Use extrapolation to derive an O(h) formula for f'(xo). Suppose that N (h) is an approximation to M for every h > 0 and that M = N(h) + K,h+ K2h? + K,h +. , for some constants K, K2, K3, .. . Use the values N(h), N (4), and N () to produce an O(h approximation to M.

Suppose that f(2)=1 and f′(x)≤5 for all values of x. Determine a lower bound for f(−2). Determine an upper bound for f(5).

Find an approximation that is accurate to three decimal places to the zero of f (x) = x³ -3x- 1 in the interval [-1,0]

Suppose that during the first few hours of a laboratory experiment, the temperature of a certain substance is closely approximated by the function f(t) = -2²+ St, where t is measured in hours, with t = 0 corresponding to the instant the experiment begins, and f(t) is the temperature (°F) of the substance after t hours. (a) Find an expression for Af/At using the interval [0, t]. What are the units for Af/At? O °F2/hr² O eF2/min O °C/min @F/hr O °C/hr (b) Use the result in part (a) to complete the following table. Round the answers to four decimal places. [0, 0.1] [0, 0.001] [0, 0.0001] Interval Af/At [0, 0.01] (c) In part (b), as the right-hand endpoint gets closer and closer to 0, what value does Af/At seem to be approaching? This target value tells us the rate at which the temperature is changing at the instant the experiment begins. °F/hr