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Calculus I

by: Alvena McDermott

Calculus I MATH 1431

Alvena McDermott
GPA 3.69


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This 31 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1431 at University of Houston taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/208407/math-1431-university-of-houston in Mathmatics at University of Houston.


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Date Created: 09/19/15
1 LGCture 17Sections 5154 Integration J iwen He Area and De nite Integral y A Sections 51 85 52 Area and De nite Integral b Area of Q 2 dCL De nite Integral and Lower Upper Sums y y Q192 Q3 9n ax0 X1X2 x3 xnb x area of shaded region is a lower sum forf gtltv area of shaded region is an upper sum forf Area of Q 2 Area of 21 Area of 22 o o o Area of n LfP mlAcrl mgAcvg o o o mnAacn b LfP g f dCL S UfP for all partitions P of a b Example ff 962 dyc y J 1 g 2 3 x 122 3 A Upper sum Lower sum 1 9 1 45 LfP i 15 41 g m 5625 9 1 1 Ufa Z5 4 91 x 12125 3 LfP lt 132dac 33 13 m 8667 g Ufa 1 Lower Upper Sums and Riemann Sums V I I I I I I I I I I l I X17 I 7C2 I 3C3v X4 F 10 X1 x2 A3 x4 395 6 7 A8 5P ffIJTArIJ1 ffIJ AJJ2 ffIJZAfL39n LfP g SP g UfP for all partitions P of Ia b 2 dz 2 1 Example f SltPgt flt gtlt gt 1133 12 lt1 m 8563 LfP m 5625 Ufa x 12125 3 1 26 x2d 33 13 m 8667 1 3 3 b LfP g SP fxdx g UfP De nite Integral as the Limit of Riemann Sums Theorem 1 b x dx lllgillrgo fiA1 l fA2 fAn 2 Sections 53 amp 54 De nite Integral and An tiderivative De nite Integral and Antiderivative a H J b Fx area from a to x and Fx h area from a to x h Therefore Fx h Fx 2 area from x to x h 2fx h if h is small and Fxh Fx 5 fx 2 h h fa Theorem 2 Let Fm E m dt Then F f for all x in a b Example for r1752 dt 3 1 Sketch the graph of 0 1 t2 dt y A X 1 2 F 12 gt07 122 SigHOfFquot 0 behavior of graph concave o concave u p T down point of inflection 1 1 t2 dt tan1 5v Note that o Fundamental Theorem of Integral Calculus Theorem 3 In general f1dx Fb Fm where is an antiderivative 0f Function Antiderivative xrl xquot r a rational number 7E l r l sinx cos x cos x sin x S602 x tan x sec x tan x sec x c502 x cot x cscx cot x csc x Example J13 x2 dx Evaluate x2 dx 1 fx x2 mlw 26 Area of the shaded region Ifo dx Step 139 Get an antiderivative for fz 12 Fz zg Step 21 312d1 F3 7 F1 1 3 1 3726 gc 71 7 3 12 gt 0 gt 12 dz area of the shaded region 1 Example 7r2 Evaluate sin 1 dm 42 2 2 smz dz Step 139 Get an antiderivative for m 7 5m Fm 7 7 0053 Step 239 12 5mm 7 mm 7 F77r2 774mm 7 MOSH2 0 7r2 sinz y 0 gt sinzdz is not an area 42 Further Properties of Integral Calculus Theorem 4 1 AbfzgzdzAbfzdzzbgzdz 2 Abafzdzaabfzdz 5 Ab dzAcfzdzzbfzdz 4 Iff 29 on m then Ab zwzzabng Examllnles Examp es 5 Evaluate 1 ll 2176z45dz 0 2 4 z ldx 1 z 3 Algi fdz 7r4 4 secz2 tan175seczdz 0 Lecture 2 Sectmn 2 2 ommnn n1 mun Sectmn 2 3 Same mun Thenvems 2W He Wm M WM WW M mm quotA iivlavamim22 m L n nhkmgihehm io Khan Izsxzvvrmzhst mg m mn evwha ev 39 sda vad 21 z m m MW 1 s dam were a m on mm Miners 5 themusthzt 39 mg on when x 5 quotan m 39x z mm m a 39 w cm 9390 calyx 013 my Usememphwwm 2M5 ragga og39m dumx mewphmlmim 2W3 MMX aggxm dumx em 0 Wu g Wu h mm H m m zsm Useihe w m 39 m n mm b hm m d W o A a a mu Momma KhmrlxL m Immaa Weszyihzi Ly x 7 L Nuveazh 5 gt u 0972963 2 gtEI mm NEIlt X 4a 09quot WW Section 22 r Defrnltlon of Limit Examles Four Basic Limits Choice of 6 Depending on the Choice of e ylk m2 lt f L L61 I 7 Li 6 T 1 bel 4 i i l I l I c 6 x1 0 x2c6 3 H Jiwen He University of Houston Math 1431 Section 24076 Lecture 2 August 28 2008 9 22 Section 22 y L 6 L1 L EK J v C lt 5 lt J6 3 C C C 63 2C 61C i 51 2C63 Which of the 639s works for the given 6 a 51 b 52 c 53 Jiwen He University of Houston Math 1431 Section 24075 Lecture 2 August 28 2008 10 22 Section 22 For which of the 6 s given does the specified 6 work a 61 b 62 C 63 Jiwen He University of Houston Math 1431 Section 24076 Lecture 2 August 28 2008 11 22 5mm hm2xr he Latsgt Wezho gumm n Dlt x72 ltamp than Maya w W m 5m Mith 73x m 5 gt n We mm gk u m wuq 44nd 097v K2434 a 9wsz x2 Latsgt mam lt xr ltamp ihen 4Q a 5m m m Latsgt Wezhmazwmm lt a then u 44 Shawnaka k Latsgt Weanzmoseznynumbev5gt suzhimi ulth Ra than Vim Section 22 W 39 BasicLimitIirn r X gtC 2 5 fx 6 Show that for C gt 0 2 26 Iim E X gtC Hmf2 xa4 Let 6 gt 0 We choose 6 minc e such that ifOltix cilt6 then i Eilte Jiwen He University of Houston Math 1431 Section 24076 Lecture 2 August 28 22008 17 22 4 m and max m Mm 39x x Aquot 39lt n o m m Hm zmqmm alw z x za m x x m m Ax u hm m and mm m am mg 39X my o thw mmm h A 4 m ax u m m mm s u 41mm m not w m hm i miy may quot H591 m m z w x in be a WWW m1 5 be W quotmquot mquot I m m Ame 2 72x73 24 42 724 73 z 2 Magma MEI577EI22EIBB


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