99年 - 99 國立臺灣師範大學_轉學生招生考試試題_地球科學系二年級：微積分#122359

科目：研究所、轉學考(插大)-微積分 | 年份：99年 | 選擇題數：0 | 申論題數：12

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (12)

(1) f(x) =66cd2732276b8.jpg
(2) g(x) = log₂(x² + x + 1)
(3) h(x) = sin-1x

(4) F(x) =66cd2766a8bf7.jpgdt 

(1) 66cd27966c0e3.jpgdx

(2)66cd27b49e446.jpg(cos3 x + 1)tan x) dx 
(3) ∫2x dx

(4)66cd27f0081b4.jpgdx

3. (15%) Suppose that the source of current in an electric circuit is a battery. Then the power output P (in watts) obtained if the circuit has a resistance of R ohms is given by
66cd280f1f81e.jpg
where E is the electromotive force in volts and r is the internal resistance of the battery in ohms. If E and r are constant, find the value of R that will result in the greatest power output. What is the maximum power output?
4. (15%) A solid ball can be obtained by revolving the semi-circular region x² + y² ≤ R², x ≥ 0, about the y-axis. Use the method of disks or cylindrical shells to find the volume of such a solid ball.

5. (15%) Let M, N, k be three positive constants, with M> N. Solve the differential equation
66cd285c31b63.jpg = k(M - y)(N - y),
under the initial condition y(0) = 0.
6. (15%) Suppose x units of labor and y units of capital are required to produce
f(x, y) = 100x3/4y1/4
units of a certain product. If each unit of labor costs $200 and each unit of capital costs $300 and a total of $60,000 is available for production, determine how many units of labor and how many units of capital should be used to maximize production.