The function is differentiable everywhere except possibly at
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The function g is differentiable everywhere except possibly at x=5. Table 3 lists symmetric sample values of g around x=5 for three different distances h.
Table 3: Symmetric samples around x=5
h 0.40 0.20 0.10
g(5−h) 7.64 7.36 7.22
g(5+h) 6.36 6.64 6.78
Assume g has no singularities on the interval [5−h,5+h] for the h values shown.
(a) For each value of h in Table 3, compute an estimate E(h) of x→5limg(x) by averaging g(5−h) and g(5+h).
(b) Examine the sequence of estimates E(0.40), E(0.20), and E(0.10). Based on these values, produce your best numerical estimate of x→5limg(x) and justify your choice.
(c) Using the two finest-step estimates, provide an empirical bound for the absolute error of your estimate from part (b). State clearly how you arrived at this bound.