b. ∫ x⁸(2x⁹-1)⁶dx

a. y =

b. f(x) = 5(x4-x)7

c. g(x) =

2.The total weekly revenue (in dollars) of the company realized in manufacturing and selling its rolltop desks is given by R(x, y) = -0.2x² - 0.25y² - 0.2xy + 224x + 180y where x denotes the number of finished units and y denotes the number of unfinished units manufactured and sold each week. The total weekly cost attributable to the manufacture of these desks is given by C(x, y) = 120x + 80y + 2000 dollars. Determine how many finished units and how many unfinished units the company should manufacture each week in order to maximize its profit. What is the maximum profit (P) realizable?

3.The present value of a perpetual stream of income that flows continually at the rate of p(t) dollars per year is given by the formula PV = where r is the interest rate compounded continuously. Using this formula, find the present value of a perpetual net income stream that is generated at the rate of p(t) = 30000 + 2000t dollars per year.

a.

5.Find the second-order partial derivatives of the function f(x, y) = x4y - xy3. Show that the mixed partial derivatives fxy and fyx are equal.

(14) Find the extreme values of f(x, y, z) = z subject to the constraints x² + y² = z² and x + 2y + z =   (16) 

(13) A logistic population model with relative growth rate 0.1 per year and carrying capacity 50 thousand can be expressed by the differential equation =0.1P(1 - ), with P in thousands and t in years. Given that the initial population is 9 thousand. Find the population size after 20 years. (If you memorized the formula, then you need to derive it for this problem.)

(12) Evaluate the definite integral xln ln(x² - 4x + 5) dx.

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