臺灣綜合大學系統109學年度學士班轉學生聯合招生考試
科目名稱:微積分B
解答:(a){u=x⇒du=dxdv=sec2xdx⇒v=tanx⇒∫π/40xsec2xdx=[xtanx]|π/40−∫π/40tanxdx=[xtanx+ln|cosx|]|π/40=π4+ln√22=π4−12ln2(b)令u=x−3⇒∫54x−3√x2−6x+13dx=∫54x−3√(x−3)2+4dx=∫21u√u2+4du=[√u2+4]|21=2√2−√5
解答:u(x,y)=(ex−y6)∫2x−2√4−t2dt⇒∂u∂x=ex∫2x−2√4−t2dt+2(ex−y6)√4−4x2⇒∂u∂x|(0,1)=∫0−2√4−t2dt+4(1−16)=π+103
解答:h(s,t)=t4s−t2+tanslns⇒u(x,y)=h(x2+y2,3x−4y)=3x−4y4(x2+y2)−(3x−4y)2+tan(x2+y2)ln(x2+y2)⇒∂u∂y=−44(x2+y2)−(3x−4y)(2y)4(x2+y2)2+8(3x−4y)+sec2(x2+y2)⋅(2y)ln(x2+y2)−tan(x2+y2)(ln(x2+y2))2⋅2yx2+y2⇒∂u∂y|(2,0)=−14−0+48+0+0=1914
解答:
y=sin(x2)⇒x2=sin−1y⇒繞y軸旋轉體積=∫sin(1)0πsin−1ydy=π[xsin−1x+√1−x2]|sin(1)0=π(sin(1)+√1−sin2(1)−1)=π(sin1+cos1−1)
解答:ex=1+x+12!x2+13!x3+⋯+1k!xk+⋯⇒xex=x+x2+12!x3+13!x4+⋯+1k!xk+1+⋯⇒∫xexdx=∫(x+x2+12!x3+13!x4+⋯+1k!xk+1+⋯)dx⇒f(x)=xex−ex+C=12x2+13x3+14⋅2!x4+15⋅3!x5+⋯+1(k+2)k!xk+2+⋯⇒f(0)=−1+C=0⇒C=1⇒f(1)=1=∞∑k=01(k+2)k!
解答:假設圓心O,則小狗坐標為P(cos(0.1t),sin(0.1t)),其中t為時間單位(分鐘)小狗每分鐘走0.1公里,⌢AD=34π公里需時34π×10=152π分鐘因此h=¯PA=√(cos(0.1t)−1)2+sin2(0.1t)=√2−2cos(0.1t)⇒h′=0.1sin(0.1t)√2−2cos(0.1t)⇒h′(152π)=0.1sin(34π)√2−2cos(34π)=√220√2+√2
解答:f(x)=x5+2x3+2x⇒f′(x)=5x4+6x2+2⇒{f(1)=5f′(1)=13f−1(f(x))=x⇒ddxf−1(f(x))=1⇒(f−1)′(f(x))f′(x)=1⇒(f−1)′(5)f′(1)=1⇒(f−1)′(5)=1/f′(1)=1/13
解答:
解答:假設圓心O,則小狗坐標為P(cos(0.1t),sin(0.1t)),其中t為時間單位(分鐘)小狗每分鐘走0.1公里,⌢AD=34π公里需時34π×10=152π分鐘因此h=¯PA=√(cos(0.1t)−1)2+sin2(0.1t)=√2−2cos(0.1t)⇒h′=0.1sin(0.1t)√2−2cos(0.1t)⇒h′(152π)=0.1sin(34π)√2−2cos(34π)=√220√2+√2
解答:f(x)=x5+2x3+2x⇒f′(x)=5x4+6x2+2⇒{f(1)=5f′(1)=13f−1(f(x))=x⇒ddxf−1(f(x))=1⇒(f−1)′(f(x))f′(x)=1⇒(f−1)′(5)f′(1)=1⇒(f−1)′(5)=1/f′(1)=1/13
解答:
積分區域為四分之一環形,如上圖著色區域;因此取{x=rcosθy=rsinθ,則原式=∫π/20∫31rer2drdθ=∫π/20[12er2]|31dθ=π4(e8−e)
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