Polynomials and Division Theory


 August Conley
 3 years ago
 Views:
Transcription
1 Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the coefficients nd n N is the degree f(x) = Know tht polynomil function is of the form: n x + n x n + n x n + + n x + Know tht polynomil of degree 0 is constnt x + 0 Know tht polynomil of degree is liner expression Know tht polynomil of degree is qudrtic expression Know tht polynomil of degree is cuic expression Use the nested form of polynomil to clculte specific vlues Know the Reminder Theorem, nmely tht if polynomil f (x) is divided y (x h), then the reminder is f (h) Apply the Reminder Theorem y using synthetic division (or long division) to find the quotient nd reminder when dividing polynomil y liner fctor of the form (x h) Use the Reminder Theorem to find the quotient nd reminder when dividing polynomil y liner fctor of the form (x + ) Know tht root of polynomil f is numer x for which 0 Show tht given numer is root of polynomil Know the Fctor Theorem, nmely tht if f (h) = 0, then (x h) is fctor of f (x); lso, if (x h) is fctor of f (x), then f (h) = 0 Use the Fctor Theorem to find ll roots of polynomil Use the Fctor Theorem to fctorise polynomils such s: 6x 5x 7x + 6 x 7x + 9 x x + x x + x 7x + 4x + 4 x 8x + 5 x 5x M Ptel (August 0) St. Mchr Acdemy
2 Higher Checklist (Unit ) Higher Checklist (Unit ) x + 4x 5x Upon fctorising polynomil f, determine the set of x vlues for which f(x) > 0 or f(x) < 0 Given polynomil f, determine, y fctorising f(x) c, the set of x vlues for which f(x) = c, f(x) > c or f(x) < c (c R) Use the Fctor Theorem to find unknown coefficients in polynomil, for exmple, find k given tht (x + ) is fctor of x + x + kx + Determine the eqution of polynomil from knowledge of where the grph crosses the x  xis nd y  xis Sketch the grph of polynomil, indicting sttionry points nd intersections with xes Qudrtic Theory Skill Achieved? Know tht qudrtic expression is of the form: x + x + c Know tht qudrtic function is of the form: x + x + c Know tht qudrtic eqution is degree polynomil equted to zero: x + x + c = 0 Recognise the generl shpe of the grph of qudrtic function (prol): ( > 0, minimum) ( < 0, mximum) M Ptel (August 0) St. Mchr Acdemy
3 Higher Checklist (Unit ) Higher Checklist (Unit ) Know tht the roots (solutions) of qudrtic eqution re otined y solving: x + x + c = 0 Clculte the roots of qudrtic eqution y using the Qudrtic Formul: ± 4c x = Use the Qudrtic Formul to solve qudrtic, expressing the roots in exct (surd) form, i.e. P ± Q R Know tht there re 4 techniques for solving qudrtics: Using grph Fctoristion Completing the Squre Qudrtic Formul Know tht the discriminnt of qudrtic def = x + x + c is: D 4c Use the discriminnt to decide ( discriminte etween) the numer nd nture of solutions of qudrtic eqution x + x + c = 0: 4c > 0 4c = 0 ( rel nd distinct) ( rel nd repeted) 4c < 0 (no rel solutions) Know tht when the discriminnt is positive, the rel roots re either oth rtionl or oth irrtionl Solve prolems involving the discriminnt such s, find the vlue(s) of k for which x 5x + (k + 6) = 0 hs equl roots Use the discriminnt to show tht given qudrtic hs rel roots for ll vlues of vrile, for exmple, ( k) x 5kx k = 0 Given generic fetures of qudrtic function, such s the sign of the discriminnt nd, identify possile grph for the function Know tht if qudrtic eqution hs distinct roots nd, then it cn lwys e written s y = k (x ) (x ) Know tht to complete the squre in qudrtic expression M Ptel (August 0) St. Mchr Acdemy
4 Higher Checklist (Unit ) Higher Checklist (Unit ) mens writing the qudrtic in the form: y = (x + p) + q Know tht the process of completing the squre cn e used to rewrite ny qudrtic expression Complete the squre in qudrtic expressions such s: x + x 8 x + 6x + x 0x + x + 6 x + 4x x + 4x x + 8x + 7 9x 8 x 9 5x 4x + x Know tht qudrtic written in the form y = (x + p) symmetry xis eqution x = p nd turning point ( p, q ) + q hs Know tht if stright line intersects prol, then the liner nd qudrtic equtions re to e equted to mke new qudrtic eqution Use the discriminnt to decide whether line intersects prol, nd if so, how mny times, using the criteri: D > 0 D = 0 ( intersection points) ( intersection point  tngency) D < 0 (no intersection) Here, D is the discriminnt of the new qudrtic eqution Know tht clculting f (0) gives the y  intercept of prol M Ptel (August 0) 4 St. Mchr Acdemy
5 Higher Checklist (Unit ) Higher Checklist (Unit ) Sketch grphs of qudrtics functions, noting specil fetures such s mximum or minimum turning point, intersections with xes nd loction nd eqution of the symmetry xis Know tht qudrtic ineqution is qudrtic eqution with the equlity replced y n inequlity Solve qudrtic inequtions y sketching the grph of the corresponding qudrtic function Solve qudrtic inequtions such s: x x + < x x < 0 x + x 0 x + 4x > 0 x 5 0 x x 0 x + x > 0 Given the x intercept(s) nd the y intercept of prol, determine the eqution of the prol, especilly if the prol hs eqution px (x q) M Ptel (August 0) 5 St. Mchr Acdemy
