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# DIFFEREN AND INTEGRAL CALCULUS M 408C

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This 9 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 408C at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 28 views. For similar materials see /class/181489/m-408c-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15

M408C Final Exam Review Instructor Guoliang Wu December 27 2010 1 Exam Time and Location If you lectures are MWF 1111250am Unique 7s 545255453054535 your nal exam WEDNESDAY DECEMBER 08 912noon in CMA A2320 If you lectures are MWF 22250pm Unique 7s 545855459054595 please pay special attention be Ch5 cause there was a mistake in the syllabuei Your correct nal exam schedul e SATURDAY7 DECEMBER 117 912noon in CMA A2320 You can double check your exam schedule on the Registrar s website 2 Covered Sections Ch2i Limits and Rates of Change 21 The Tangent and Velocity Problems 22 The Limit of a Function 23 Calculating Limits Using the Limit Laws 24 not cov ered 2 5 Continuity Ch3i Derivatives 3 1 Derivatives and Rates of Change 32 The Derivative as a Function 33 Differentiation Formulas 3 4 Derivatives of Trigonometric Functions 35 The Chain Rule 36 Implicit Differentiation 37 Rates of Change in the Natural and Social Sciences Ch4i Applications of Differentiation 41 Maximum and Minimum Values 42 The Mean Value Theorem 43 How Derivatives Affect the Shape of a Graph F g Limits at In nity Horizontal Asymptotes 45 Summary of Curve Sketching 4 7 Optimization Problems 49 Antiderivatives i Integrals 5 1 Areas and Distances 52 The De nite Integral 53 The Fundamental Theorem of Calculus 5 4 Inde nite Integrals and the Net Change The orem 55 The Substitution Rule i Applications of Integration 61 Areas between Curves 62 Volumes i Inverse Functions Exp Log and Inverse Trig 7 1 Inverse Functions 72 Exponential Functions Their Derivatives 7 3 Logarithmic Functions 74 Derivatives of Logarithmic Functions 76 Inverse Trigonometric Functions i Techniques of Integration 81 Integration by Parts 82 Trigonometric Integrals 8 3 Trigonometric Substitution 8 4 Integration of Rational Functions by Partial Fractions 85 Strategy for Integration 3 Recommended Review Mate rials 11 Homework assignments 7 Quest and Written 21 Practice Exam posted later weekend l24 l25 Check Blackboard 31 Midterm exams 41 Other practice problems from the textbook andor TA andor SLCC 4 Review Problems for 123 Most of these questions are selected from written home work Those with a checkmark are intended to be discussed in lecture 12 7 21 11 l39 x71gtg1 12 7 4x 4 W75 21 lim x7gt74 z 4 3 Evaluate the limit 1 1 3x 3 311 1 412 3x4 41 Use the Intermediate Value Theorem to show that there is a root of the equation 2 sin I 3 7 217 in the given interval 07 l 51 Find fa where a f 2 and z2l fa I721 61 Find fz where z y 0 and 12 sin7 if I f 07 fx 0 11101 71 Find f z where 35 71751 81 Let 12 if I g 2 7 7211 ifz gt 21 Find the values of m and b that make f differen tiable everywhere 9 Differentiate y csc 99 cot 91 101 H H H 93 H F H 5 H 03 H 7 H 5 0 H to Find the derivative of the function 72 y CLer 12b2gt 7 where a and b are constants 1 Find y by implicit differentiation z4y4 11 1 If a tank holds 5000 gallons of water which drains from the bottom of the tank in 40 minutes then Torricelli s Law gives the volume V of water re maining in the tank after t minutes as t 2 V5000177 0lttlt40 lt 40gt H 1 Find the rate at which water is draining from the tank after 10 minutes 2 At what time is the water owing out the fastest The slowest Find the absolute maximum and absolute mini mum of the function IV 4 7 12 on 712 Find the limit 1 915 7 z xAHPoo 13 l Sketch the curve sin I y2cosz A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building xFind that satis es f t9 sint9 cos 9 37 f0 41 Find the derivative g z of the function 2 E l idti 91 2t4 1 Evaluate the integral 7r4lcos2t9d6 1 cos2 9 0 O i Find the area of the region enclosed by the curves yz3 yz32i 0 H i Find the volume of the cap of a sphere With ra dius 7 and height hi 0 0 i Find f 12 Where 13 3sinz2coszi i Find the absolute minimum value of the function 0 CA3 61 7 0 91 171 gt i Find the limit 0 g ln12 71 7lnz 1 7 lnz lim 100 0 01 i For each of the following functions nd the do main and then differentiate it 251 lnlnln 252 91 sin 121 1 0 on i Evaluate the integral z lnz 