1. Find the derivative of the function f(x) = (x + 1)5. 2. Find the derivative of the function f(x) = x sin(x) 11. (1.5 points) Consider the equation xy + 10 = 5y. (a) (0.5 point) Use Implicit Differentiation to find dy: dx (b) (0.5 point) Evaluate dy at the point (3,5). dx (c) (0.5 point) Find the equation of the tangent line at (3,5). 12. (1.5 points) A manufacturing company wants to construct a rectangular box with a square base and an open top as shown below. Suppose the box needs to have a volume of 500 cm. Follow the steps below to find the dimensions of the box that will minimize the surface area (and therefore minimize the manufacturing costs). y X X (a) (0.2 point) Write an equation for the surface area of the box in terms of x and y. (b) (0.2 point) Find a constraint equation which relates x and y. (c) (0.2 point) Use the constraint equation to write the surface area in terms of a single variable. (d) (0.4 point) Find the critical numbers of the surface area equation. (e) (0.5 point) Find the dimensions that will minimize the surface area. 3. Find all critical numbers of the function y = f(x) = (2x + 1)6(x 1)5. 4. Find the Absolute Max and Min of the function f(x) = 2x + 3x - 12x on the interval [1,3]. 5. Verify that the conditions of the Mean Value Theorem hold. Next find the number c which satisfies the conclusion of the Mean Value Theorem for the function f(x) = x on the interval [1,2].