上學期期中考 1.(15%)Find the indicated limit 2|x-1|-x+1 (a) lim ------------- x→1 2|x-1|+x-1 [1/x] (b) lim x(-1) where [ ] is the greatest integer function x→0 1-cosθ sin(--------) θ (c) lim ----------------- θ→0 θ 2.(10%)Use the ε-δ definition of a limit, prove that lim √(x+1) = 2 x→3 3.(10%)Give an example to show that the following is false: If lim f(x)=L and lim g(x)=M then lim g(f(x))=M x→a x→L x→a 4.(10%)Give an arbitrary number c, show that the equation x+sinc=c always has a solution 5.(25%)(a)Discuss the continuity and differentiable at x=0 for the following functions f(x) and g(x), 1 2 1 f(x)= sin--- , x≠0 g(x)= x sin--- , x≠0 x x and = 0 , x=0 = 0 , x=0 xcosx (b)Let f(x)=tan√(-----------) , where a is a constant a^2+x^2 find f'(x) 6.(10%)Use the Mean Value Theorem to show that √(1+x) < 1+0.5x 3 2 7.(10%)Find a cubic function f(x)=ax +bx +cx+d that has a local maximum value 3 at -2 and a local minimum value 0 at 1 8.(10%)Find all equations of asymptotes of the graphs of 4x^2-2x+5 y=------------- 2x^2+x-3 9.(10%)Assume s''(t)=10sint+3cost, s(0)=0, s(2π)=12 Find s(t) x^2 y^2 10.(16%)Given the ellipse E: ----- + ----- = 1 a^2 b^2 (a) Show by implicit differentiation that the tangent line to E at the point (Xo,Yo) is Xo*x Yo*y ------- + ------- =1 a^2 b^2 (b) Find the area of the largest rectangle that be inscribed in E 喔..沒期末考的考卷 但是期末考總分有150喔! 分數是依照小考*25% 作業*5% 期中*35% 期末*40%算的 但其實期末考破百的人不少喔..也就是說後面那個40很多人都拿到破40的分數喔 所以說再加上作業的5%....學期成績大約是用130分去扣的..蠻容易拿高分的@@" 期末考佔非常重..但我覺得期末考的題目比期中簡單 期末幾乎都是偏計算的題目...積分積得出來就好啦 不像期中有些證明..有些又要求從基本定義下手 我是雙修生..雖然以前就修過微積分了..但還是又要再修一次@@"" 但也因為這樣..這學期靠這科賺了不少分數喔! --

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