Download 微積分期中考 每題十分 共十題 總分100分

微積分期中考
每題十分 共十題 總分 100 分
日期:95.11.23
1. Find equations for these lines that is tangent to the graph of f for the given value of
x.
1
(a) f ( x)  x 2 2 x  3; x  1
(b) f ( x) 
; x 1
(2 x  1)6
2. Find the give limit or show it does not exist . If the limit is infinite, indicate whether
it is   or -  .
(a) lim x3
x2  4x  3
x3
x
(c) lim x
(d) lim x
x3  x 2
1  3x  1  2 x
x
(b) lim x0
x2  x
2 x
3. List all the values of x for which the give function is not continuous
f ( x) 
(a)
x 2  2 x 1
x2  4 x 5
2  3x if x  1

(b) f ( x)   2
 x  2 x  2 if x  1
4. Discuss the continuity of the function
 x  5 i fx  5
f ( x)  
5 if x  5

5. Find the derivative of the given function
(a) 2 x 3  3 x
(c)
(b) ( x 2  3) 5 (3x  1) 3
3x  5
(1  2 x ) 3
6. Find
(d) ( x 5  4 x 3  7) 7
dy
for given value of x
dx
(a) y   3  4 2  5  2 ;   x 2  1 for x=1
2
(b) y  500 3 ,  
2 x 4  3x  149
for x=3
x2
7. A bookstore can obtain a certain gift book from the publisher at a cost of $3 per
book. The bookstore has been offering the book at price of $15 per copy, and at this
price, has been selling 200 copies a month. The bookstore is planning to lower its
price to stimulate sales and estimates that for each $1 reduction in the price, 20
more books will be sold each month. Express the bookstore’s monthly profit from
the sales of this book as a function of the selling price, draw the graph, and estimate
the optimal selling price.
8. A manufacturer can sell a certain product for $80 per unit. Total cost consists of a
fixed overhead of $4,500 plus production costs of $50 per unit.
(a) How many units must the manufacturer sell to break even?
(b) What is the manufacturer’s profit or loss if 200 units are sold?
(c) How many units must the manufacturer sell to realize a profit of $900?
9. An importer of Brazilian coffee estimates that local consumers will buy
4,374
approximately D(p)=
pounds of the coffee per week when the price is p
p2
dollars per pound. It is also estimated that t weeks from now, the price of Brazilian
coffee will be p(t)= 0.02t 2  0.1t  6 dollars per pound.
(a) At what rate will the demand for coffee be changing with respect to price when
the price is $9?
(b) At what rate will the demand for coffee be changing with respect to time 10
weeks from now? Will the demand be increasing or decreasing at this time?
10. Bea Johnson, the owner of the Bea Nice boutique, estimates that when a particular
500
p5
kind of perfume is priced at p dollars per bottle, she will sell B(p)=
p3
bottles per month at a total cost of C ( p)  0.2 p 2  3 p  200 dollars
(a) Express Bea’s profit P(p) as a function of the price p per bottle.
(b) At what rate is the profit changing with respect to p when the price is $12 per
bottle? Is profit increasing or decreasing at that price?