Help! math problems

aquarius_xuUID: 2143301

1.Let f(x)=X^X with 0<x<1. Determine the minimum value of f on the interval
A &frac12;  B1/e  C1/p D &frac14;  E 0
I think the value of x is 1/e, but not the value of f.yet the answer is B.

2. which curve cannot be a mapping of the unit circle |z|=1 under transformation w=(az+b)/(cz+d) where a,b,c,d are real numbers, ad-bc&sup1; 0.

A circle B line C ellipse D hyperbola.

3.which function has unique fixed points on the stated intervals?

I f(x)=e-x x:[1/3,1]  II g(x)= p+1/2 SinX, x:[0,2p]  III h(x)=x3-1, x:[1,2]

4.The temperature distribution u(x,t) in the bar obey teh partial differential equation ut=kuxx, k is constant. Suppose the temperature of the bar at x=0is fixed at 0 degree. While the temperature at x=L is fixed at 1 degree. At time t=0, the temperature distribution is u(x,0)=f(x). what is the average temperature in the bar in  the limit as t->infinite
A 0  B1/10 integral f(x)dx from 0 to infinite C 5  D 10

5. the differential equation dP/dt=P(1-P)(P-2) possesses the three equilibrium solutions P1=0, P2=1.P3=2. which if these are stable.
My question is what is the difference between a equilibrium solution and a stable one.

6.suppose intergral  f(x)dx from 0 to infinite  exist. Which statements are false
I lim from 0 to infinite  f(x)=0  II intergral from 0 to infinite | f(x)|dx exists  III intergral from 0 to infinite [f(x)]^2dx  exists

Notes: x^2 means x squares

I am sorry, the symbols may not be easily recognized. but I hope you know them.

clonedUID: 2108386

原帖由 aquarius_xu 于 2006-10-9 19:14 发表
1.Let f(x)=X^X with 0<x<1. Determine the minimum value of f on the interval
A &frac12;  B1/e  C1/p D &frac14;  E 0
I think the value of x is 1/e, but not the value of f.yet the answer is B. ...

第一题:同意x=1/e;

第二题:w=(az+b)/(cz+d); z=(b-dw)/(cw-a); 又|z|=1; |(b-dw)/(cw-a)|=1; |dw-b|=|cw-a|;

第三题:不动点是f(x)=x;I f(x)=e-x=x; x=e/2 不在(1/3,1)里;
II g(x)= p+1/2sinx=x; 另G(X)=p+1/2sinx-x; G`(X)=1/2COSx-1在[0,2p]上单调递减. 故Gmax=G(0)=P; Gmin=G(2P)=1/2sin(2p)-p<=0; 所以存在一个G(~)=0(0<~<2P)
III h(x)=x^3-1=x; 令H(x)=x^3-x-1 H`(X)=3x^2-1 在[1,2]上单调递增,而H(1)=-1,H(2)=4, 所以一定有个H(~)=0(1<~<2)
选 II,III

nasamizUID: 2132258

都是REA上面的，NO4的解释看不懂啊！

pazaerUID: 2220935

第六题挺有趣的,若f是可测函数就好玩了,哈哈!

qfjpzmUID: 2125368

觉得4题的题目有错误,应该是在L 处为10degree.