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Hints 203 Show that the roots of the cubic equation 64t3 - 192t2 - 60t - 1 = 0 are cos3(2r/7) sec(6?r/7), cos3(4a/7) sec(2?r/7 ...
204 6. Symmetric Functions of the Zeros 1.11. (a) Express p and r in terms of the roots to get two equations for (a + b) and (c ...
Approximations and Inequalities 7.1 Interpolation and Extrapolation A scientist wishes to know the index of refraction of pure w ...
206 7. Approximations and Inequalities The coefficients a, b are to be determined from the conditions that ~(50) = 1.32894 and r ...
7.1. Interpolation and Extrapolation 207 Then verify that p(t) = CzO bipi(t) is the desired polynomial. This is called the Lagra ...
208 7. Approximations and Inequalities Let f(n) be a function defined on the positive integers. As in Exercises 7 and 8, constr ...
7.1. Interpolation and Extrapolation 209 (f) Manipulating formally, we have Ek = (I+A)k. Expand the right side by the binomial t ...
210 7. Approximations and Inequalities Investigate finding an approximate value of fi from a table of powers of 2 with integer ...
7.1. Interpolation and Extrapolation 211 where, for any nonnegative integer m, and any u, U ( > = U(U-l)(u-2)...(U-7n+l) u(m) ...
212 7. Approximations and Inequalities The polynomial obtained should be the same as the Lagrange polynomial of degree n determi ...
7.2. Approximation on an Interval 213 (k=O,1,2,... , n). Verify the table 1 1 2 t2 3 (3t2 + 1)/4 4 (-4t4 + 7P)/3 5 (-125t4 + 290 ...
(^214) 7. Approximations and Inequalities two functions can be defined in many ways, and we will content ourselves here with a v ...
7.2. Approximation on an Interval^215 measure the degree of closeness between f(t) and p(t). This measures how far apart the val ...
216 7. Approximations and Inequalities --y=at+b Suppose that f(t) is a given function and that p(t) is a polynomial of degree n ...
7.2. Approximation on an Interval 217 Argue that the graph of q must cross that of p in at least n + 1 places. (Use a sketch.) D ...
218 7. Approximations and Inequalities Comment: It turns out that when the function f(t) is continuous in t. Let us get some ins ...
7.2. Approximation on an Interval 219 (a) Show that the mapping f -+ B(f, n; t) is linear: i.e. B(f+g,n;t) = B(f,n;t)+B(g,n;t) ...
220 7. Approximations and Inequalities (e) the Bernstein polynomial of degree 2 (modified to the interval) (f) the Taylor approx ...
7.3. Inequalities 221 (a) Verify the identity x6 + y6+z6+u6+~6+w6-6xyzuvw = ~(x2+y2+,a)[(y2-z2)2+(%2-x2)2+(x2-y2)2] + i(u2 + v ...
222 7. Approximations and Inequalities (b) Let u = w,-r/(w,-1 + wn), v = w,/(w,-1 - w,) and a = uu,,-1 + vu,. Verify that wlul + ...
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