1. Algebraically find all ER that satisfy the following equations. 46.2+8 (a) 2. 45 26 (b) In(x+2) In(x-4) = 2. T 2. Consider the function : R R defined by f(x) = 2 cos(x + +2 sin(x). (a) Show for all x R that f(x) = 2sin(x + ++) (b) Find all [0,4x] that satisfy (z) = 0. (c) Sketch the curve y = f(x) over x [0,47]. Include in your sketch the coordinates of all of the x- and y-intercepts, and the coordinates of the endpoint of the curve. 3. Calculate the first derivatives of each of the following functions. Show all of your calculations. (a) f(x)=9r (6x+8). (b) g(x)=4eIn(2+1). (c) h(x) = cos(2 sin(x)). (d) q(x) = tan(x) 4. Once upon a time, in a busy factory, there was a team working on packaging a product called "Perfect Cylinders." These cylinders had a diameter d and height h. The challenge for the team was to find the best way to package these cylinders. They wanted to use the least amount of material while keeping the cylinders safe. After thinking on the problem, they came up with an idea: using spheres as packaging containers. To achieve this, they needed to find the smallest possible sphere that could fully enclose each cylinder. Using the fact that the volume of the circular cylinders is 47, determine the radius R of the smallest sphere that contains the cylinder, by first finding an expression for R in terms of h. Verify that the radius R is minimal using the Second Derivative Test. R h d/2