Math Xb Spring 2004Worksheet: Applications of Trig DerivativesMarch 15, 20041. A television camera is positioned 4000 feet from the base of a rocket launching pad. The angle ofelevation of the camer a has to change at the correct rate in order to keep the rocket in sight. Also, themechanism for fo c using the camera has to take into account the increasing distance from the camerato the rising rocket. Assume that the rocket rise s vertically and its speed is 600 feet per second whenit has risen 3000 feet.(a) How fast is the distance from the televis ion camera to the rocket changing at that moment?(b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle ofelevation changing at that same moment?2. Two sides of a triangle are 4 meters and 5 meters in length and the angle between them is increasingat a rate of 0.006 radians per second. Find the rate at which the area of the triangle is increasing whenthe angle between the sides of fixed length is π/3.3. A plane flying with a constant speed of 300 kilometers per hour passes over a ground r adar station atan altitude of 1 kilometer and climbs at an angle of 30◦. At what rate is the distance from the planeto the radar station increasing a minute la ter?4. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point Cdiametrically opposite A on the other side of the lake in the shortest possible time. She can walk atthe rate o f 4 miles per hour and row a boat at 2 miles per hour. How should she proceed?2 2ABCΘ5. Find the area of the largest rectangle that can be inscribed in a semicircle of radius r. (Hint: Writethe area of the recta ngle as a function of θ .)rΘ6. Let f(θ) = sin2θ, 0 ≤ θ ≤ 2π.(a) Find the intervals on which f increases and decreases.(b) Find the loc al maximum and minimum values of f.(c) Find the intervals on which f is concave up and concave down.(d) Find the inflection points of f .(e) Use your answers to parts (a–d) to sketch the graph of f . Use your graphing calculator to checkyour work.7. Follow the instructions in the previous question for f(t) = t + cos t, −2π ≤ t ≤