CALCULUS BC NAME_________________________________ TEST ON SERIES PERIOD____DATE______________________ Show all work, and circle your answers. 1. The function fhas derivatives of all orders for all real numbers x. Assume 14,12,13, and 15.ffff (a) Write a third-degree Taylor polynomial for (b) The fourth derivative of fsatisfies the inequality 46fxfor all xin the closed interval [-2,0]. Use the Lagrange error bound to determine the amount of error if you were to approximate0.5f(c) If 21gxfx, write the fourth-degree Taylor polynomial for gabout x f = 0. _________________________________________________________________________________________ 2. The Taylor series about x= 3 for a certain function fconverges to fxfor all xin the interval of convergence. The nth derivative of fat x= 3 is given by 1!3and 31.nnnffn(a) Write the first three nonzero terms and the general term of the Taylor series for fabout x= 3.(b) Show that the first four terms of the series found in (a) can be used to approximate 4with an error less than 12000. Justify your answer. 1 5 n f
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