For more important questions visit :


 Brianne McDaniel
 3 years ago
 Views:
Transcription
1 For mor important qustions visit : www4onocom CHAPTER 5 CONTINUITY AND DIFFERENTIATION POINTS TO REMEMBER A function f() is said to b continuous at = c iff lim f f c c i, lim f lim f f c c c f() is continuous in (a, b) iff it is continuous at c c a b f() is continuous in [a, b] iff (i) f() is continuous in (a, b) (ii) lim f f a, a (iii) lim f f b b Trigonomtric functions ar continuous in thir rspctiv domains Evr polnomial function is continuous on R If f () and g () ar two continuous functions and c R thn at = a (i) f () ± g () ar also continuous functions at = a (ii) g () f (), f () + c, cf (), f () ar also continuous at = a, (iii) f g is continuous at = a providd g(a) 0 f () is drivabl at = c in its domain iff 39 XII Maths
2 f f c f f c lim lim, and is finit c c c c Th valu of abov limit is dnotd b f (c) and is calld th drivativ of f() at = c d u v u dv v du d d d du dv v u d v v d u d d du If = f(u) and u = g(t) thn f u g t Chain Rul If = f(u), = g(u) thn, dt du dt d d du f u d du d g u d sin, cos d d d d d d tan, cot d d sc, cosc d d d, log d d f () = [] is discontinuous at all intgral points and continuous for all R Z Roll s thorm : If f () is continuous in [ a, b ], drivabl in (a, b) and f (a) = f (b) thn thr ists atlast on ral numbr c (a, b) such that f (c) = 0 XII Maths 40
3 Man Valu Thorm : If f () is continuous in [a, b] and drivabl in (a, b) thn thr ists atlast on ral numbr c (a, b) such that f b f a f c b a f () = log, ( > 0) is continuous function VERY SHORT ANSWER TYPE QUESTIONS ( MARK) For what valu of, f() = 7 is not drivabl Writ th st of points of continuit of g() = What is drivativ of 3 at = 4 What ar th points of discontinuit of f Writ th numbr of points of discontinuit of f() = [] in [3, 7] 3 if 6 Th function, f 4 if if R, find is a continuous function for all tan3, 0 7 For what valu of K, f sin K, 0 is continuous R 8 Writ drivativ of sin wrt cos 9 If f() = g() and g() = 6, g () = 3 find valu of f () 0 Writ th drivativ of th following functions : (i) log 3 (3 + 5) (ii) log (iii) 6 log, 4 XII Maths
4 (iv) sc cosc, (v) 7 sin (vi) log 5, > 0 SHORT ANSWER TYPE QUESTIONS (4 MARKS) Discuss th continuit of following functions at th indicatd points, 0 (i) f at 0, 0 sin, 0 (ii) 3 g at (iii) cos f 0 at (iv) f() = + at =, (v) f at 0 For what valu of k, f 0, 3 k 3 5, is continuous 3 For what valus of a and b a if f a b if b if is continuous at = XII Maths 4
5 4 Prov that f() = + is continuous at =, but not drivabl at = 5 For what valu of p, p f sin is drivabl at = 0 6 If tan tan, 0, find d 7 If sin tan thn? d 8 If = 5 + thn prov that 5 0 d 9 If a thn show that d 0 If a thn show that d If ( + ) m + n = m n thn prov that d Find th drivativ of tan wrt sin 3 Find th drivativ of log (sin ) wrt log a (cos ) 4 If + + = m n, thn find th valu of d 5 If = a cos 3, = a sin 3 thn find d d 43 XII Maths
6 6 If = a t (sint cos t) = a t (sint + cost) thn show that at is d 4 7 If sin thn find d thn find d log 8 If log 9 Diffrntiat wrt d 30 Find, if cos cos 3 If sin sin tan whr sin sin find d 3 If sin log a thn show that ( ) a = 0 33 Diffrntiat log log, w r t 34 If sin = sin (a + ) thn show that d sin a sina 35 If = sin, find d d in trms of 36 If a d thn show that b d, 4 b 3 a cos 37 If a d,, show that a 0 d d 38 If 3 = 3a 3 thn prov that d a 5 d 39 Vrif Roll's thorm for th function, = + in th intrval [a, b] whr a =, b = 40 Vrif Man Valu Thorm for th function, f() = in [, 4] XII Maths 44
7 ANSWERS = 7/ R 3 4 = 6, 7 5 Points of discontinuit of f() ar 4, 5, 6, 7 i four points Not : At = 3, f() = [] is continuous bcaus lim f 3 f k 4 8 cot (i) log3 3 5 log (ii) log (iii) 6 ( ) 5 (iv) 0 (v) 7 7 (vi) log 5 log (i) Discontinuous (ii) Discontinuous (iii) Continuous (iv) continuous (v) Discontinuous k = 3 a = 0, b = 5 p > cot log a 4 log log d log 45 XII Maths
8 5 d 4 cosc sc d 3a 7 d log log log log log log log log d log tan logcos d tan logcos d Hint : sin cos for, 33 log log log log, 35 sc tan For mor important qustions visit : www4onocom XII Maths 46
Calculus II (MAC )
Calculus II (MAC2322) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationAP Calculus BC AP Exam Problems Chapters 1 3
AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationCONTINUITY AND DIFFERENTIABILITY
MCD CONTINUITY AND DIFFERENTIABILITY NCERT Solvd mpls upto th sction 5 (Introduction) nd 5 (Continuity) : Empl : Chck th continuity of th function f givn by f() = + t = Empl : Emin whthr th function f
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More information( ) Differential Equations. Unit7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its yintrcpt is 8. y
More informationINTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)
Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationExponential Functions
Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.
