polynomial approximation. Calculate the error of each approximation. 5 Exercise. Find the fourth-order Taylor polynomial approximation for the function, g(x) = 2x cos(x), centered at x = T. (Use the value of n as stored by your calculator.) Evaluate g(3) and g(3.14) using both the explicit formula for the function and the polynomial approximation. Calculate the error of each approximation. 6 Exercise. Derive the closed formula for the Maclaurin series representation of the function,9 ( x ) = 2 x cos ( x ) 9 ( 0 ) = 0 9' ( x ) = 2 cos ( x ) - 2 x sin (x ) 9 ( 0 ) = 2 9" ( x ) = 2 ( - sinx ) - ( 2sinx + 2 x cosx ) " ( 0 ) = 0 = - 4sinx-2xCoSX 9" ( x ) = - 4cosx - (2cosx - 2xsinx ) 9 111 ( 0 ) = - 6 = - 6cosx + 2x Sinx 9 4) ( x ) = 6 sinx + (2sinx + 2xcosx) 9 ( 4 ) ( 0 ) = 0 = 8 sinx + 2 xcosx h = 0 f ( m ) ( c ) ( x - c ) " = 9 ( 0 ) + 9 ' ( o ) ( x - AT ) + 8 " ( 0 ) AT 2 1 ( x - T ) 2 9" (0 ) TT ( X - IT ) 3 + 9 LA ) ( 0 ) TT ( x - TT ) + = 0+2 (x - TT ) + 0 + ( =LT 4 Le ( X - TT ) 3 ) + 0 = 2 ( X-1 ) + (- AT (X - 1 ) 3 ) = - 1X 3+ 31 2x2 - 31 x + 2x + 14- 2AT 9 ( 3 ) = 6 cos ( 3 ) 2 - 5.94 9 ( 3 . 14 ) = 6. 28 cos ( 3. 14 ) = - 6. 28\fSo, R,, (x) = f(x) - P,,(x). The absolute value of R,,(x) is called the error associated with the approximation. That is, Error = [R, (x) | = If(x) - Pn(x)1. The next theorem gives a general procedure for estimating the remainder associated with a Taylor polynomial. This important theorem is called Taylor's Theorem, and the remainder given in the theorem is called the Lagrange form of the remainder. THEOREM 9.19 Taylor's Theorem If a function f is differentiable through order n + 1 in an interval / containing c, then, for each x in 1, there exists z between x and c such that 2 ! f ( x ) = f ( c ) + f' ( c ) ( x - c ) + 10 ( x - c)2 +... + 1( 0) ( x - c )" + Rx(x) n! where R. (x) = f(n+1) (z) (n + 1)! (x - c)n+1 O A proof of this theorem is given in Appendix A. One useful consequence of Taylor's Theorem is that R , ( x ) | 5 x - can+1 (n + 1)! max f(n + 1) ( z)| where max f("+ 1)(z)| is the maximum value of f(n+1)(z) between x and c. For n = 0, Taylor's Theorem states that if f is differentiable in an interval I containing c, then, for each x in 1, there exists z between x and c such that f (x) = f (c) + f' (z ) (x - c) or f' ( z ) = (x ) - f ( c ) * - C Do you recognize this special case of Taylor's Theorem? (It is the Mean Value Theorem.) When applying Taylor's Theorem, you should not expect to be able to find the exact value of z. (If you could do this, an approximation would not be necessary.) Rather, you are trying to find bounds for f("+ 1)(z) from which you are able to tell how large the remainder R,,(x) is