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2022年9月10日 星期六

109年台綜大轉學考-微積分B詳解

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)limx2x+4x7=2+427=65(b)limx04ln(17x)x=limx0(4ln(17x))(x)=limx028/(17x)1=28limx0(17x)4/x=limx0e4ln(17x)/x=e28
解答(a){u=xdu=dxdv=sec2xdxv=tanxπ/40xsec2xdx=[xtanx]|π/40π/40tanxdx=[xtanx+ln|cosx|]|π/40=π4+ln22=π412ln2(b)u=x354x3x26x+13dx=54x3(x3)2+4dx=21uu2+4du=[u2+4]|21=225
解答u(x,y)=(exy6)2x24t2dtux=ex2x24t2dt+2(exy6)44x2ux|(0,1)=024t2dt+4(116)=π+103
解答h(s,t)=t4st2+tanslnsu(x,y)=h(x2+y2,3x4y)=3x4y4(x2+y2)(3x4y)2+tan(x2+y2)ln(x2+y2)uy=44(x2+y2)(3x4y)(2y)4(x2+y2)2+8(3x4y)+sec2(x2+y2)(2y)ln(x2+y2)tan(x2+y2)(ln(x2+y2))22yx2+y2uy|(2,0)=140+48+0+0=1914
解答
y=sin(x2)x2=sin1yy=sin(1)0πsin1ydy=π[xsin1x+1x2]|sin(1)0=π(sin(1)+1sin2(1)1)=π(sin1+cos11)
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+)dxf(x)=xexex+C=12x2+13x3+142!x4+153!x5++1(k+2)k!xk+2+f(0)=1+C=0C=1f(1)=1=k=01(k+2)k!
解答OP(cos(0.1t),sin(0.1t)),t()0.1AD=34π34π×10=152πh=¯PA=(cos(0.1t)1)2+sin2(0.1t)=22cos(0.1t)h=0.1sin(0.1t)22cos(0.1t)h(152π)=0.1sin(34π)22cos(34π)=2202+2
解答f(x)=x5+2x3+2xf(x)=5x4+6x2+2{f(1)=5f(1)=13f1(f(x))=xddxf1(f(x))=1(f1)(f(x))f(x)=1(f1)(5)f(1)=1(f1)(5)=1/f(1)=1/13
解答
{x=rcosθy=rsinθ=π/2031rer2drdθ=π/20[12er2]|31dθ=π4(e8e)
解答{f(x,y,z)=exyzg(x,y,z)=x3y3+z324{fx=λgxfy=λgyfz=λgzg=0{yzexyz=λ(3x2)(1)xzexyz=λ(3y2)(2)xyexyz=λ(3z2)(3){(1)/(2)y/x=x2/y2(2)/(3)z/y=y2/z2(3)/(1)x/z=z2/x2{y3=x2z3=x3g=03x324=0x=2{y=2z=2f(2,2,2)=1/e8

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