5. Cloudy with a Chance of Calculus In the town of Swallow Falls, Flint Lockwood has made a startling breakthrough. His new invention, the FLDSMDFR, is producing an almost endless supply of food. The rate of food production (megatons of food/month) can be modeled by the following differential equation: F' (t) = 0.5t +2t+5 e a. (6 pts) Flint is trying to figure out the total amount of food being produced over the next two months (from t = 0 to t = 2 months) to see if the town will be buried. Using a Riemann sum, approximate this value. Use a step size of At 0.5. On the graph of F'(t) below, show what you've calculated with this sum. 5- F'(t) 01 t b. (4 pts) Find the exact amount of food being produced over the next two months (t = 0 to t = 2). 80 6. Bacteriophage lambda (a virus), regulates its decision to initiate lysis in an infected cell or remain dormant (lysogeny) using a single transcription factor, Cl. As long as enough CI is present in the infected cell, lysis is repressed. But if CI levels drop below a critical level, lysis is initiated. This system can be modeled by the following differential equation, which includes both CI production and degradation: CI' = [input] - - r * CI Both production and degradation (r = 0.5) are represented in the graph below: Cl production/degradation 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 L 0 0.5 1 1.5 2 intracellular Cl 2.5 3 3.5 a. (3 pts) Draw a single phase portrait for r = 0.5 that communicates the stability of all equilibrium points. 9 b. (6 pts) Draw a bifurcation diagram using the following values of r: 0.3, 0.5, 0.8 c. (2 pts) What kind of bifurcation is represented above? lasso saddle-node pitchfork transcritical 10