1. A ship leaves port and travels south at 30 mi/hr. At the same time another boat which is 60 miles due east of the port, approaching the port to dock, and traveling at a rate of 40 mi/hr. After half an hour, how fast is the distance between the boats changing? Are they moving closer together or further apart? 2. As the wheel of radius r cm in the figure rotates, the rod of length L attached to point P drives a piston back and forth in a straight line. Let x be the distance from the origin to point at the end of the rod as shown. (a) Use the Pythagorean Theorem to show that L2 = (xr cos 0) + r sin 0. == (b) Differentiate the equation in part (a) with respect to t to show that 0 = 2(x-r cos 0) (d+rsin0) + 2r2 sin cos de (c) Calculate the speed of the piston when 0=, assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute. Piston moves back and forth L x Optimization Lab Preparation b . Show 2a' 1. Recall that a quadratic function is one that can be written in the form f(x) = ax2 + bx + c, where a, b, and c are real-valued constants. Use this form to answer the following questions. (a) In college algebra, students are taught that the vertex is located at x = - using calculus methods that this value is the only possible location for a vertex. (b) In college algebra, students are taught that the value f (-2) is the maximum value attained by a quadratic function whenever a < 0 and that this is the minimum value whenever a > 0. Show using calculus methods that these statements are true. 2. Consider the real-valued function f(x) = x + (a) Establish the domain for f. (b) Identify any critical points. 100 Xx (c) Identify any inflection points and find intervals where the graph is concave up and concave down. (d) Use your results in parts (b) and (c) to establish the location of any local extrema (Second Derivative Test).