(20 points). Let f(x)=2xcos(2x)-(x-2)^(2) and x_(0)=0.
(a) Find the Taylor polynomial P_(3)(x) and use it to approximate f(0.4).
(b) Use the error formula in Taylor's Theorem to find an upper bound for the error
|f(0.4)-P_(3)(0.4)|. Compute the actual error.
(10 points). A function f:[a,b]->R is said to satisfy a Lipschitz condition with
Lipschitz constant L on a,b if, for every x,yin[a,b], we have |f(x)-f(y)|<=L|x-y|.
(a) Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an
interval a,b, then finC([a,b]).
(b) Show that if f^(')(x) exists and is bounded on a,b by L, then f satisfies a
Lipschitz condition with Lipschitz constant L on a,b.
(5 points). Find the rate of convergence of the sequence \alpha _(n)=sin((1)/(n^(2))), as n->\infty sin(x)<=x, for 0<=x<=(\pi )/(2) 10^(-2) to the solution of x^(3)+x-4=0 in the interval
1,2 using the Bisection method, then find an approximation of the root with given
accuracy.
