102年 - 102 國立暨南國際大學_轉學生入學考試試題_資管系三、財經系三：微積分#124611

(a) y cos x²dxdy =________

(b)dxdy =_________.

(c)dx =_________.

(d)dx =______.

(a) Evaluate the limit of thef(x)_______.

(b) Find f'(x) =_____.

(a) Let u = g(x) = 4 + 3x and y = f(u) = √u. Then = ______.

(b) Let w = ln. Find the differential dw of w.

(a) =__________.

(b) - ... =___________.

(a) Find relative maximum = _____ of f.

(b) Find relative minimum = _____of f.

1. The total weekly revenue (in dollars) of the Country Workshop realized in manufacturing and selling its rolltop desks is given by
R(x,y) = -0.2x² - 0.25y² - 0.2xy + 200x + 160y
where xdenotes 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) = 100x + 70y + 4000
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 realizable? (10分)

2. Find the particular solution of differential equation
y' = 2 - y
that satisfies the condition y(0) = 3. (10分)

3. Use the second Taylor polynomial of f(x) = ln x at x = 1 to approximate ln 1.1 and find a bound for the error in the approximation. (10分)

4. John Mills - the proprietor of Mills Engine Company, a manufacturer of model air-plane engines - finds that it takes x units of labor and y units of capital to produce
f(x,y) = 100x^3/4y^1/4
units of the product. If a unit of labor costs $100, a unit of capital costs $200, and $200,000 is budgeted for production. Determine how many units should be expended on labor and how many units should be expended on capital in order to maximize production. (10分)