Suppose that a population develops according to the logistic equation dP dt = 0.06P − 0.0006P2, where t is measured in weeks. (a) What is the carrying capacity? M =  What is the value of k? k =  (b) A direction field for this equation is shown in the figure below. On the coordinate plane the horizontal axis is labeled t and the vertical axis is labeled P. There is a slope field in the first quadrant on the graph. The slopes are nearly horizontal at P = 0. As P increases, the slopes go up and right becoming more steep until about P = 50, then go up and right becoming less steep, become horizontal near P = 100, and go down and right becoming more steep until they exit the window near P = 150. Where are the slopes close to 0? (Enter your answers as a comma-separated list.) P =        Where are the slopes largest? (Enter your answers as a comma-separated list.) P =        Which solutions are increasing? (Enter your answer using interval notation.) P0 ∈        Which solutions are decreasing? (Enter your answer using interval notation.) P0 ∈

Kleiber's Law states that the metabolic needs (such as calorie requirements) of a mammal are proportional to its body weight raised to the 0.75 power.¹ Surprisingly, the daily diets of mammals conform to this relation well. Assuming Kleiber's Law holds, answer the following questions. (a) Write a formula for C, daily calorie consumption, as a function of body weight, W. NOTE: Use k for the constant of proportionality. C(W) = (b) If a human weighing 150 pounds needs to consume 1800 calories a day, estimate the daily calorie requirement of a horse weighing 740 lbs and of a rabbit weighing 11 lbs. NOTE: Round your answers to the nearest calorie. Daily calorie requirement of a horse = Daily calorie requirement of a rabbit = calories Choose one calories (c) On a per-pound basis, which animal requires more calories: a mouse or an elephant? ¹S. Strogatz. "Math and the City." The New York Times. May 20, 2009. Kleiber originally

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3. A simple pendulum is a weight on the end of a string or rod, which when given an initial push, swings back and forth under the influence of gravity. The period T of a pendulum is the time taken for two swings (left to right and back again). The formula for the period is T = 27 dT (a) Find d 9 where is the length of the pendulum (in meters) and g is the acceleration due to gravity (in m/sec2). Note that g is a positive constant. lo zulev lie (x)) (8) (b) When the length of the pendulum increases, does the period increase or decrease? Justify your answer. (0) al bar nob

Show that e dt = 317