微積分期末考 考試日期:96.06.28
1.(a) Evaluate(2分)
(b)Evaluate(2分)
(c) Evaluate(2分)
(d) Evaluate(2分)
2. (a) Find the average value of the given function over the specified interval (2分)
(b) Sketch the indicated region R and find its area by integration.
R is the region bounded by the curve and the line (2分)
(c)Evaluate(2分)
(d)Evaluate(2分)
(e)Evaluate(2分)
3. (a) Find the general solution of the given differential equation.
(5分)
(b) Find the particular solution of the given differential equation that satisfies the indicated condition: ;when(5分)
4.The length of time (days) between the breakdown of a particular piece of machinery is exponentially distributed with density function
(a)What is the probability that the machine will operate for more than 20 days without a breakdown?(3分)
(b)What is the probability that the machine will operate between 5 and 15 days without a breakdown?(3分)
(c)What is the expected life of a randomly selected machine?(3分)
5. It is estimated that years from now, a certain investment will be generating income at the rate of dollars per year. If the income is generated in perpetuity and the prevailing annual interest rate remains fixed at 5% compounded continuously,find the present value of the investment.(5分)
6. Suppose the priceof a particular commodity varies in such a way that its rate of change with respect to time is proportional to the shortage D-S, where andare the demand and supply functions for the commodity, respectively. That is,, Findfor the case where and , if the price is $4 whenand is $3 when .(10分)
7. A manufacturer has determined that whenthousand units of a particular commodity are produced, the price at which all the units can be sold isdollars per unit, whereis the demand function
(a)At what price are 5000units demanded?(4分)
(b)Find the consumers’ surplus when 5000 units are demanded.(4分)
8. Money is transferred into an account at the rate of dollars per year for 10 years, whereis the number of years after 2000. If the account pays 5% interest compounded continuously, how much will be in the account at the end of the 10-year investment period (in 2010)?(5分)
9. Records indicate thatmonths after the beginning of the year, the price of bacon in local supermarkets was dollars per pound. What was the average price of bacon during the first 6 months of the year?(10分)
10. Magda wants to expand and renovate her import store, and is presented with two plans for making the improvements. The first plan will cost her $40000 and the second will cost only $25000. However, she expects the improvements resulting from the first plan to provide income at the continuous rate of $10000 per year, while the income flow from the second plan provides $8000 per year. Which plan will result in more net income over the next 3 years if the prevailing rate of interest is 5% per year compounded continuously?(10分)
11. In a certain state, it is found that the distribution of income for lawyers is given by the Lorentz curve , wherewhile that of surgeons is given by , where .Compute the Gini index for each Lorentz curve. Which profession has the more equitable income distribution?(5分)
12. Afterhours on the job, one factory worker is producing units per hour, while a second worker is producing units per hour.
(a)If both arrive on the job at 8:00A.M.,how many more units will the first worker have produced by noon than the second worker?(5分)
(b)Interpret the answer in part(a) as the area between two curves.(5分)