1dzi 27 csc4 z cot6 zdz 12 1 28 d 1272122 I 1 29 7d z E I The next page Will be attached to the nal exami Limit Formulas 111 I zny nrn7l cosz 7 7s1nz tanz sec 1 2 cot z 7csc21 secz secz tanz cscz 7 cscz cotz 61y 1 a a lna7 a gt 0 lnz l 3 1 arcsmz 7 17 71 lt I lt1 arccosz 7J1i 77l lt I lt1 arctan z 1 112 arccot z 7 Basic Integrals dz zn1 CJL 7l f dxdz ln z C femdz e C faxdz fsinzdz 7coszC fcoszdz sinz C fsec2 zdz tanz C fcsc21dz 7cotz C fsecztanzdz secz C fcsczcotzdz 7cscz C Vii 7 arcsin C 1 man a o fseczdz ln sechrtanz C ftanzdz ln secz C fcsczdz ln cscz7cotz C fcotzdz ln sinz C Algebraic Formulas 13 y3 7 I y12 7 Iy f 13 7 y3 7 I 7 yac2 zy y2 acy3 13 312y31y2 y3 I 7 y3 7 IS 7 312y 3zy2 7 y3 Trigonometric Formulas csc9 sec 6959 tan 7 53 cot9 7 3 sin2 9 cos2 l ltan2 9 sec2 9 lcot2 9 csc2 9 sin79 7 sin9 cos79 cos9 tan79 7 tan9 sin7r2 7 9 cos9 cos7r2 7 9 sin9 tan7r2 7 9 cot 9 sin7r 7 9 sin9 7 sin7r 9 7sin27r 7 9 cos7r 7 9 7cos9 cos7r 9 7 cos27r 7 9 sin 7r 7 9 7 cos9 sin7r 9 cos n 7 9 7 sin9 7cos7r 9 Addition and Subtraction Formulas sinz y sinz cosy cosz siny sinz 7 y sinz cosy 7 cosz siny cosz y cosz cosy 7 sinz siny cosz 7 y cosz cosy sinz siny tanltx y tanxtany WWI 7 y anyy sinz cosy lsinz 7 y sinz sinz siny cosz 7 y 7 cosz cosz cos y cosz 7 y cosz Double and HalfAngle Formulas sin 21 2 sin 1 cos 1 cos 2x cos2 2 z2cos217ll72sin21 z 7 sin 7 tanx tan2z 7 17mquot 1 1121 1 cgsh COSz I 1cgs2x Functions of Angles martinez om3232 7 HW07 7 freeman 7 54450 1 This print out should have 20 questions Multiple choice questions may continue on the next column or page 7 find all choices before answering 001 100 points Consider the following functions F1ac sin2ac 77r4 x S 7r4 F2ac 2x 7l 1 F3ac lxl 713x31 Which have an inverse on the given domain 8 1 1 None of these 4 2 All of these 3 F2 and F3 only 4 F2 only 5 F1 only 8 6 F1 and F2 only 2 7 F3 only 4 lt 8 F1 and F3 only 78 74 74 002 100 points 78 Suppose f is a 1 1 function With domain A and range B What is the range of fil 8 LB 2 Can7t be determined 3A 003 100 points 8 If the graph of f is martinez 01113232 7 HW07 7 freeman 7 54450 2 004 100 points Find the inverse of at 6x5 6 1 7 y 5 2 y 65 3 y 675 75 4 y 6 5 5 y 6 76 7y 5 005 100 points Find the inverse of 13ac 574 1W 5x71 3x4 51 3x4 51 43 5x71 43 371 45 006 100 points Find the value of g71 When 9 is the inverse of the function f de ned by 7 33 fi3 ac ii 007 100 points martinez om3232 7 HW07 7 freeman 7 54450 When 9 is the inverse of ne 36x75 find the value of g15 Hint f2 15 1 g159 1 2 15 7 9 18 2 3 15 7 9 9 4 g1518 1 5 15 7 9 9 9 6 15 7 9 2 008 100p0ints Find the solution of the exponential equa tion 262 4gx3 2 x 73 3 none of these 4x76 009 100 points Find the value of the derivative of f at at 71 When at 5x 263 1 f71 653 2 f71 523 3 f71 57253 4 f71 56e 3 5 f71 253 6 f71 5435 3 010 100 points Determine if the limit 61 l39 1 7 7 exists and if it does compute its value 2 1 limit 77 3 2 limit does not exist 3 limit E 3 4 limit 0 5 limit g 3 011 100 points Find the derivative of at 3 420 7 2e 1 fx 7272e 2 fac 34xe 3 fx 32xi w 4 fac 77295 5 fx 4575363 6 far 37425 012 100 points Find the derivative of f When at 3 7 4gt62I5 m7mw3 far 10 7 8x 62I5 far 10 8x 62I5 w7ampk rm 7 lt2 8x 6 lf is de ned by H 0 find the value of F4 2 1 F4 41 2 2 F4 7616 5 3 F4 41 2 4 F4 7616 7 5 F 4 54 013 100 points 5 8 67 372 Determine the integral 3 I 153 O 014 100 points I 3ac egigxzdx 3 727222 7 77 O 45 martinez oni3232 7 HW07 7 freeman 7 54450 3 3 I 7 6372I2C 3 4 I 7163 2I2O 5 I Ha o 3 4 3 727222 7 O 2 6 015 100 points Evaluate the definite integral 1112 x 6 dac 0 36I4 11 ltm7 gt 21 ltW7 gt 31 M 41 E gt 5 I M7 6 I m 016 100 points Find the value of at when 1 x 5log2910g4 64 1x719 2718 319 4x7l7 517 martinez om3232 7 HW07 7 freeman 7 54450 017 100 points Find the derivative of f When f 6 in sin36 1 fQ 7tan36 3 2 f a sin36 3 fa 3cot36 1 7 cos 36 5 fQ cot36 6 fa 3tan36 018 100 points Differentiate the function at cosin3ac 1 WW 73sinn3ac 2 WW sin13ac 3 far 7sinin3 4 WW 3sinn3ac i 1 5 fa 7 cosin3ac 6 WW isini3x 019 100 points Determine the derivative of f When at 37 1 fac 3964 2 fac 31m1 3 fx inac3 x39 4 fx 31nx1 x39 5 fx 3353904 6 far inx1 37 020 100 points Determine the integral 4 3 I 732inx dx ac 1 I 321nac40 1 2 2 I 1n321n0 1 4 7 321nac 0 4 I in32inx20 1 2 5 I 7 1nx321nx 0 1 4 32in 0 7 I 7in32in20 s I 732in40

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