Problem 1. Use the formula for the derivative of ln(x) using the method of inverse functions to compute the limit as h approaches 0 of (1/h) ln(1 + h). Next consider the quantity (1 + h)^1/h. Find a continuous function that transforms the limit of (1/h) ln(1 + h) into (1 + h)^1/h Use this to evaluate the limit as h approaches 0 of (1 + h)^1/h. Finally, let h be 1/n for a positive integer n, and expand (1 + (1/n))ⁿ using the Binomial Theorem. For a given positive integer m, as n tends to , what is the limit of the mth summand of the Binomial Theorem expansion of (1 + (1/n))ⁿ ? This gives a series of fractions that converges to e.

Problem 2. A wire is bent into the shape of a parabola y = x² . A bead slides along the wire, falling along the parabola in the first quadrant towards the origin. At the moment when the bead is at coordinates (a, a² ), the distance from the bead to the origin is decreasing at a rate of b, i.e., increasing at a rate of b. Compute the rates of change of both the x-coordinate and y-coordinate at that moment. Double-check your answer by comparing the ratio of these rates of change to the slope of the tangent line to the parabola at (a, a²).