According to an economic model, the budget b(t) at time t ≥ 0 in a household is chosen to maximise the lifetime utility U [b] = for d dt e-tu(c(t)), where u(c) ≥ 0 is the household utility function, ß> 0 is the discount rate, a constant, and c(t) is the household consumption satisfying the budgetary equation b'(t) = yb(t) + wc(t), b(0) = bo. In this equation, y> 0 is the bank interest rate, w> 0 is the household wage (both assumed constant in this model), and bo > 0 is the initial household budget b(0). Note that b(t) may be negative if the household is in debt, but c(t) > 0. The initial household consumption is c(0) = co > 0. The budget b(t) is subject to a No-Ponzi condition lim et b(t) 20 t-→∞ (which prevents the household financing current consumption through indefinitely borrowing and rolling over debt). Throughout this question you may assume that the usual theory of the calculus of variations is valid for this model on the infinite time interval t = [0, ∞).

4. Monthly sales of a particular computer are expected to decline at a rate of d(t) = −25t 2/3 computers per month, where t is time in months. Currently, monthly sales are 2,000 computers. Determine monthly sales 8 months from now.

The total sales of a company (in millions of dollars) t months from now are given by S(t) = 0.04t³ + 0.5t² + 2t + 5. (A) Find S'(t). (B) Find S(5) and S'(5) (to two decimal places). (C) Interpret S(14)=240.76 and S'(14) = 39.52. (A) S'(t) =

A research group developed the following mathematical model relating systolic blood pressure and age: P(x) = a + bln(x + 1), where P(x) is pressure, measured in millimeters of mercury, and x is age in years. By examining Guilford County hospital records, they estimate the values for Guilford County to be a = 37 and b = 25. Using this model, estimate the rate of change of pressure with respect to time after 15 years. Round to the nearest hundredth (2 decimal places). millimeters per year