6 Higher Checklist (Unit ) Higher Checklist (Unit ) Integrtion Skill Achieved? Understnd the concept of integrtion in wys: The opposite of differentition (ntidifferentition) Clculting res Know tht if there re functions F nd f such tht: F (x) then F is clled n ntiderivtive (k primitive) of f Know tht not ll functions hve ntiderivtives Know tht if function f hs n ntiderivtive F, then f is integrle nd F is lso clled n integrl of f Know tht if function is integrle, then it hs infinitely mny integrls Know tht the process of working out F is clled integrtion Know tht integrls re of types: Definite Indefinite Know the nottion for n indefinite integrl of f : F = f dx Know the Fundmentl Theorem of Clculus, nmely, tht if: F (x) then the definite integrl of f is worked out s, f (x) dx = F () F () ( x ) Know the result: n x dx = n + x n + + C ( n ) M Ptel (August 0) 6 St. Mchr Acdemy
7 Higher Checklist (Unit ) Higher Checklist (Unit ) where C is the constnt of integrtion Know the integrtion rules : n x dx = n x dx ( is constnt) (f (x) ± g (x ) dx = f (x) dx ± g (x) dx Know how to simplify rtionl expressions efore integrting Know tht the integrl of the zero function is constnt, i.e.: 0 dx = C Know tht the integrl of constnt function is liner function, i.e.: k dx = k dx = kx + C Know tht the integrl of liner function is qudrtic function, i.e.: A ( Ax + B) dx = x + Bx + C Know tht the integrl of qudrtic function is cuic function, i.e.: ( Ax Bx C A + + ) dx = Integrte functions such s: B x + x + Cx + D 0 x 4 x x 5 x x 5 M Ptel (August 0) 7 St. Mchr Acdemy
8 Higher Checklist (Unit ) Higher Checklist (Unit ) 4 x x x x 5 ( x ) ( x + ) x 4 x x Evlute definite integrls exctly, for exmple: x dx 0 x dx Given the vlue of definite integrl nd one of the limits, evlute the other limit, for exmple: t 4 sin ( x + 5 5) dx = 0 Know tht the region of integrtion is the set of x vlues over which function is to e integrted Know tht the re ounded (enclosed) y the curve y =f (x), the lines x = nd x = nd the x xis is given y the definite integrl: Are = f (x) dx (ove x xis: f (x) > 0, x [, ]) M Ptel (August 0) 8 St. Mchr Acdemy
9 Higher Checklist (Unit ) Higher Checklist (Unit ) Are = f (x) dx or (elow x xis: f (x) < 0, x [, ]) Know tht oth the ove res re positive Know tht for function tht is oth positive nd negtive over the region of integrtion, the intersection point(s) of the grph with the x xis must e known efore integrting Know tht for function tht is oth positive nd negtive over the region of integrtion, the totl re ounded the curve, the x  xis nd the integrtion limits is found y: Clculting ll res ove the x xis (ech of which is positive) Clculting ll res elow the x  xis (ech of which is negtive) Ignoring the negtive signs for ll negtive res Adding ll the (now positive) ove res Clculte the re ounded y the x xis, stright line nd given limits (one of which my hve to e found) Clculte the re enclosed y the x xis, prol nd limits (which my hve to e found) Clculte the re enclosed y the x xis, cuic grph nd limits (which my hve to e found) Know tht the re ounded (enclosed) y two curves is otined y integrting top function ottom function etween two limits nd : A = (f (x) g (x ) dx = f (x) dx g (x) dx where f (x) g (x) nd x Know tht to find the re enclosed etween curves, the intersection points of the two grphs must e known nd they re the limits of integrtion Find the re enclosed etween: A prol nd stright line A stright line nd cuic grph M Ptel (August 0) 9 St. Mchr Acdemy
10 Higher Checklist (Unit ) Higher Checklist (Unit ) A prol nd cuic grph prols Clculte the re enclosed etween other curves, for exmple: x nd g (x) = x x (x [0,]) Know the grphicl integrtion rules : f (x) dx = f (x) dx c f (x) dx + c f (x) dx = f (x) dx ( c ) Know tht (first order ordinry) differentil eqution is n eqution involving the first derivtive of function: dy dx = g (x) Know tht solving differentil eqution mens finding the originl function y y integrtion Know tht initil conditions re required to solve for the constnt of integrtion Solve differentil equtions such s: dy dx = 4x 6x when x = nd y = 9 M Ptel (August 0) 0 St. Mchr Acdemy
