課程名稱︰微積分乙 課程性質︰必修 課程教師︰陳俊全 開課學院：醫學院 開課系所︰醫學系、牙醫系 考試日期（年月日）︰114/12/18 考試時限（分鐘）：120 試題 : Part I: Answer all the problems (1.-3.) below 1.(10%) True or False. (a) There is a differentiable function f such that f(-2)=-2, f(2)=6, and f'(x)<1 for all x. (b) If f(x) is integrable on the closed interval [a,b], then there exists a b number c∈[a,b] such that∫ f(x)dx= f(c)(b-a). a x t^2 √(1+u^4) (c) Let f(x)=∫∫ ───── du dt with x>0 and t>0. Then f(x) has a local 1 √t u minimum at x=1. f'(x) (d) If lim ─── can not be determined, according to L'Hopital's Rule, x→∞ g'(x) f'(x) lim ─── does not exists. x→∞ g'(x) t (e) If f(x) is continuous on [0,1] and f(0)=0, then ∫√(1+[f'(x)]^2)dx≧ √(t^2+[f(t)]^2) for 0≦t≦1. 0 2.(45%) ∞ (a)(6%) Evaluate the integral ∫x^(-2) ln(1+x^2) dx. 1 sin^2(x) (b)(6%) Evaluate the integral ∫───── dx. 1+sin^2(x) (c)(7%) Compute the volume of the solid obtained by rotating y=(x^2+4x+7)^(-¼) , 0≦x≦2, about the x-axis. (d)(6%) Find the limit 6 6*2 6*3 6n lim (─────+─────+─────+...+─────). n→∞ n^2+3*1^2 n^2+3*2^2 n^2+3*3^2 n^2+3*n^2 1 1 (e)(6%) Evaluate lim ( ─-────). x→0+ x arctanx (f)(7%) Show that |(66)^⅓ -4-(1/24)|<1/(9*256). (g)(7%) Prove that (1+x)^p≧1+px if p≧1 and x≧0. 3.(15%) The Euler's gamma function Γ(x) is defined as ∞ Γ(x)=∫ t^(x-1) e^(-t) dt, x>0. 0 (a)(3%) Compute Γ(1). (b)(6%) Show that Γ(x+1)=xΓ(x) for x>0 ∞ (c)(6%) It is well known that ∫ e^(-x^2/2)dx=√(2π). Use this fact to compute Γ(1/2) -∞ ─────────────────────────────────────── Part Ⅱ: Choose 2 of the following 5 problems (4.~8.) and solve them. 4.(15%) (1) Assume that f is continuous on [0,1] and n∈N. Evaluate the limit 1 lim ∫f(x)x^n dx. x→∞ 0 (2)Assume that f' exists and is continous on [0,1] and n∈N. Evaluate the limit 1 lim n∫f(x)x^n dx. x→∞ 0 (3) If we only assume that f is continuous on [0,1], does 1 lim n∫f(x)x^n dx exists? x→∞ 0 5.(15%) Solve the differential equation N'(t)=N(t)(N(t)-2)(1-N(t)/5). Show that if N(0)>2, then lim N(t)=5. t→∞ 6.(15%) Let α∈R. Use Taylor's Theorem to show that α(α-1) α(α-1)...(α-n+1) (1+x)^α=1+αx+────x^2+...+──────────x^n+... for |x|<1. 2! n! In the following problems, you can use the property that a closed interval [a,b] is compact: every sequence in [a,b] has a subsequence which converges to a point in [a,b]. 7.(15%) Show that a continuous function f on [0,2] is uniformly continuous and 2 1 2 lim ∫ f(x)[1-─sin^2(nx)]dx=∫f(x)dx. n→∞ 0 n 0 8.(15%) Show that a continuous function on [0,2] is integrable. Happy Winter Vacation!! --

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