102年 - 102 東吳大學轉學生招生考試試題_經濟學系：微積分#116023

科目：研究所、轉學考(插大)-微積分 | 年份：102年 | 選擇題數：0 | 申論題數：12

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (12)

1.The demand and supply equations for an e‐book reader are given by

p = 195 -3.8x

q = 150 +3.2x

where p is the price in dollars and x represents the number of units in millions. Find the equilibrium
point for this market.

2.Find an equation of the tangent line to the graph of y= at x=1.

3.The demand function for a product is modeled by

p=1/(0.000001x³+0.01x+1)

where p is measured in dollars and x is measured in thousands of units. Find the rate of change of the demand x with respect to the price p when x=100.

4.Show that the graph of the normal probability density function

has points of inflection at x=±1.

5.The marginal cost of producing x units of a product is modeled by.

It costs $50 to produce one unit. Find the total cost of producing 200 units.

6.Find the area of the region bounded by the graphs of f(x)=3x³-x²-10x and g(x)=-x²+2x.

7.A manufacturer’s production is modeled by the cobb‐Douglas function

Where x represents the units of labor ( at $ 150 per unit ) and y represents the units of capital ( at $ 250 per unit ). The total cost for labor and capital cannot exceed $ 50,000. Will the manufacturer’s maximum production level exceed 16,000 units? ( Hint: Use the method of Lagrange Multipliers and the funciton . F(x,y,λ)=f(x,y)-λ(150x+250y-50,000).)

8.A. Find the limit.

B. Evaluate

by interpreting it in terms of areas.

C. Use the Fundamental Theorem of Calculus to find the derivative of the function.

D. Use logarithmic differentiation to find the derivative of the function.

E. Evaluate the integral.