More informationIntegration by Parts
Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'(
More informationSouthern Taiwan University
Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:
More informationAP Calculus MultipleChoice Question Collection
AP Calculus MultiplCoic Qustion Collction 985 998 . f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and nondrivability
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationMATHEMATICS (B) 2 log (D) ( 1) = where z =
MATHEMATICS SECTION I STRAIGHT OBJECTIVE TYPE This sction contains 9 multipl choic qustions numbrd to 9. Each qustion has choic (A), (B), (C) and (D), out of which ONLYONE is corrct. Lt I d + +, J +
More informationCLASS XII CBSE MATHEMATICS CONTINUITY AND DIFFERENTIATION
CLASS XII CBSE MATHEMATICS CONTINUITY AND DIFFERENTIATION sin5 + cos, if 0 ) For what value of k is the function f() = { 3 k, if = 0 ) For what value of k is the following function continuous at =? ; f
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 512
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by AddisonWsly.
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN3198354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. WarmUp: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() =  Objctivs, Day #1 Studnts will b
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &D) AND CALCULUS. TIME : hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.If th portion
More informationPrelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed  2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd  hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationTrigonometric functions
Robrto s Nots on Diffrntial Calculus Captr 5: Drivativs of transcndntal functions Sction 5 Drivativs of Trigonomtric functions Wat you nd to know alrady: Basic trigonomtric limits, t dfinition of drivativ,
More informationALevel Mathematics DIFFERENTIATION I. G. David Boswell  Math Camp Typeset 1.1 DIFFERENTIATION OF POLYNOMIALS. d dx axn = nax n 1, n!
ALevel Mathematics DIFFERENTIATION I G. David Boswell  Math Camp Typeset 1.1 SET C Review ~ If a and n are real constants, then! DIFFERENTIATION OF POLYNOMIALS Problems ~ Find the first derivative of
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationRELATIONS AND FUNCTIONS
For more important questions visit : www.4ono.com CHAPTER 1 RELATIONS AND FUNCTIONS IMPORTANT POINTS TO REMEMBER Relation R from a set A to a set B is subset of A B. A B = {(a, b) : a A, b B}. If n(a)
More informationMath 120 Answers for Homework 14
Math 0 Answrs for Homwork. Substitutions u = du = d d = du a d = du = du = u du = u + C = u = arctany du = +y dy dy = + y du b arctany arctany dy = + y du = + y + y arctany du = u du = u + C = arctan y
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationMAXIMAMINIMA EXERCISE  01 CHECK YOUR GRASP
EXERCISE  MAXIMAMINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6.
