微積分期中考
總分 100 分
考試日期:97.04.15
1. Find the absolute maximum and absolute minimum of f ( x) 
x2
x 1
on the
interval – 2  x  –1/2 ? (5 分)
300  p 2
units of a certain commodity are demanded
60
when p dollars per unit are charged.
2. Suppose that x 
(a)Determine where the demand is elastic, inelastic, and of unit elasticity with
respect to price.(2 分)
(b)Use the results of part (a) to determine the intervals of increase and decrease of
the revenue is function and the price at which revenue is maximized. (2 分)
(c)Find the total revenue function explicitly and use its first derivative to determine
its intervals of increase and decrease and the price at which revenue is
maximized. (2 分)
3. A store uses 600 cases of electronic parts each year. Each case costs $1,000. The
cost of storing one case for a year is 90 cents and the ordering fee is $30 per
shipment. How many cases should the store order each time to minimize total cost?
Assume the orders are planned so that a new shipment arrives just as the number of
cases in the store reaches zero. Also assume the parts are consumed at a constant
rate. (5 分)
4. The owner of a novelty store can obtain joy buzzers from the manufacturer for 25
dollars apiece. He estimates he can sell 50 buzzers when he charges 40 dollars
apiece for them and that he will be able to sell 3 more buzzers for every one dollar
decrease in price. What price should he charge in order to maximize profit? (5 分)
5. If the demand for a commodity is D(p) = 28 – 5p, where p is the price, and the total
cost is C ( p)  p 2  4 p .
(a)At what price should be to maximize the profit? (3 分)
(b)What is the level of maximum profit? (3 分)
6. Evaluate the given expressions.


(a)  27  8 


2
3
4
3

1
3
2
 16  4  125 
(b)   

 81   8 
(2 分)

2
3
(2 分)
Use logarithm rules to simplify each expression.
1 1 
(c) ln   2  (2 分)
x x 

3  x2
(d) ln x e

(2 分)
7. How much money should be invested today at 7 percent compounded quarterly so
that 5 years from now it will be worth $5,000? (5 分)
8. Solve for x:
4
(a)
 x  3 3
 16 (2 分)
(b) 5  1  4e2 x 1 (2 分)
(c) ln  x  2  3  ln  x  1 (2 分)
a
1  b
(d) Find ln 
 if ln b  6 and ln c  2 . (2 分)
a  c 
9. An economist estimates that the gross national product (GNP) of a certain country
is G  G0 e kt , where G0 and k are positive constants. If the GNP is 100 billion in 1990
and 180 billion in 2000, what will it be in the year 2010? (5 分)
10. Find
df ( x )
dx
3 2 x  x
(a) f ( x)  x e
(b) f ( x)  x ln e
2
x2
(2 分)
(2 分)
(c) Find the equation of the tangent line to f ( x)  x  ln x at x = e (2 分)
dy
e3 x (2 x  5)
(d) Use logarithmic differentiation to find
, where y 
dx
(6  5 x) 4
1
2
(2 分)
11. Determine where the given function is increasing and decreasing, and where its
graph is concave upward and concave downward. Sketch the graph, showing as
many key features as possible (high and low points, points of inflection,
asymptotes, intercepts, cusps, vertical tangents). (10 分)
y
4
1  e x
12. It is estimated that t years from now, the population of a certain country will be
P t  
20
million.
2  3e 0.06t
(a) What is the current population? (3 分)
(b) What will be the population 50 years from now? (3 分)
(c) What will happen to the population in the long run? (3 分)
13. Suppose your family owns a rare book whose value t years from now will be
V  t   200e
2t
dollars. If the prevailing interest rate remains constant at 6% per
year compounded continuously, when will it be most advantageous for your
family to sell the book and invest the proceeds? (5 分)
14. Suppose that for a particular semelparous organism, the likelihood of an
individual surviving to age x years is p  x   e0.2x and that the number of female
births to an individual at age x is f  x   5x0.9 . What is the ideal age for
reproduction for an individual organism of this species? (5 分)
15. What professors select texts for their courses, they usually choose from among the
books already on their shelves. For this reason, most publishers send
complimentary copies of new texts to professors teaching related courses. The
mathematics editor at a major publishing house estimates that if x thousand
complimentary copies are distributed, the first-year sales of a certain new
mathematics text will be approximately f  x   20  15e0.2 x thousand copies.
(a) How many copies can the editor expect to sell in the first year if no
complimentary copies are sent out? (3 分)
(b) How many copies can the editor expect to sell in the first year if 10,000
complimentary copies are sent out? (3 分)
(c) If the editor’s estimate is correct, what is the most optimistic projection for the
first-year sales of the text? (4 分)