11 Higher Checklist (Unit ) Higher Checklist (Unit ) Addition Formule Skill Achieved? Know the ddition formule (k compound ngle formule): sin ( x ± y ) = sin x cos y ± cos x sin y cos ( x ± y ) = cos x cos y sin x sin y Use compound ngle formule to expnd expressions such s: sin ( F G ) cos ( 9 ) Use compound ngle formule in reverse to simplify nd find exct vlues of extended expressions such s: π cos 4 π cos 4 + π sin 4 π sin 4 sin 70 cos 40 cos 70 sin 40 Apply compound ngle formule to find exct vlues when given the exct vlues of sin x, cos y, cos x nd sin y Given rightngled tringle with sides nd one ngle x, find the exct vlues of sin x nd cos x Apply compound ngle formule to find exct vlues y decomposing ngles, for exmple: sin 7π = sin π π + 4 cos 5 = cos (45 0) Apply compound ngle formule to prove trigonometric identities such s: cos ( x + y ) cos ( x y ) = cos x cos y sin x sin y tn ( x + y ) = tn x + tn y tn x tn y sin ( A + B) + sin ( A B) = sin A cos B Know the Pythgoren Identity : M Ptel (August 0) St. Mchr Acdemy
12 Higher Checklist (Unit ) Higher Checklist (Unit ) sin x + cos x = Given n exct vlue for sin x, where x is n cute ngle, find the exct vlues of cos x nd tn x Given n exct vlue for cos x, where x is n cute ngle, find the exct vlues of sin x nd tn x Given n exct vlue for tn x, where x is n cute ngle, find the exct vlues of sin x nd cos x Know the doule ngle formule : sin x = sin x cos x cos x = cos x = cos x = sin x sin x Given n exct vlue for sin x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Given n exct vlue for cos x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Given n exct vlue for tn x, where x is cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Rerrnge doule ngle formule in the form: sin x = ( cos x) cos x = ( + cos x) Know tht doule ngle formule re used to chnge etween terms of the form sin x nd sin x (nd similrly for cosine) Given n exct vlue for cos x with x cute, pply doule ngle formule to find the exct vlues of sin x nd cos x Apply doule ngle formule (nd possily) the Pythgoren Identity to solve trigonometric equtions such s: cos x + cos x = (x [0, 60]) sin x cos x = 0 (x [0, 80]) cos x + 0 cos x = 0 (x [0, π]) cos x 5 cos x 4 = 0 (x [0, π)) M Ptel (August 0) St. Mchr Acdemy
13 Higher Checklist (Unit ) Higher Checklist (Unit ) Use rerrngements of doule ngle formule to prove trigonometric identities such s: sin x + cos x = tn x 4 cos α 4 sin α = cos α cos x + cos x = tn x 4 sin x = (cos 4x 4 cos x + ) 8 Comine doule ngle formule with compound ngle formule to find expressions nd exct vlues for quntities such s, sin x, cos x, sin 4x nd cos 4x The Circle Skill Achieved? Know tht circle is the set of points in the plne ll of which re the sme distnce from given point; the given point is clled the centre nd the common distnce is clled the rdius Know tht the eqution of circle with centre ( 0, 0) nd rdius r is: x + y = r Given the eqution of circle in the ove form, write down the centre nd rdius Write down the eqution of circle of given rdius nd centre ( 0, 0) Know tht the eqution of circle with centre (, ) nd rdius r is: ( x ) + ( y ) = Write down the eqution of circle of given rdius nd centre ( ), r Given the eqution of circle in the ove form, write down the centre nd rdius Sketch circle of given rdius nd centre M Ptel (August 0) St. Mchr Acdemy
14 Higher Checklist (Unit ) Higher Checklist (Unit ) Know tht the expnded form (k generl eqution) of circle is: x + y + gx + fy + c = 0 where the centre is ( ) g, f nd the rdius is g + f c, ssuming tht g + f c > 0 Rerrnge circle eqution tht is not in expnded form to one tht is in expnded form Given the eqution of circle in expnded form, clculte the rdius nd centre Write down the eqution of circle in expnded form given the rdius nd centre Given missing vrile in the generl eqution, determine which vlues render the eqution tht of circle, for exmple, determine the vlues of k for which the following eqution represents circle: x + y + 4kx ky k = 0 Know the condition for when the expnded form does not represent circle, nmely: g + f c 0 Given the centre of circle nd point on it, determine the eqution of the circle Given the eqution of circle, determine whether or not point lies on the circle Decide whether point lies on, in or outside circle of given rdius Know tht line nd circle my intersect: At no points At exctly point (tngent) At points Know tht if line is to intersect circle, then the line eqution in the form y = mx + c is sustituted for y into the circle eqution, which upon simplifying produces qudrtic eqution Ax + Bx + C = 0. If this eqution hs no solutions ( B 4AC < 0) then the line does not intersect the circle M Ptel (August 0) 4 St. Mchr Acdemy
