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# Class Note for MATH 1432 at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 30 views.

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Date Created: 02/06/15

Lecture 3Section 73 The Logarithm Function7 Part II Jiwen He Section 7 2 Highlights X area of shaded region 2 Lx YA yzy 2 y Inv 1 I In 2 1 I I gt 2 3 5 x 1 Properties of the Log Function 1 l d l o lnz 1 gdt lnz g gt 0 o lnl 07 lne ll 0 lnzy lnz lny7 lnzy lnz 7 lnyi o lnzr Tlnz7 lneT Ti 0 domain 0007 range 70000 0 lim lnz7oo7 lim lnzoo 170 EH00 lnz o hm 07 hm IT lnz 0 0700 17 x70 Limits 1 Differentiation and Graphing 11 Chain Rule Differentiation Chain Rule Theorem 1 d l d d lnuz form 575 gt 0 d d du ldu Proof By the Chain rule7 E nu 7 lnu 7 1 d 2 7 1 d 2 7 1 7 2x Examples 2 o ln Jrz71I2dzlz711221 7 11 forallz 5 lz2gt0i d 1 d 3 1 o lnl31713IUSIgt71313713I7forallz gt 7E lt l31gt0i Example Example 3 Find the domain off and nd if lnz4 12 Solution 0 For I E domainf7 we need zxil 12 gt 07 thus I gt 0 0 Before differentiating f7 simplify it lnx4 952 11196 lnltv4 952 11196 5 1114 952 0 Thus 1 1 1 1 1 1 1 x 7 7 4 2 7 7 2 7 1124zzx 112412 x x4z2 12 Graphing Example VA point of inflection minimum value E 225 vertical asymptote x 1 GUIb f 77777 770 I I I f39 l 4 2 x deCreaSeS 3 increases fquot O 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 77 I I graph 1 concave up 2 concave down X 4 Example 4 Let ln Specify the domain of On What intervals does f increase Decrease Find the extrem values of fDetermine the concavity and in ection points Skectch the graph specifying the asymptotes Solutior For x 1 gt 0 We need at gt 1 thus 96 7 domainf 100 0 Simplify 4lnz7lnz 71 Then i 7 L 0 Thus f1 on1and1 on 00 0 Atz f x0 Thus f 41n4731n3225 is the only local and absolute minimum 4 1 z 7 23x 7 2 7 7 77 7 7 o Fromf 7 z 171 We have f x2 I 71V 1206 71y 0 At 95 2 f x O domainf is ignored Then the graph is concave up on 1 2 concave doWn on 200 0 The point 2 24ln2 Z 2277 is the only point of in ection 0 From g 7 i We have lim 700 lim 0 171 1700 0 From 4lnz 7 lnz 71We have limJr oo lim 00 171 00700 0 The line at 1 is a vertical asymptote Example VA I L 1 2 x L e e fxxnx fa 7 7 7 7 770 I I l 1 2 J decreases 8 increases Example 5 Let zln x Specify the domain of f and nd the inter cepts On What intervals does f increase Decrease Find the extrern Values of f Determine the concavity and in ection points Skectch the graph Solution 0 lnx is de ned only for x gt 07 thus domainf 07 0 There is no yintercepti Since f1 1 ln1 07 z l is the only zintercepti o Wehavefzzllnzllnzi z oForf z0Wehave llnz0 lnzil zli e Thus f l on 07 andT on o Atz 0 Thus ln 7 m 70368 is the only local and absolute minimumi 0 From llnzwe have f z i gt 07 for all z gt 0 o Then7 the graph is concave up on 000 0 There is no point of in ectioni 0 From l lnz7 we have 133 Hz foo 131010 1 oo 0 From zlnz7 we have 133 1 07 113010 1 00 Quiz Quiz 1 lnl 7 a 71 b 07 c 1 2i lne 7 a 07 b 17 c e 2 ln 21 Properties fr1nlzl7 z y o J A yn X ynx gt X y I n M Graph The graph has two branches 3 1117967 x lt O and y 111957 at gt 07 each is the mirror image of the other Theorem 6 d 1 1 Eltln x ltgt dzln x 0 d d 1 Proof 0 For x gt 07 01195 d d 1 d 1 o For x lt 07 EOIKix D Power Rule z 195 Power Rule 111n107 ifna 71 1nd1 n lnl1lC7 ifn 711 E1ample 7 I1d1 li d1lnl1lilCl 12 1 12 1 22 Chain Rule Differentiation Chain Rule Theorem 8 d 