103年 - 103 國立暨南國際大學_轉學生入學考試試題_經濟系二、資管系二、財金系二：微積分#123439

科目：研究所、轉學考(插大)-微積分 | 年份：103年 | 選擇題數：0 | 申論題數：14

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所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (14)

1. Use a Taylor polynomial to estimate

dx with an error less than 10^-2. (10 分)

(a) f(x，y) ye^x3; R is bounded by x=

, x=1, y=0, and y=2.

4. Find $h'(0)$ if $h(x) = g(f(x)), g(x) = x+

, and f(x) = e^x. (10 分)

5. Find the absolute maximum value and the absolute minimum value, if any, of the function. (10 分)

(a) Find the points on the graph of f at which the slope of the tangent line is equal to -2.

(b) Find the equation(s) of the tangent line(s) of part (a).

7. 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 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) = 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)

8. The temperature (in °F) in Boston over a 12-hr period on a certain December day was given by

T = -0.05t³ + 0.4t² + 3.8t + 5.6 (0 ≤ t ≤ 12)

where $t$ is measured in hours, with t = 0 corresponding to 6 a.m. Determine the average temperature on that day over the 12-hr period from 6 a.m. to 6 p.m. (10 分)

9. HAL Corporation invests P dollars/year (assume this is done on a frequent basis in small deposits over the year so that it is essentially continuous) into a fund earning interest at the rate of r%/year compounded continuously. Then the size of the fund A grows at a rate given by

Suppose A = 0 when t = 0. Determine the size of the fund after $t$ yr. What is the size of the fund after 5 yr if P = $50,000$ and r = 12%/year? (10 分)