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What is limx->0 ((e^2x)-1)/(tanx)

Let f be the function defined by f(x) = x + ln x. What is the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4]?

For what value of k does the graph of f(x) = 2x^2+k/x has a point of inflection at x = −1.

A rectangle is drawn in the first quadrant so that it has two adjacent sides on the coordinate axes and one vertex on the curve y = − ln x. Find the x-coordinate of the vertex for which the area of the rectangle is a maximum.

Let y = f(x) be the solution to the differential equation dy/dx=y-x The point (5,1) is on the graph of the solution to this differential equation. What is the approximation of f(6) if Euler’s Method is used given ∆x = 0.5?

Why does the TC curve have the same slope as the VC curve?