107年 - 107 國立臺灣大學_碩士班招生考試_國際企業研究所乙組：微積分(C)#114199

試卷資訊

選擇題 (0)

申論題 (12)

1.A one-story storage building is to have a volume of 250,000 cubic feet. The roof costs $32 per square foot, the walls cost $10 per square foot, and the floor costs $8 per square foot. Find the dimensions that minimize the cost of the building.

(1)Rewrite the above double integral by exchanging the order of integration.

(2)Evaluate the double integral derived in (1).

3.Find the least squares approximation h(x) = ax + ax² for f(x) = e^-x, 0 ≤ x ≤ 1. [Hint: Solve a by minimizing the integral of the squared difference between f(x) and h(x).]

(1)Set up and solve a linked pair of differential equations to determine respectively the amount of money in the savings account and stock fund t years from now.

(2)How long does it take for the savings account to be exhausted? How much money is in the stock fund at this time point?

5.If = 0, what are the values of r and s?

6.Find In sec 5x.

7.Evaluate the definite integral .

8.Find the area of an ellipse by using the parametric equations x = 5 cosθ, and y = 6 sinθ.

9.Find the interval of convetgence for the power series .

10.Solve the differential equation x²dy + 2xydx + 2xdx + dy = 0.