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[12 pts] Evaluate the limits. Hint: write them as definite integrals and then evaluate the integrals. 2 (a) lim i=1 (b) lim - 3 n 2+ 32 n-too n L n 1 + 32 n (c) n 1 n-+00 i= 1 Vn2 + 12 [12 pts] Let f(x) = 24 - x3 +1. (a) Show that the graph of the function f lies above the x-axis. (b) Consider the region bounded by the graph of f, the x-axis, the line x = a, and the line x = a + 1. What is the value of a for which the area of this region reaches its minimum? What is the value of this minimum? [12 pts] (a) Let f be a continuous function such that (arctan x)? = f(t) dt. Find f(1). (b) Evaluate the limit lim Jo sin(t2) dt + evidt (c) Let f(x) = esintat. Find the second degree Taylor polynomial for f centered at zero. Use it to approximate the integral / esint at