1 d d for 1 575 f 01 d d du 1 du Proof By the Cham rule7 7 7 D d 3 1 d 3 7312 E1ample59 o ln 171 71m1 d 1 7 1 d d 1 oltln172 Elnl171 i lnl172l 1717 1 1 7 2 23 Logarithmic Differentiation Logarithmic Differentiation Theorem 10 Let 91 911 gg1 Then W I 911 921 ms7navgt 5 gt 5 ltglltzgt g2ltzgt we Proof 0 First Write 111M9011n1911139192Ilquot3919n11 1n191111n19211quot391n19n1l 0 Then differentiate MI 911 y W m 921 91 911 921 97M 0 Multiplying by 91 gives the result D Examples Examples 11 Find if 0 91 11 71z 72z 73 7 I2 1321 7 52 0 91 W Solution lngz ln z ln z71 1n z72 4r1n z73 i mm1gg1 1 gz7z I71 I72 173 gzzz71z72z73ltIi1zi2ri3gti lngz 31n 121 21n 2175 721n z25 i g z 7 21 2 7 21 gz7312122175 212 gz 12 1321 7 52 61 4 7 4x 12 2 z21 2175 12 Quiz cont Quiz cont 3 11131111 z a 7007 b 07 C 00 4 lim lnz z a 7007 b 07 C 00 3 Integration and Trigonometric Functions 31 uSubstitution Integration uSubstitution Theorem 12 gI n z z gaymelwltna 0 10 Proof Let u 91 thus du g zdz then gz 7 lunu n z gzdzud 1110 1go 2 Example 13 Calculate Let u 17 413 thus du ilZIQdI 7 1 th id 7 i 7d 7 i11107 i111 4310 en 17435317 12 uu 12D 12n I Examples uSubstit ut ion ln 1 Examples 14 o 7dzi 39 md 2 6122 o Sidzi 1 z zl Solution 1 Set u lnz7 du idzi Then I lnz 7 712 71 2 sziudu72u C72lnz 0 SetulE7 du I Then 1 T m 1 d 21d 21HC 211f0 I iu nu n I 1 1E u Setuzgzl7 du312ld11Atzlu3atz2ulli Then 2 2 11 Slidz 2 ldu 2ln uH1 2lnll iln31 1 z zl 3 u 0 Natural log an39ses only When integrating a quotient Whose numerator is the derivative of its denominator or a constant multiple of it gI 7 1111 111 n I gltxdzud 111c 1M Ho 11 32 Trigonometric Functions Integration of Trigonometric Functions d Recall that lex Coszdz sinz C 42gt d7sinz coszi z d sinzdz 7coszC 42gt d700sz 7sinzi 1exsec2zdz z d tanz C 42gt d7tanz setin Cs02zdz 7cotz C 42gt z 7cotz 7csc2 zi lex secz tanzdz secz C 42gt 7secz dz dz secz tanzi Cscz cotzdz 7csczC 42gt jcscz 7cscz cotzi z New Integration Formulas Integration of Trigonometric Functions tanzdz 71n 00sz Ci Cotzdz lnisinz Ci seczdz lnisecztanziCi Csczdzlnicscz7cotz Ci Proof Set u cosz7 du 7 sinzdz7 then sinz 1 tanzdz dz7 7du7ln u C cosz u 71n cosz C Set u sinz7 du coszdz7 then CotzdzCSxdzlduln u c s1nz u lnisinz C Set u secz tanz7 du secz tanz sec2 z dz7 then sec z tanz sec z tanz sec2 z seczdz sec dz dz secztanz secztanz 1 7du ln u C lnisecztanzi 0 u Set u cscz 7 cot z7 du 7 cscz cotz csc2 z dz7 then cscz 7 cotz 7cscz cotz cs02z cscz dz csc dz dz cscz 7cotz cscz 7 cotz 1 7duln u Clnicscz7cotz Ci u 12 d Examples l u 2 Examples 15 0 13 It 0 zsecz2dzl tanln x M I Solution Set u 1tan317 du 3seC2 31 dz 860231 1 1 id 7 id 1tan31 I Su u lnlulCln 1tanSIlCl Setu12du21dzz 2 1 1 zsecz dz secudu ln secutanulC 1 Eln secz2tan12 Cl Set u lnz7 du idz wdztanudulnlseculc I ln seclnz Cl Outline Contents 1 Differentiation 3 11 Chain Rule l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 3 1 2 Graphing l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 4 2 mm 7 21 Properties l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 7 22 Chain Rule l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 9 213 Log Differentiation l l l l l l l l l l l l l l l l l l l l l l l l l l 9 3 Integration 10 3 1 uSubstitution l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 10 3 2 Trigonometric Functions l l l l l l l l l l l l l l l l l l l l l l l 12 13

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