1. A box with a square base and open top must have a volume of 55296 cm3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(x)= ______ Next, find the derivative, A'(x). A'(x)= ____________ Now, calculate when the derivative equals zero, that is, when A'(x)=0. [Hint: multiply both sides by x2.] A'(x)=0 when x=_________ We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x)= ______________ Evaluate A"(x) at the x-value you gave above.____________ NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.) 2. Use the Method of Midpoint Rectangles (do NOT use the integral or antiderivative) to approximate the area under the curve f(x)=x²+2x+6 from x=3 to x=23. Use n=5 rectangles to find your approximation. Answer = _____________