课程名称︰微积分甲上 课程性质︰ 课程教师︰曾琇瑱 开课学院：（非一二组）..生农吗..？还是公卫 开课系所︰（非一二组）..生农吗..？还是公卫 考试日期（年月日）︰98/11/06 五 考试时限（分钟）：110 mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.(5%) Find g(x) and h(x) such that f(x)=g(h(x)) where 5 ____ f(x)=_________+3√x+4 _____ √x+4 2.(9%) Find the follow limits _ √x-1 (a) lim _______ x→1 x-1 ∣x∣ (b)lim - _____ x→0- x (c)lim 2e^(x-x^2) x→+∞ 3.(8%) Show that lim 1/x = 1 x→1 4.(5%) Make a rough sketch of the derivative of the given graph of f(x) 1 _____ 5.(15%) Let f(x) = _______ + √x^2+x - x answer the follow problems x^2-3x (a) Find the domain of f(x) (b) Find the asymptotes of f(x) (c) Find f'(x) 6.(5%) Determine whether f(x) is continuous(give the proof) or discontinuous (give your reason) at x=3 where f(x)=2x^2 -x if x<3 and f(x)=3-x if x≧3 7.(18%) Find dy/dx of the follow function (a) y= [(5x^2 +2)^3]coshlnx _____ √1+e^x (b) y=tan^-1 ____________ __________ √1+(sinx)^2 (c) 4cosxsiny=1 8.(10%) If f(x)+(x^2)[f(x)]^3 = 10 and f(1)=2 find f'(1) 9.(10%) Use the definition of derivation to prove that ln(1+x) lim _______ = 1 x→0 x 10.(15%) (a) State the Intermediate Value Theorem (b) Show that there is a root of the equation sinx=2x-(x^3-1)^-1 between -π/2 and π/2 (c) Suppose that f is continuous on the interval [1,5] and the only solution of the equation f(x)=6 are x=1 and x=4 If f(2)=8 explain why f(3)>6 --

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