(a) Let r(t) = (√√t² – 1, e¹t, ln(2t — 4)). (i) Find the domain of r(t). (ii) Find lim3 r(t). (iii) Find r'(t).

Find T4(x) Taylor polynomial of degree 5 of the function f(x)=cos(x^3) at a=0

Find the linearization L(x) of the function at a. f(x)= sin(x), a= π\4

15) Annual U.S. imports from a certain country in the years 1996 through 2005 could be approximated by I(t) = t2 + 3.5t + 48    (1 ≤ t ≤ 9) billion dollars, where t represents time in years since 1995. Annual U.S. exports to the country in the same years could be approximated by E(t) = 0.5t2 − 1.4t + 13    (0 ≤ t ≤ 10) billion dollars. Assuming that the trends shown in the above models continue indefinitely, calculate the limits lim t→+∞ I(t)  and  lim t→+∞  I(t)/E(t) algebraically. (If an answer does not exist, enter DNE.) lim t→+∞ I(t)  =         lim t→+∞  I(t) E(t)  =        Interpret your answers. In the long term, U.S. imports from the other country will (select) (be rounded or rise without bound) and be  times U.S. exports to the other country. Could the given models be extrapolated far into the future? Yes or No