2 + 6x (1 point) Let f(x) Compute f(I) f(1) f'(x) = f' (1) f" (x) = f" (1) = 1. f" (I) = fill (1) = f (iv) (x) = f(iv) (1) = f(0) () = f( ) (1) We see that the first term does not fit a pattern, but we also see that f() (1) = |for k > 1. Hence we see that the Taylor series for f centered at 1 is given by f(x) = 8 +>(x-1)* (1 point) The Taylor series for f(x) = In(sec(x)) at a = 0 is En , Cn T". Find the following coefficients. Co 2 C1 C2 C3 CA (1 point) Use sigma notation to write the Taylor series about x = To for the function. e, = 3. Taylor series = KO Use partial fraction decomposition to find the power series representation of 5 f(x) = (x - 3) (x + 1) 4. The power series representation for f(a) is help ( formulas) Use the identity y y", for lyl help ( formulas ) sin () Find the Maclaurin series of f(x) 2 2n41 Hint: g(x) = sin(x) = > (-1)" (2n + 1)! The Maclaurin series of f(x) is help (formulas) Find the Maclaurin series of f(x) = cos(x). Hint: Power reducing identity cost x = + cos (21) The Maclaurin series of f(x) is 1 + help (formulas) n 1 Find a power series for F(x) = / edt. Hint: f(x) = ex? S n! , integrate term by term. F(x) = > help (formulas)I Find a power series for F(@) = cot (:) dt. 0 1+ +2 OO 1 Hint: f(@) = - ( - 1)" He' integrate term by term. 1+a2 n O F(E) = help (formulas) Find the first three nonzero terms of the Maclaurin series for f(x) = In(cos(x)). help (formulas)