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Differential Rules and Applications in Calculus Explained

School
University of Alberta**We aren't endorsed by this school
Course
MATH 100
Subject
Mathematics
Date
Dec 12, 2024
Pages
12
Uploaded by MagistrateProton22759
Chapter3DifferentialRules&ApplicationsofDifferentiationCRITICALNUMBERSOFTHEFUNCTION1)f(x):2x-3x25.1g(x)=Ex9)f(x)=4x2- 9xx(2x+3f(x)=26xg(x)=xf'(x)=12x218x-12let/'(x)=0g'(x)=-('(x)=G(dX2-3x2)26X=0letf'(x)=0=le+q'(x)=00=6(2x23x2):X=0=6(2x2-4x+x2)-4.14X=t6.)g(x):(x+110=6(2x(X2)+1(x2))4+132)((x)=5+8Xq'(x)=2x+1=0X=1/2ā†‘'(X)=8X-2=jX=2letf(x)=0letq'(x)=0:.NONE8:X+110)S(t)=2t"+3t2-6t+43)((x)=X"3x+1X=-1s'(t)=6t2+6t-6f(x)=3x23s'(l)=6(+2+t-1)letf(x)=0M)((x)=516x2xlets'(t)=00:3x23l'(X)=6-6x20:6ItIt-17=X=IX=b=4a)2aX==18)/(H)=273+372+6t+4if(t)=Gt2+6t+64.)((t)=+3+6t2+3t-1f'(t)=G(+2+t+1)f'(l):3t2+12t+3letfll):0letf'(t)=0:32+12t+311)((t):+414t3+atJa!"I:4516213t+11X=b=m4as(t)=4+3+12+2+47x=-12!4(3)(3)X=UNDEFINEDs(t):02(3)X=0X:Elit:X=NONE*
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12)fil15.7VIX1:FV(X)=X(x2)1)ru+1) Iv'(x)=(X2)+(xt-(x2i)flo:v'(x)=(x-2)+ 2v'(X)=2(X2)+XI'Ir):L(X-2)'lalet!'(r)=0v'(x)=2x-4+xG(X2)10=p+1,v2=1r=If~'(x)=V'(x)=0v'(X):GNE13.)f(0):sir128)0:3x4X=2+'(0):</sin1281)Ā·Cos(20):2x=%'10):4sinGECOS2t-(indoĀ·16)T(X1=x 2(2x-17"i'(X)=<x(2x-1)+(x-2. (ax-17.".2]:<(sin40)T'(x)2x(2x1)%+4x212++3(2X17"i'(x)=6x(2x1)+4x14)g(8)=Osint3(2X1)"q10)=1+costi'(x)-xletgf)=gO=ItcostCost=1i'(x):16x"Ex-=T3(2x-1)+'(x)=DNEX=i+hi,neIi'(x)=2x(8x3)3(2X-1)"3X=-ItI'l=0X=0020
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MeanValueTheorem19.1f(x)=1X2,10,371.)((x1=x2-4x+5,(1,5]Man=f (3)-f(0)3 -fMan=((b)-fla)baMan:1-1312-11-01=/(5)fill35-110-2=52415)+5[12-4(1)+5]:19=4f'(x)=2xf'(c=2=25-20151+454-2=Ef(x)=GX-4=8/4=2c=22f'()=2c4=G2)-4=h2=2+4āŠ„18)f(x):x 3-2X+1(-2,3)f'(l)=3x2-2Man=f(3)-f(-2)f'(c)=3c2-2=53(2)32=5+2Max=332(3771-((-2)"2(2)+1)E5:FleMay=27 -4+1-(8+4+1)5Man=276+1+84-1=255
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21)f(x):+,11,2)20.1f(x)=2x3+x=-X1,(0,2]Man=f(27-f(0)Man:Ef()20Man=((2)3+1213-2-1-1-17Ā·It:+=2May=16+43+1=a2f(x)=x(0)=f'(x)=6x2+2x-1f'(l)=6c=+2.1=&t'(c):t=62+2cq-1=0Gc2+2c-10=02)(3x2+c5)=0=-1:c="c=b=b2Ha11,27:cre2a==IIF41311+5)2(3)"
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24)62.)f(x)=x,(1,4)fiscont(2.5112f'(x)14(2,5)Man=f(4)fillshowthat3[f(5)-f(e)<123-Man=FrhypothesisofMeanValueTheorem!'(2)=/(5)(12)l'x1=I'lltest5-2:f'(c)=1151-f(a)=f(x)3:f(d)-(15)-f(2)1fla)q=c323.7/x):ItETF;(2,971:((2)Man=f(q)-f(2)31(15) - f(2)112MmanLetmas=iI'(x)=1.(x-172ā†‘'Ill=31c-1):12-12":-(1:c=(1)2+1