More informationSAFE HANDS & IITian's PACE EDT15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT (JEE) SOLUTIONS
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writup th problm natly and
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6  86   X Th natural logarithm function ln in US log: function
More informationIncreasing and Decreasing Functions and the First Derivative Test
Section 3.3 Increasing and Decreasing Functions and the First Derivative Test 3 Section 3.3 Increasing and Decreasing Functions and the First Derivative Test. f 8 3. 3, Decreasing on:, 3 3 3,,, Decreasing
More informationMATH 1080 Test 2SOLUTIONS Spring
MATH Tst SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b= c=1 4 1 1or 1 1 Quadratic
More information3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationMAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative
MAT 7 Tst Rviw (Spring ) Tst on April in PSA Sction.7 Implicit Drivativ Rmmbr: Equation of t tangnt lin troug t point ( ab, ) aving slop m is y b m( a ). dy Find t drivativ y d. y y. y y y. y 4. y sin(
More informationCLASS XII MATHEMATICS. Weightage (Marks) (i) Relations and Functions 10. (ii) Algebra 13. (iii) Calculus 44
CLASS XII MATHEMATICS Units Weightage (Marks) (i) Relations and Functions 0 (ii) Algebra (iii) Calculus 44 (iv) Vector and Three Dimensional Geometry 7 (v) Linear Programming 06 (vi) Probability 0 Total
More informationy »x 2» x 1. Find x if a = be 2x, lna = 7, and ln b = 3 HAL ln 7 HBL 2 HCL 7 HDL 4 HEL e 3
. Find if a = be, lna =, and ln b = HAL ln HBL HCL HDL HEL e a = be and taing the natural log of both sides, we have ln a = ln b + ln e ln a = ln b + = + = B. lim b b b = HAL b HBL b HCL b HDL b HEL b
More informationHyperbolic Functions Mixed Exercise 6
Hyprbolic Functions Mid Ercis 6 a b c ln ln sinh(ln ) ln ln + cosh(ln ) + ln tanh ln ln + ( 6 ) ( + ) 6 7 ln ln ln,and ln ln ln,and ln ln 6 artanh artanhy + + y ln ln y + y ln + y + y y ln + y y + y y
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: 5 Eampls  6 ar copyrightd
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationANSWER KEY 1. [A] 2. [C] 3. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A] 10. [A] 11. [D] 12. [A] 13. [D] 14. [C] 15. [B] 16. [C] 17. [D] 18.
ANSWER KEY. [A]. [C]. [B] 4. [B] 5. [C] 6. [A] 7. [B] 8. [C] 9. [A]. [A]. [D]. [A]. [D] 4. [C] 5. [B] 6. [C] 7. [D] 8. [B] 9. [C]. [C]. [D]. [A]. [B] 4. [D] 5. [A] 6. [D] 7. [B] 8. [D] 9. [D]. [B]. [A].
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationChapter two Functions
Chaptr two Functions  Eponntial Logarithm functions Eponntial functions If a is a positiv numbr is an numbr, w dfin th ponntial function as = a with domain  < < ang > Th proprtis of th ponntial functions
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th yaxis as long as
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initialvalu problm for th wav uation using th Fourir transform
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationas a derivative. 7. [3.3] On Earth, you can easily shoot a paper clip straight up into the air with a rubber band. In t sec
MATH6 Fall 8 MIDTERM II PRACTICE QUESTIONS PART I. + if
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationThe function y loge. Vertical Asymptote x 0.
Grad 1 (MCV4UE) AP Calculus Pa 1 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Dfinition of (Natural Eponntial Numr) 0 1 lim( 1 ). 7188188459... 5 1 5 5 Proprtis of and ln Rcall th loarithmic
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationGrade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Gra (MCV4UE) AP Calculus Pag of 5 Drivativ of a Function & Diffrntiabilit Th Drivativ at a Point f ( a h) f ( a) Rcall, lim provis th slop of h0 h th tangnt to th graph f ( at th point a, f ( a), an th
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More informationCombinatorial Networks Week 1, March 1112
1 Nots on March 11 Combinatorial Ntwors W 1, March 111 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More information10. The DiscreteTime Fourier Transform (DTFT)
Th DiscrtTim Fourir Transform (DTFT Dfinition of th discrttim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th plan is th acclration vctor of th particl
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th khighr Mahlr masur m k P )
More informationNARAYANA I I T / P M T A C A D E M Y. C o m m o n P r a c t i c e T e s t 1 6 XII STD BATCHES [CF] Date: PHYSIS HEMISTRY MTHEMTIS
