课程名称︰微积分一 课程性质︰数学系大一必修 课程教师︰余正道老师 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2018/10/29 考试时限（分钟）：180 试题 : (以下的ε均代表属於) (满分125分) 1.(a)[10%] Determine if the limit lim(1/n+1/(n+1)+...+1/2n) converges;if it does, find the value. n→∞ (b)[10%] Fix a1>b1>0. Define sequences(an), (bn) by an+1=(an+bn)/2, bn+1=√anbn. Show that lim an=lim bn. n→∞ n→∞ (c)[10%] Prove the following: Let (an)∞ be a bounded sequence. n=1 Then there exists a convergent subsequence of (an). 2. Consider the polynomial f(x)=x^n+an-1x^n-1+...+a1x+a0εR[x] of degree n>=1 (a)[10%] Show that lim f(x)=∞. x→∞ (b)[5%] Suppose n is odd. Show that f(x) has at least one (real) root. 3.(a)[10%] Suppose f(x) is continuous. Let f(x) if f(x)≧0 f+(x)={ 0 if f(x)＜0. Show that f+(x) is continuous. (b)[5%] Show that if f(x) on [a,b] has a continuous derivative, then f(x) can be written as a (finite) sum of monotone functions. 4.(a)[10%] Determine the form of a rational function r(x) for which xr'(x) lim ------=0. x→∞ r(x) (b)[10%] Find the formula of the functions fn(x) on x＞0, n=1,2,...defined by x fn-1(t) f0(x)=1, fn(x)=∫ ---------dt 1 t 5.[10%] Show that the function f(x)=e^x is transcendental: For any n, if ai(x)εR[x] such that anf^n+...+a1f+ao=0,then ai=0 for all i=0,1,...,n. 6. Let f(x) be defined on [a,b]. Recall that f(x) is "integrable" if the Riemann sums converge. (a)[5%] Give an example of "non"-continuous f(x) which is integrable. Justify your answer. (b)[15%] Let a＜c＜b. Prove that f(x) is integrable if and only if it is integrable on [a,c] and [c,b]. 7.[15%] Suppose f(x) has f''(x) on [a,b]. Show that f''(x)≧0 on [a,b] if and only if the line segment connecting(x1,f(x1)) and(x2,f(x2)) lies above the graph y=f(x) for any a≦x1,x2≦b. --

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