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:limx→∞x+cosxx−cosx=limx→∞(1+2cosxx−cosx)=1+0=1
解答:f(x)=|x|xx2+1⇒{f(x)=x2x2+1=1−1x2+1,if x≥0f(x)=−x2x2+1=−1+1x2+1,if x≤0⇒{limx→∞f(x)=1,if x≥0limx→∞f(x)=−1,if x≤0⇒水平漸近線為y=±1解答:F(x)=∫x0cos(t3/2)dt⇒F′(x)=cos(x3/2)⇒F″(x)=−32√xsin(x3/2)因此F″(x)=0⇒x=0,x3/2=nπ,n∈Z;依題意找最小正值的x,因此x=π2/3
解答:假設長寬高分為x,y,z⇒{xyz=12000⇒z=12000/xycost=2xy+8xz+6yz⇒cost=f(x,y)=2xy+8x⋅12000xy+6y⋅12000xy=2xy+1y96000+1x72000欲求f(x,y)的極小值,即{fx=0fy=0⇒{2y−72000⋅1x2=02x−96000⋅1y2=0⇒{2x2y=72000⋯(1)2xy2=96000⋯(2)(1)(2)⇒xy=34⇒x=34y代入(2)⇒y3=64000⇒y=40⇒x=30⇒f(30,40)=2400+9600040+7200030=7200
解答:令{x=rcosθy=rsinθ,又(cos2θ−sin2θ)2≥0⇒cos4θ+sin4θ≥2cos2θsin2θ⇒(xy)px4+y4=r2p(sinθcosθ)pr4(cos4θ+sin4θ)≤r2p(sinθcosθ)pr4(2cos2θsin2θ)=12r2p−4(sinθcosθ)p−2⇒lim(r,θ)→(0,0)={0,p>212r4−2p(sinθcosθ)2−p≠0,p≤2⇒{f is continuous for all(x,y), if p>2f is discontinuous at (0,0), if p≤2
解答:∫ln50∫5ex1lnydydx=∫51∫lny01lnydxdy=∫511dy=4
解答:f(x,y)=10xye−(x2+y2)⇒{fx=(10y−20x2y)e−(x2+y2)fy=(10x−20xy2)e−(x2+y2)⇒{fxx=(−60xy+40x3y)e−(x2+y2)fxy=(10−20x2−20y2+40x2y2)e−(x2+y2)fyy=(−60xy+40xy3)e−(x2+y2)⇒g(x,y)=fxxfyy−f2xy因此{fx=0⇒10y(1−2x2)=0⇒x=±1/√2或y=0fy=0⇒10x(1−2y2)=0⇒x=0或y=±1/√2⇒局部極值的候選點{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/e⇒共有3種不同的局部極值
解答:∫∞0∫∞0kxye−(x2+y2)dxdy=∫∞0[−12kye−(x2+y2)]|∞0dy=∫∞012kye−y2dy=[−14ke−y2]|∞0=14k=1⇒k=4
乙、計算、證明題:共3題,每題12分,共36分
解答:依題意:在x1p1+x2p2=L的條件下最大化U(x1,x2)因此x2=L−x1p1p2⇒∂∂x1x2=−p1p2⇒U(x1,x2)=U(x1,L−x1p1p2)⇒U′(x1)=0⇒∂∂x1U+∂∂x2U∂∂x1x2=0⇒∂∂x1U−∂∂x2Up1p2=0⇒∂U/∂x1∂U/∂x2=p1p2,故得證解答:a.級數:∞∑n=1(−1)nan,其中an=ln(1+1n)(1)limn→∞an=ln1=0(2)an−an+1=ln(n+1n)−ln(n+2n+1)=ln(n+1)2n(n+2)=lnn2+2n+1n2+2n=ln(1+1n2+2n)>0⇒an>an+1由(1)及(2)可得∞∑n=1(−1)nln(1+1n)收斂;又limn→∞n∑k=1ln(1+1k)=limn→∞ln2⋅3⋅4⋯(n+1)1⋅2⋅3⋯n=∞⇒∞∑n=1|(−1)nln(1+1n)|=∞∑n=1ln(1+1n)發散,因此原級數條件收斂b.∞∑n=1an收斂⇒limn→∞an=0⇒limn→∞n√|(3+sin(an)5)n|=limn→∞|3+sin(an)5|≤limn→∞3+|sin(an)|5=35<1⇒∞∑n=1(3+sin(an)5)n絕對收斂,即收斂,故得證
解答:水槽長度固定,因此容量大小取決於截面積;截面積=一個長形及兩個全等三角形=cosθ+2×12sinθcosθ⇒截面積f(θ)=cosθ+sinθcosθ⇒f′(θ)=−sinθ+cos2θ−sin2θ=−2sin2θ−sinθ+1因此f′(θ)=0⇒(2sinθ−1)(sinθ+1)=0⇒sinθ=1/2(−1不合,θ≠3π/2)⇒θ=30∘有最大的容量
==================== END ===========================
未公告答案,解題僅供參考
沒有留言:
張貼留言