3. Consider the problem of maximizing the function
S
4
Its graph is shown in Figure 1. A visual inspection seems to indicate that the maximum occurs somewhere just below = 1.
Figure 1: The graph of f(x) = sin (x) -
This maximization problem cannot be solved exactly since the equation f'(x) = 0 cannot be solved exactly in closed form, but we can attempt to approximate the maximum (along with error bounds) by applying Taylor polynomials.
(a) Find the 2nd-degree Taylor polynomial approximation p2 of f centered at x = 1.
(b) Use a computer to graph f(c) and p2(x) on the same set of axes to visualize how closely they match each other near x = 1. (An online graphing calculator like Desmos might be useful.) (c) While it is impossible to exactly maximize f, it is easy to exactly maximize p2 (there is a closed form formula for where the vertex of a parabola is located). Show that p2(c) is maximized when x = places).
(d) We don't know the exact maximum of f for comparison, but we would like to have some idea of how close our previous estimate was to the maximum of f. Apply Taylor's Tr2
bound to four decimal places.
