436-536W06HW1 - starting with an “integer” guess 7 Use...

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MATH 436/536 - HW 1 Due Wed January 25 th . 1) Find a formula for the nth derivatives for y = sin( π x). 2) Give the 7 th order Taylor polynomial ) ( 7 x P approximating the function f(x) = logwith center at x = π /3. Also, graphically find the largest interval about x = π /3 in which the absolute error in this approximation is at most .001. 3) Graphically, find the smallest order K so that the Kth Taylor polynomial ) ( x P K will approximate the function f(x) = arctan(x) with center at x = 0 in which the absolute error in this approximation is at most 11 10 - . 4) Find from the Lagrange (derivative) error formula the smallest order K so that the Kth Taylor polynomial ) ( x P K will approximate the function f(x) = cos(x) with center at x = π /4 on the entire interval [0, π /2] in which the absolute error in this approximation is at most 16 10 - . 5) Use Newton's method to find to 25 decimal places starting with an integer guess. 6) Use Newton's method to find arcsin(.419) and arcsin(1.419) to 25 decimal places
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Unformatted text preview: starting with an “integer” guess. 7) Use Newton's method to find ) 1835 ( log 6 to 25 decimal places starting with an integer guess. 8) Graph the butterfly parametric curve given by x = r(t) cos t, y = r(t) sin t for 0 < t < 2 π where r(t) = e- 2 cos(4 t). You can make this much fancier by adding to r(t) the term sinand graphing to 24 π which is desired. 9) Graph the butterfly parametric spacecurve given by x = r(t) cos t, y = r(t) sin t and z(t) = r(t) for 0 < t < 2 π where r(t) = e- 2 cos(4 t). You can make this much fancier by adding to r(t) the terms ) 24 ( cos 2 1 ) 12 ( sin 3 5 t t-and graphing to 48 π which is desired. 10) Graph the butterfly parametric surface given by x(t, s) = (r(t) cos t)*s , y(t, s) = (r(t) sin t)*s and z(t, s) = r(t)*(1 + ) 24 ( cos 24 t )*s for 0 < t < 48 π and 0 < s < 1 where r(t) is as in problem 9....
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