2. Complete the following tasks to tackle Question 5 by other means. a. Denote by x and y the dimensions of a plot of land measured in meters. Use the information about the rope's length to write an identity involving x and y. We refer to this identity as the constraint for x and y. b. Let A denote the area, measured in square meters, of such a rectangle. Write A first in terms of x and y, and then use the constraint for x and y to express A as a function of x only. Call this new expression A(x). What is the domain of A(x)? c. Use a calculator to graph A(x). What do you see? Use your calculator's zoom and trace features to find the value of x for which A(x) is the largest. How does your answer compare to the answer you obtained in Problem 1? 3. Derivatives at Work Return to the function A(x) from Problem 2. You should have seen that the domain of A(x) is the interval (0,50) and that the graph of A is a parabola. Use calculus to determine the coordinates of the highest point on this parabola, which corresponds to the value of x that gives the largest area. a. What slope does the tangent line have at the highest point on the graph of A? What does this information tell you about the derivative A'(x)? b. Compute A'(x) and graph it. Find the value xo in the interval (0,50) with the property that A'(x) = 0. c. What is the sign of A'(x) to the left of xo? How about to the right of xo? What sort of information do you obtain about A(x) from the sign of A'(x)? What can you say about the value of A(xo)? How does this relate to Question 5 in Classroom Discussion 4