UMUC MATH 140, FALL 2017 FINAL EXAM KOBI SNITZ (1) The ceiling function dxe rounds a number up to the nearest integer above it. The floor function bxc rounds x down to the nearest integer below it. The nearest integer function bxe rounds a number to the nearest integer and half integers are rounded up. For example b3.5c = 3 d5.5e = 6 b3.5e = 4 Determine if the following limits exists and their values if they do exist. limx1 bxc limx1.5 dxe limx3.5 bxe (2) Use the formal definition of the limit to show that the limits above either exist or not and their values. Recall that, if a limit lim f (x) = l xa exists then for every \u000f there is a such that if |x a| < then |f (x) l| < \u000f. (3) Differentiate the function f (x) = exp(x2 ) 2x + sin(x) (4) let f (x) = cos(x3 ), Find the critical numbers of f on the interval [2, 2] and the local and global minimum and maximum points. Note that the degrees here are in radians. (5) find the limit sin(3x3 ) lim x0 2 sin3 (x) (6) evaluate Z 4 x2 4x dx 1 1