(a) Let f(x) = e ², x > 0. Show that, for every n ≥ 1, the n'th derivative f(n)(x) is of the form Pn(1/x) e for some polynomial P₁ (depending on n). · (b) Define g(x) = = 0 e if x ≤ 0 if x > 0. Use part (a) to prove that g(n) (0) = 0 for all n ≥ 1. [Hint: You may want to use the fact that lim F(1/h) = _lim F(t), for any function F.] Conclude that function g of part (b) is not equal to the sum of its Maclaurin series. h→0+ (c)

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