(1)Suppose that f : R R is a continuous function so that limx f(x) = 1 and lim x f(x) = 1. Prove that for all M (1, 1), there is a solution to f(x) = M. Is it necessarily true that there exists x R so that f(x) = 1? Explain.

(2)Let f(x) = 1/ x^n , where n 1 is a positive integer. Use the definition of derivative to find f' (x).

(3)Find and prove an explicit formula for the n^th derivative of f(x) = 1 /2x + 7 .

(6) Let f(x) = cos(x), where x is measured in degrees, not radians. Find f' (x).

(7)Consider the curve given by cos(x + y) + xy^2 = e^( xy ). Find the equation of the tangent line to this curve at the point (1, 0).