. (D). (A). (D). (D) 5. (B) 6. (A) 7. (A) 8. (A) 9. (B). (A). (D). (B). (B). (C) 5. (D) NARAYANA I I T / P M T A C A D E M Y C o m m o n P r a c t i c T s t 6 XII STD BATCHES [CF] Dat: 8.8.6 ANSWER PHYSIS
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationDISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P
DISTRIBUTION OF DIFFERENCE BETWEEN INVERSES OF CONSECUTIVE INTEGERS MODULO P Tsz Ho Chan Dartmnt of Mathmatics, Cas Wstrn Rsrv Univrsity, Clvland, OH 4406, USA txc50@cwru.du Rcivd: /9/03, Rvisd: /9/04,
More informationAnswers & Solutions. for MHT CET2018 PaperI (Mathematics) Instruction for Candidates
DATE : /5/8 Qustion Booklt Vrsion Rgd. Offic : Aakash Towr, 8, Pusa Road, Nw Dlhi5 Ph.: 75 Fa : 77 Tim : Hour Min. Total Marks : Answrs & Solutions for MHT CET8 PaprI (Mathmatics) Instruction for
More information(HELD ON 21st MAY SUNDAY 2017) MATHEMATICS CODE  1 [PAPER1]
QUESTION PAPER WITH SOLUTION OF JEE ADVANCED  7 (HELD ON st MAY SUNDAY 7) FEEL THE POWER OF OUR KNOWLEDGE & EXPERIENCE Our Top class IITian facult tam promiss to giv ou an authntic answr k which will
More informationMETHODS OF DIFFERENTIATION. Contents. Theory Objective Question Subjective Question 10. NCERT Board Questions
METHODS OF DIFFERENTIATION Contents Topic Page No. Theor 0 0 Objective Question 0 09 Subjective Question 0 NCERT Board Questions Answer Ke 4 Sllabus Derivative of a function, derivative of the sum, difference,
More informationMATHEMATICS XII. Topic. Revision of Derivatives Presented By. Avtar Singh Lecturer Paramjit Singh Sidhu June 19,2009
MATHEMATICS XII 1 Topic Revision of Derivatives Presented By Avtar Singh Lecturer Paramjit Singh Sidhu June 19,2009 19 June 2009 Punjab EDUSAT Society (PES) 1 Continuity 2 Def. In simple words, a function
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Timspac: 9:: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationJOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH 241) Final Review Fall 2016
JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw Fall 06 Th Final Rviw is a starting point as you study for th final am. You should also study your ams and homwork. All topics listd in th
More informationCopyright PreCalculusCoach.com
Continuit, End Behavior, and Limits Assignment Determine whether each function is continuous at the given values. Justif using the continuit test. If discontinuous, identif the tpe of discontinuit as
More informationReview of Exponentials and Logarithms  Classwork
Rviw of Eponntials and Logarithms  Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial
More informationAP Calculus BC Problem Drill 16: Indeterminate Forms, L Hopital s Rule, & Improper Intergals
AP Calulus BC Problm Drill 6: Indtrminat Forms, L Hopital s Rul, & Impropr Intrgals Qustion No. of Instrutions: () Rad th problm and answr hois arfully () Work th problms on papr as ndd () Pik th answr
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationMATH1120 Calculus II Solution to Supplementary Exercises on Improper Integrals Alex Fok November 2, 2013
() Solution : MATH Calculus II Solution to Supplementary Eercises on Improper Integrals Ale Fok November, 3 b ( + )( + tan ) ( + )( + tan ) +tan b du u ln + tan b ( = ln + π ) (Let u = + tan. Then du =
More information: f (x) x k, if x 0 continuous at x = 0, find k.
ASSIGNMENT ON CONTINUITY AND DIFFERENTIABILITY LEVEL 1 ( CBSE AND OTHER STATE BOARDS) 1 Test the continuit of the function f() at the ; 0 origin : f () 1; 0 Show that the function f() given b sin cos,
More informationReview Exercises for Chapter 3. Review Exercises for Chapter r v 0 2. v ft sec. x 1 2 x dx f x x 99.4.
Review Eercises for Chapter 6. r v 0 sin. Let f, 00, d 0.6. v 0 00 ftsec changes from 0 to dr 00 cos d 6 0 d 0 r dr 80 00 6 96 feet 80 cos 0 96 feet 8080 f f fd d f 99. 00 0.6 9.97 00 Using a calculator:
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUELRATESPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A, B and C.
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Tim: 3hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A, B and C. SECTION A Vry Short Answr Typ Qustions. 0 X = 0. Find th condition
More informationCalculus II Solutions review final problems
Calculus II Solutions rviw final problms MTH 5 Dcmbr 9, 007. B abl to utiliz all tchniqus of intgration to solv both dfinit and indfinit intgrals. Hr ar som intgrals for practic. Good luck stuing!!! (a)
More information