國立臺灣海洋大學113學年度碩士班考試入學招生考試試題
考試科目:微積分
學系組名稱:運輸科學系碩士班不分組
解答:f′(x)=limh→0f(x+h)−f(x)h=limh→02(x+h)2−5(x+h)−2x2+5xh=limh→04hx+2h2−5hh=limh→0(4x+2h−5)=4x−5解答:f(x)=x3−2|x|3+1⇒{limx→∞f(x)=1limx→−∞f(x)=−1⇒ horizontal asymptotes: y=1,y=−1
解答:y=tan2(sin3t)⇒dydt=2tan(sin3t)sec2(sin3t)3sin2tcost=6tan(sin3t)sec2(sin3t)sin2tcost
解答:f(x,y)=xcosy+yex⇒{∂f∂x=cosy+yex∂f∂y=−xsiny+ex⇒{∂2f∂x2=yex∂2f∂x∂y=−siny+ex∂2f∂y2=−xcosy
解答:∫20dx√|x−1|=∫10dx√1−x+∫21dx√x−1=[−2√1−x]|10+[2√x−1]|21=2+2=4
解答:∫π/40√1+cos(4x)dx=∫π/40√1+2cos2(2x)−1dx=√2∫π/40cos(2x)dx=√2[12sin(2x)]|π/40=√22
解答:x=3tanu⇒dx=3sec2udu⇒I=∫1x2√x2+9dx=∫3sec2u9tan2u⋅3secudu=∫secu9tan2udu=∫cosu9sin2udu=−19sinu+c=−√x2+99x+c
解答:∬Rex+2ydA=∫ln21∫ln30ex⋅e2ydxdy=∫ln212e2ydy=4−e2
解答:f(x)=2x−3x2/3(a)domain of f:(−∞,∞)limx→∞f(x)=∞,limx→−∞f(x)=−∞⇒No asymptotes(b)f′(x)=2−2x−1/3=2(1−13√x)=0⇒x=1⇒{f(0)=0f(1)=−1⇒critical points: (0,0),(1,−1)(c){f′(x)>0x<0f′(x)<00<x≤1f′(x)>0x≥1⇒{f is increasing, if x<0,x≥1f is decreasing, if 0<x≤1(d)from (c), we havelocal minimum: -1; local maximum: 0(e)f″(x)=23x4/3>0,∀x≠0⇒f(x) is concave up ,x∈(−∞,0)∪(0,∞)(f):
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