09-10(2)english标准答案A

中南民族大学试卷标准答案

试卷名称 数学专业英语 （A卷共5页）

only element is x..

8、假定这个结论对n-1个指数函数成立，我们将证明此结论对n个指数函数也成立。 Assume it is true for n-1 exponential functions, we will prove it is true for n exponential functions also.

四、Translate the theorem into Chinese and prove (10’)

The Cauchy Mean Value Theorem Suppose that the function

f:[a,b]?R and

g:[a,b]?R are continuous, and that

differentiable. Moreover, assume that

f:(a,b)?R and g:(a,b)?R are

g?(x)?0 for all x in (a,b)

Then there is a point

x0 in the open interval (a,b) at which

f?(x0)f(b)?f(a) ?g?(x0)g(b)?g(a)Translation: 柯西中值定理：设函数f:[a,b]?R与g:[a,b]?R，他们在[a,b]上连续，在(a,b)上可导，且g?(x)?0 for all x in (a,b)，那么

f?(x0)f(b)?f(a) ?x0?(a,b)， s.t. ?g?(x0)g(b)?g(a)2、Linearity with respect to the integrand. If both fand g are integrable on[a,b], so is

c1f?c2g for every pair of constants c1and c2.

g积分的线性性 如果f与在[a,b]上都可积，那么对任意对常数c1与c2，c1f?c2g也可

积。

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