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INCREASING/DECREASING26)f(x)=X5+4x425.)f(x)=x3+2x2- X+1f'(x)=3x2+4x-1f(x)=5x"+12x2let!'(x)=0letf(x)=05x*+12x2=00=3x2+4x-1x(5x21(2)=0Ā·x=-undefine:fisincreasingOnall#S2) 7f(x)=Glanx-lan=Xf(x)=2secix-(2-tanxĀ·secix]=Isec*x(1-tanx]intervalf(x)inc/desf(x)=01- tanx=01=tanxx=Ļ€(4+In10.EnincCON(A vI+Yintervalf"(X)12)dec28)y=6x22x3xY7- 0,162)CDy=12x6x2-4X(162,062)+wlat,)incy"=1212x12x21062,2)CDcurveupalety"=0intervalsā†‘(-2)=3(2)2+4(2)10=12-12x-12x2=1281=30:12(1X-x2)I.0:-12(x21x-1)+107:-*:11,14111(1)X=- 1618033989:INC:1-c,-2- m)&(2,a)2(1)X=06180339893IDEC:1-2-Els,-2+/**(-115/a
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29)y=X(1+x)"y"=(1+x 7"2(8+8x=2x+Rx-(2x(x))d4).2xIt4)((tx)3y=2x(1+x)X2911+x7(01xA s2(1+x)"(1+x)%::YXx)1"OXTy)=4X+4x2X22(1+x)3/g':elEy"=((1+x(3.(4+6x)(6 ---(1+x)x.(4x+3xz)]:(2(1+x)3/2)2eleforfunctionf(x]y"=(8+12x)(1+x)*-(((2x+axz(1+x)"]Y(1+x)3y"=(8+12x)(1+x73/(12x19x2)(1+x)"24(1+x)3y"=(1+x)'/18+12x)(1+x)12x-ax2)4(1+x)3
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30.)y:x)eintervaly"(-20.-1)-CDy=(1+x)(20(1+x)Ā·x)(-1,2)=>LD(1+x)Y(2,c)ā†‘Chy=(+2x+x=-X-2x2curveuponinterval(1+x)4(G,c)9:y"=(11x)"-2x-(4(1+x)3(1x2)](1+x):y"=2x((+x)"-((+x(3(44x4](1+x):y"=(1+x))2x(1+x)-4+4x*](1+x)y"=-2x-2x24+4xz(1+x)5y":2x2-2x-4y":(1+x)5y"=0(x2)(X+1)X=2X=+
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31)y=Xintervalsy1cu/cpx2-3(-0,3)ā‘”1-3,0)=>y=(x23)(3xx(2x-x3]10.3)#13,c)#(x2-3)2y=3xY9x22x4(x23)2y=X*-9x2(x=3)2y"=(x2=3)2Ā·(4x-18x)-(2(x2-3).2x(x*-9x1](x2- 3)4y"=(x23)2(4x3-18x)-4x(x23)(X"9x?)(x2-3)4y"=(x2 -3)((x2 -3)(4x3-18x)-4x(X*-9x2)](x2-3)4y"=(x23)(#518x312x2-54x4x5+36x3](x2-3)4y"=Gx354X=bx(x2-9)(x?- 3)3(x2 -3)3y"=0X=0,3,3
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CURVESKETCHINGE.7INC/DEC32)y=1+xy=Itx21X21-x2A)DOMAINy=(1-xz)(2x)(2x(1+x2)]11-x2)2Domainy=X F11,XERy=2x2x3)2x2x3)(1-x2)2B.)INTERCEPTXinty1=Gx-x+2x+2x3lety=0noxint0:1+x2(l-xe)2letX=0:Ho,yin:yeX=Iintervalsy'INC/DECc)Symmetry=f(-x)=f(x):loā†‘Co,1tINCsymmetricabouttheyaxis(1,a)tincwD)ASIMPTOTESF.)localminInm10,1)HAy=lim1+x2-alim=AX+1-1-x 2X=1lim
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6)concavity5:y"=11-X12.4-(2/1-X4-2x.4x](1-y2)4y"=4(1x 2)21-16x2(1X)(l-X2)4y"=4(1x22+16x211-x2(l-x2)"y":4(1X2(1x"+4x)11-x2)Yy"=4)cxi+4x)(l-x2)3y"=H4x2+164(ly2)3x==16!4(4-4)214)y"-
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407hyperbolaye-X==H49.71"(x)=1closestto(2.0f(x)=X+C41)53.)[4)f(x)=+x+(x+4INTROTOINTEGRATION50.)f"(x)=sinxANTIDERIVATIVESf(x)=cosX+Cfindf(x)f(x):-Sinx+(x+D47.)+"(x)=X*+x351)&"(x)=24Xf(x)=x+f"(x)=24(t)x+2((x)=!(4)x"++(f)x5+xx+Df(x)=12x+ef(x)=(x"+2x+Df(x)=+(x+Df(x)=x+Exit52.)g(X):E+z=248)f"(x)=GOx"45x2f(x)=y+f"(x):2x+=Ef(x)=10x515xf(x)=2.Ext+exf(x)=12x5-15x3+Cf(x)=1x+exf(x)=12x"Ex+exf(x):.x+Ex+Dxf(x)=2x"-15x"+(x+bf(x):<Ex+Bx
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