MATH 1A Final: MATH 1A Harvard Fall1999 Final
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Instructors: yum-tong siu (course head), peter clark, kim froyshov. |x|(x + 1) (x 2)(x + 3) (b) lim x . |x|(x + 1) (x 2)(x + 3) 1 + x ex x2 x 0+ hxsin xi . lim. 2: consider the function f (x) = for x < . |x| cos x for x . 2 x for x > . 3 (a) identify all points x where f (x) is not continuous. Justify your answer. (b) identify all points x where f (x) is not di erentiable. 3: calculate the following derivatives. (a) (b) d dx hx2 + cos x + ln xi . d dx. 3(cid:21) . (c) (d) d dx z 7 x2 sin(cid:16)t3(cid:17) dt. (hint: use the fundamental theorem of calculus. ) Justify your answer. (b) calcuate the derivative of f 1 (as a function of y) at y = 1. (note that y = 1 when x = 0. )
