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2022年9月4日 星期日

109年台聯大轉學考-微積分A2詳解

台灣聯合大學系統109學年度學士班轉學生考試

科目:微積分
類組別:A2

甲、簡答題:共8題,每題8分,共64分

解答limxx+cosxxcosx=limx(1+2cosxxcosx)=1+0=1
解答f(x)=|x|xx2+1{f(x)=x2x2+1=11x2+1,if x0f(x)=x2x2+1=1+1x2+1,if x0{limxf(x)=1,if x0limxf(x)=1,if x0y=±1
解答F(x)=x0cos(t3/2)dtF(x)=cos(x3/2)F(x)=32xsin(x3/2)F(x)=0x=0,x3/2=nπ,nZ;xx=π2/3
解答x,y,z{xyz=12000z=12000/xycost=2xy+8xz+6yzcost=f(x,y)=2xy+8x12000xy+6y12000xy=2xy+1y96000+1x72000f(x,y){fx=0fy=0{2y720001x2=02x960001y2=0{2x2y=72000(1)2xy2=96000(2)(1)(2)xy=34x=34y(2)y3=64000y=40x=30f(30,40)=2400+9600040+7200030=7200
解答{x=rcosθy=rsinθ,(cos2θsin2θ)20cos4θ+sin4θ2cos2θsin2θ(xy)px4+y4=r2p(sinθcosθ)pr4(cos4θ+sin4θ)r2p(sinθcosθ)pr4(2cos2θsin2θ)=12r2p4(sinθcosθ)p2lim(r,θ)(0,0)={0,p>212r42p(sinθcosθ)2p0,p2{f is continuous for all(x,y), if p>2f is discontinuous at (0,0), if p2
解答ln505ex1lnydydx=51lny01lnydxdy=511dy=4
解答f(x,y)=10xye(x2+y2){fx=(10y20x2y)e(x2+y2)fy=(10x20xy2)e(x2+y2){fxx=(60xy+40x3y)e(x2+y2)fxy=(1020x220y2+40x2y2)e(x2+y2)fyy=(60xy+40xy3)e(x2+y2)g(x,y)=fxxfyyf2xy{fx=010y(12x2)=0x=±1/2y=0fy=010x(12y2)=0x=0y=±1/2{A(0,0)B(1/2,1/2)C(1/2,1/2)D(1/2,1/2)E(1/2,1/2){g(A)=10>0g(B)=g(C)=g(D)=g(E)=400/e2>0{f(A)=0f(B)=f(E)=5/ef(C)=f(D)=5/e3
解答00kxye(x2+y2)dxdy=0[12kye(x2+y2)]|0dy=012kyey2dy=[14key2]|0=14k=1k=4

乙、計算、證明題:共3題,每題12分,共36分

解答:x1p1+x2p2=LU(x1,x2)x2=Lx1p1p2x1x2=p1p2U(x1,x2)=U(x1,Lx1p1p2)U(x1)=0x1U+x2Ux1x2=0x1Ux2Up1p2=0U/x1U/x2=p1p2
解答a.:n=1(1)nan,an=ln(1+1n)(1)limnan=ln1=0(2)anan+1=ln(n+1n)ln(n+2n+1)=ln(n+1)2n(n+2)=lnn2+2n+1n2+2n=ln(1+1n2+2n)>0an>an+1(1)(2)n=1(1)nln(1+1n);limnnk=1ln(1+1k)=limnln234(n+1)123n=n=1|(1)nln(1+1n)|=n=1ln(1+1n)b.n=1anlimnan=0limnn|(3+sin(an)5)n|=limn|3+sin(an)5|limn3+|sin(an)|5=35<1n=1(3+sin(an)5)n
解答==cosθ+2×12sinθcosθf(θ)=cosθ+sinθcosθf(θ)=sinθ+cos2θsin2θ=2sin2θsinθ+1f(θ)=0(2sinθ1)(sinθ+1)=0sinθ=1/2(1,θ3π/2)θ=30

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