11.æ±å½æ°y=x³-2x²+x+5å¨åºé´[-1ï¼1]ä¸çæ大å¼åæå°å¼
.解ï¼ä»¤yâ²=3x²-4x+1=(3x-1)(x-1)=0ï¼å¾é©»ç¹x₁=1/3ï¼x₂=1ï¼x₁æ¯æ大ç¹ï¼x₂æ¯æå°ç¹ï¼
x₁ï¼x₂â[-1ï¼1]ï¼æ
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çæ大å¼=y(1/3)=1/27-2/9+1/3+5=139/27ï¼æå°å¼=y(1)
=1-2+1+5=5ï¼å¨åºé´å·¦ç«¯ç¹x=-1æ¶y(-1)=-1-2-1+5=3ï¼æ
å¨[-1ï¼1]ä¸çæ大å¼ä¸º139/27=5.148ï¼
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12ãæ±ä¸å®æåï¼
(1)ãâ«[2x/(1+x²)]dx=â«d(1+x²)/(1+x²)=ln(1+x²)+C
(2)ãâ«(xe^x+cosx)dx=â«x(e^x)dx+â«cosxdx=â«xd(e^x)+sinx=xe^x-â«(e^x)dx+sinx=xe^x-e^x+sinx+C
=(x-1)e^x+sinx+Cï¼
13ãå·²ç¥y=xln[x+â(1+x²)]ï¼æ±dy/dx
解ï¼dy/dx=ln[x+â(1+x²)]+x[1+x/â(1+x²)]/[x+â(1+x²)]=ln[x+â(1+x²)]+x/â(1+x²)
14ãæ±æéx➔âlim[(4x²-3)³(3x-2)⁴/(6x²+7)⁵]ãåååæ¯çxçæé«æ¬¡æ°é½æ¯10次ï¼æ
å
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å
¶æé«æ¬¡æ°é¡¹çç³»æ°æ¯ã
解ï¼åå¼=(4³Ã3⁴)/6⁵=5184/7776=162/243=2/3
11ãæ±å®ç§¯åã0ï¼1ãâ«(2xsinx²+xe^x)dx
解ï¼åå¼=ã0ï¼1ã[â«sinx²d(x²)+â«xd(e^x)]=[-cosx²+(x-1)e^x]â£ã0ï¼1ã=-cos1-(-cos0-1)=2-cos1
12ãæ±å½æ°f(x)=4x³-5x²+6å¨å
¨é¨å®æ°åä¸çæå¼ã
解ï¼ä»¤dy/dx=12x²-10x=2x(6x-5)=0ï¼å¾é©»ç¹x₁=0ï¼x₂=5/6ï¼x₁æ¯æ大ç¹ï¼x₂æ¯æå°ç¹ï¼
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æ大å¼f(x)=f(0)=6ï¼æå°å¼f(x)=f(5/6)=4Ã(5/6)³-5Ã(5/6)²+6=523/108ï¼
13ãæ±ä¸å®åâ«x³e^(x²)dx
解ï¼åå¼=(1/2)â«x²d(e^x²)=(1/2)[x²e^(x²)-2â«xe^(x²)dx]=(1/2)[x²e^(x²)]-(1/2)â«d(e^x²)
=(1/2)[x²e^(x²)]-(1/2)(e^x²)+C=(1/2)(x²-1)e^(x²)+C
14ãæ±æ²çº¿y=x³+3x²-x-1çå¹å¸åºé´ä¸æç¹åæ
解ï¼ä»¤y'=3x²+6x-1=0ï¼y"=6x+6=6(x+1)ï¼ç±äºå¨(-âï¼-1]ä¸y"<0ï¼æ
æ²çº¿å¨åºé´(-âï¼-1]ä¸åä¸å¹ï¼å¨[-1ï¼+â)ä¸y">0ï¼æ
æ²çº¿å¨åºé´[-1ï¼+â)ä¸åä¸å¹ãæç¹åæ 为(-1ï¼2)ã
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