15 Higher Checklist (Unit ) Higher Checklist (Unit ) If this eqution hs exctly solution ( B 4AC = 0) then the line is tngent to the circle If this eqution hs distinct solutions ( B 4AC > 0) then the line crosses the circle in two plces In the ltter two cses, once n x  coordinte is otined, the y  coordinte is found y sustituting the x  coordinte into the eqution y = mx + c Find the intersection point of circle nd line tht is tngent to it Find the intersection points of circle nd line tht intersects it more thn once Find the equtions of the tngent lines to circle from given point outside the circle Find the eqution of tngent line to circle given point lying on the circle Know tht circumscried circle (k circumcircle) is circle tht psses through ll vertices of polygon Know tht every tringle hs circumscried circle Know tht the perpendiculr isectors of tringle meet t the circumcircle s centre Given the vertices of tringle, determine the eqution of the circle tht psses through these points Given the equtions of circles, clculte the sum of their rdii nd clculte the distnce etween their centres Know tht for circles: If the sum of their rdii is greter thn the centre  centre distnce, then the circles intersect t points (nd vice vers) If the sum of their rdii equls the centre  centre distnce, then the circles intersect t point (nd vice vers) If the sum of their rdii is less thn the centre  centre distnce, then the circles do not intersect (nd vice vers) Given the equtions of circles, determine whether or not they intersect M Ptel (August 0) 5 St. Mchr Acdemy
Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationFINALTERM EXAMINATION 9 (Session  ) Clculus & Anlyticl GeometryI Question No: ( Mrs: )  Plese choose one f ( x) x According to PowerRule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationStage 11 Prompt Sheet
Stge 11 rompt Sheet 11/1 Simplify surds is NOT surd ecuse it is exctly is surd ecuse the nswer is not exct surd is n irrtionl numer To simplify surds look for squre numer fctors 7 = = 11/ Mnipulte expressions
More informationThe discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+
.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationTABLE OF CONTENTS 3 CHAPTER 1
TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationR(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of
Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More information( β ) touches the xaxis if = 1
Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I  Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More information03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t
APDF Wtermrk DEMO: Purchse from www.apdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationEvaluating Definite Integrals. There are a few properties that you should remember in order to assist you in evaluating definite integrals.
Evluting Definite Integrls There re few properties tht you should rememer in order to ssist you in evluting definite integrls. f x dx= ; where k is ny rel constnt k f x dx= k f x dx ± = ± f x g x dx f
More informationCh AP Problems
Ch. 7.7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationTrigonometric Functions
Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationSection 7.1 Area of a Region Between Two Curves
Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region
More informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationIf deg(num) deg(denom), then we should use longdivision of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)
Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationHigher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6
Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationA sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.
Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.
More informationAlg. Sheet (1) Department : Math Form : 3 rd prep. Sheet
Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( 5, 9 ) ) (,
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationMATH1050 CauchySchwarz Inequality and Triangle Inequality
MATH050 CuchySchwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for relvlued functions of one rel vrile) Let I e n intervl, nd h : D R e relvlued
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationUS01CMTH02 UNIT Curvature
Stu mteril of BSc(Semester  I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT 1 Curvture Let f : I R be sufficiently
More informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationy = f(x) This means that there must be a point, c, where the Figure 1
Clculus Investigtion A Men Slope TEACHER S Prt 1: Understnding the Men Vlue Theorem The Men Vlue Theorem for differentition sttes tht if f() is defined nd continuous over the intervl [, ], nd differentile
More informationMATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. Note: This question paper consists of three sections A,B and C. SECTION A
MATHEMATICS PAPER IIB COORDINATE GEOMETRY AND CALCULUS. TIME : 3hrs M. Mrks.75 Note: This question pper consists of three sections A,B nd C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. X = ) Find the eqution
More information