = 0 large positive z) and prO- pose reformulation of the function (e.g using Taylor serics_ rationalization trigonometric identities; etc:) to remedy the problem: This is problem # 12 from the texthook Please also see pages 48-49 for examples and more details_ f()=1+ COS f(r) = IT In(/1) f (s) =1 2 sin? (b) f(z) = e +sing = 1 (d) f() = V2+1-V22+4 (f) f(c) = In(r+ Vze+1)
Answers
Polynomial Approximations Use the polynomial
approximations of the sine and cosine functions in
Exercise 100 to approximate the following function
values. Compare the results with those given by a
calculator. Is the error in the approximation the same
in each case? Explain.
$$
\begin{array}{lll}{\text { (a) } \sin \frac{1}{2}} & {\text { (b) } \sin 1} & {\text { (c) } \sin \frac{\pi}{6}}\end{array}
$$
$$
\begin{array}{lll}{\text { (d) } \cos (-0.5)} & {\text { (e) } \cos 1} & {\text { (f) } \cos \frac{\pi}{4}}\end{array}
$$
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