1.3.3　导数的实际应用

1以长为10的线段AB为直径画半圆,则它的内接矩形面积的最大值为(　　)

A.10 B.15 C.25 D.50

答案：C

2某公司生产某种产品,固定成本为20 000元,每生产一件产品,成本增加100元,已知总收益R(单位:元)与年产量x(单位:件)的关系是R={■(400x"-" 1/2 x^2 "," 0≤x≤400"," @80" " 000"," x>400"," )┤则当总利润P最大时,每年的产量是(　　)

A.100件 B.150件

C.200件 D.300件

解析：由题意,总成本为C=20 000+100x.

　　所以总利润P=R-C

　　={■(400x"-" 1/2 x^2 "-" 100x"-" 20" " 000"," 0≤x≤400"," @80" " 000"-" 100x"-" 20" " 000"," x>400"," )┤

　　则P'={■(300"-" x"," 0≤x≤400"," @"-" 100"," x>400"," )┤

　　令P'=0,得x=300,

　　易知当x=300(件)时,总利润最大.

答案：D

3要做一个圆锥形的漏斗,其母线长为20 cm,要使其体积最大,则高为(　　)

A.√3/3 cmB.(10√3)/3 cm

C. (16√3)/3 cmD.(20√3)/3 cm

解析：设圆锥的高为x cm,则底面半径为√(20^2 "-" x^2 ) cm,

　　其体积为V=1/3 πx(202-x2)(0

　　V'=1/3 π(400-3x2),令V'=0,

解得x1=20/3 √3,x2=-20/3 √3(舍去).