> It amazes me that MIT freshmen would have so much > difficulty with this exam.
I too am rather amazed, but not because they're essentially precalculus level, but because the difficulty level doesn't seem to exceed what shows up on average level precalculus tests. I can somewhat believe this now, but if you had shown this to me when I was in high school, I would have thought this was a joke (except the two or three "find the derivative" questions, which could be a problem for someone whose high school didn't offer calculus and who didn't have a college campus within driving distance). I had this view (and still do, but nowhere near as much) that those who got into places like MIT, Pinceton, Harvard, etc. were geniuses who probably could have learned calculus (and Russian, and molecular biology, etc.) in middle school if they had the opportunity. These were the 1500+ SAT people, the math olympiad people, the people who entered projects in the Westinghouse Science Talent Search, etc. ... the kinds of people who showed up at a high school like mine maybe once a decade.
Now for the obligatory nitpicks of the problem statements.
Problem A1: They should say that k is an integer. Otherwise, a correct answer would be (1/2) x 10^(log 2x), where x equals the given numerical expression.
Problem B1: They should say that a and b are real numbers. Otherwise, a correct answer would be a + 0*i, where a equals the given expression(s).
Problem B2: Instead of saying "Find all possible values of x which solve the equation x^4 - 13x^2 + 36 = 0", I would be less vague and explicitly ask for all complex solutions.