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- Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.Supply: p = 1/4q2 + 12 Demand: p = 68 − 5q − 2q2 (a) Sketch the first-quadrant portions of those functions on the same set of axes. Label the market equilibrium point E.Exercise IV: Partial derivatives II 1. If you deposit an amount S at the bank at a continuous yearly interest rate r, then the amount in your account after t years will be A(S, t) = Se". JA Find and as and interpret these quantities. 2. A company's production output P is given in tons, and is a function of the number of workers N and the value of the equipment V, in units of $25, 000. The production function for the company is P(N,V) = 5N³/4y/4. The company currently employs 80 workers, and has equipment worth $750, 000. (a) What are N and V in this case? (b) Find Py and Pv, then the values of P, PN, and Py at the values of N and V that you found in (a). (c) Give units and explain what each answer means in this context.
- Consider a theoretical model for mutual inhibition in ecological systems where n > 0 measures the strength of mutual inhibition d dt x(t) = (1/2)" (1/2)" +yn - x d dt (t) = (1/2)" - (1/2)" +x" -y Plot the nullclines and determine the equilibria points for n = 1 and n = 3. Interpret these equilibrium in the ecological setting. Explicitly calculate the local stability of the equilibrium point where x* = y* for gen- eral n. When does the system undergo a qualitative change in behaviour?6. State whether the equation is ordinary or partial, linear or nonlinear, and give its order. * (x + y) dx + (3x² – 1) dy = 0 ordinary, linear in x, order 1 ordinary, linear in y, order 1