问题 解答题

设函数f(x)=ex,其中e为自然对数的底数。

(1)求函数g(x)=f(x)-ex的单调区间;

(2)记曲线y=f(x)在点P(x0,f(x0))(其中x0<0)处的切线为l,l与x轴、y轴所围成的三角形面积为S,求S的最大值。

答案

解:(1)由已知g(x)=ex-ex,

所以g'(x)=ex-e,

由g'(x)=ex-e=0,得x=1,

所以,在区间(-∞,1)上,g'(x)<0,

函数g(x)在区间(-∞,1)上单调递减;

在区间(1,+∞)上,g'(x)>0,

函数g(x)在区间(1,+∞)上单调递增;

即函数g(x)的单凋递减区间为(-∞,1),

单调递增区间为(1,+∞)。

(2)因为f'(x)=ex

所以曲线y=f(x)在点P处切线为l:

切线l与x轴的交点为(x0-1,0),

与y轴的交点为

因为x0<0,

所以

S'=

在区间(-∞,-1)上,函数S(x0)单调递增,在区间(-1,0)上,函数S(x0)单调递减,

所以,当x0=-1时,S有最大值,此时

所以,S的最大值为

单项选择题
单项选择题

There are two factors which determine an individual’s intelligence. The first is the sort of brain he is born with. Human brains differ considerably, some being more capable than others. But no matter how good a brain he has to begin with, an individual will have a low order of intelligence unless he has opportunities to learn. So the second factor is what happens to the individual—the sort of environment in which he is reared. If an individual is handicapped environmentally, it is likely that his brain will fail to develop and he will never attain the level of intelligence of which he is capable.

The importance of environment in determining an individual’s intelligence can be demonstrated by the case of the identical twins, Peter and Mark Being identical, the twins had identical brains at birth, and their growth processes were the same. When the twins were three months old, their parents died, and they were placed in separate foster homes. Peter was reared by parents of low intelligence in an isolated community with poor educational opportunities. Mark was reared in the home of well to do parents who had been to college. He was read to as a child, sent to good schools, and given every opportunity to be stimulated intellectually. This environmental difference continued until the twins were in their late teens, when they were given tests to measure their intelligence. Mark’s IQ was 125, twenty five points higher than the average and fully forty points higher than his identical brother. Given equal opportunities, the twins, having identical brains, would have tested at roughly the same level.

The best statement of the main idea of this passage is that ().

A.human brains differ considerably

B.the brain a person is born with is important in determining his intelligence

C.environment is crucial in determining a person’s intelligence

D.persons having identical brains will have roughly the same intelligence