This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
|
|
- Pamela Park
- 6 years ago
- Views:
Transcription
1 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion Wht you should lredy know Squres, squre roots, cues nd cue roots of integers How to sustitute vlues into lgeric expressions How to solve simple lgeric equtions Quick check 1 Write down the vlue of ech of the following. 5 2 «««81 c 3 3 d 3 «««64 2 Clculte the vlue of y if x = 4. y = 3x 2 y = 1 «x 519
2 22.1 Direct vrition This section will introduce you to: direct vrition nd show you how to work out the constnt of proportionlity Key words constnt of proportionlity, k direct proportion direct vrition The term direct vrition hs the sme mening s direct proportion. There is direct vrition (or direct proportion) etween two vriles when one vrile is simple multiple of the other. Tht is, their rtio is constnt. For exmple: 1 kilogrm = 2.2 pounds There is multiplying fctor of 2.2 etween kilogrms nd pounds. Are of circle = πr 2 There is multiplying fctor of π etween the re of circle nd the squre of its rdius. An exmintion question involving direct vrition usully requires you first to find this multiplying fctor (clled the constnt of proportionlity), then to use it to solve prolem. The symol for vrition or proportion is. So the sttement Py is directly proportionl to time cn e mthemticlly written s Py Time which implies tht Py = k Time where k is the constnt of proportionlity. There re three steps to e followed when solving question involving proportionlity. Step 1: set up the proportionlity eqution (you my hve to define vriles). Step 2: use the given informtion to find the constnt of proportionlity. Step 3: sustitute the constnt of proportionlity in the originl eqution nd use this to find unknown vlues. 520
3 CHAPTER 22: VARIATION EXAMPLE 1 The cost of n rticle is directly proportionl to the time spent mking it. An rticle tking 6 hours to mke costs 30. Find the following. the cost of n rticle tht tkes 5 hours to mke the length of time it tkes to mke n rticle costing 40 Step 1: Let C e the cost of mking n rticle nd t the time it tkes. We then hve: C t C = kt where k is the constnt of proportionlity. Note tht we cn replce the proportionlity sign with = k to otin the proportionlity eqution. Step 2: Since C = 30 when t = 6 hours, then 30 = 6k 30 = k 6 k = 5 Step 3: So the formul is C = 5t When t = 5 hours C = 5 5 = 25 So the cost is 25. When C = = 5 t 40 5 = t t = 8 So the mking time is 8 hours. EXERCISE 22A In ech cse, first find k, the constnt of proportionlity, nd then the formul connecting the vriles. T is directly proportionl to M. If T = 20 when M = 4, find the following. T when M = 3 M when T = 10 W is directly proportionl to F. If W = 45 when F = 3, find the following. W when F = 5 F when W = 90 Q vries directly with P. If Q = 100 when P = 2, find the following. Q when P = 3 P when Q = 300 X vries directly with Y. If X = 17.5 when Y = 7, find the following. X when Y = 9 Y when X =
4 CHAPTER 22: VARIATION The distnce covered y trin is directly proportionl to the time tken. The trin trvels 105 miles in 3 hours. Wht distnce will the trin cover in 5 hours? Wht time will it tke for the trin to cover 280 miles? The cost of fuel delivered to your door is directly proportionl to the weight received. When 250 kg is delivered, it costs How much will it cost to hve 350 kg delivered? How much would e delivered if the cost were 33.25? The numer of children who cn ply sfely in plyground is directly proportionl to the re of the plyground. A plyground with n re of 210 m 2 is sfe for 60 children. How mny children cn sfely ply in plyground of re 154 m 2? A plygroup hs 24 children. Wht is the smllest plyground re in which they could sfely ply? Direct proportions involving squres, cues nd squre roots The process is the sme s for liner direct vrition, s the next exmple shows. EXAMPLE 2 The cost of circulr dge is directly proportionl to the squre of its rdius. The cost of dge with rdius of 2 cm is 68p. Find the cost of dge of rdius 2.4 cm. Find the rdius of dge costing Step 1: Let C e the cost nd r the rdius of dge. Then C r 2 C = kr 2 where k is the constnt of proportionlity. Step 2: C = 68p when r = 2 cm. So 68 = 4k 68 = k k = 17 4 Hence the formul is C = 17r 2 When r = 2.4 cm C = p = 97.92p Rounding off gives the cost s 98p. When C = 153p 153 = 17r = 9 = r 2 17 r = ««9 = 3 Hence, the rdius is 3 cm. 522
5 CHAPTER 22: VARIATION EXERCISE 22B In ech cse, first find k, the constnt of proportionlity, nd then the formul connecting the vriles. T is directly proportionl to x 2. If T = 36 when x = 3, find the following. T when x = 5 x when T = 400 W is directly proportionl to M 2. If W = 12 when M = 2, find the following. W when M = 3 M when W = 75 E vries directly with ««C. If E = 40 when C = 25, find the following. E when C = 49 C when E = 10.4 X is directly proportionl to ««Y. If X = 128 when Y = 16, find the following. X when Y = 36 Y when X = 48 P is directly proportionl to f 3. If P = 400 when f = 10, find the following. P when f = 4 f when P = 50 The cost of serving te nd iscuits vries directly with the squre root of the numer of people t the uffet. It costs 25 to serve te nd iscuits to 100 people. How much will it cost to serve te nd iscuits to 400 people? For cost of 37.50, how mny could e served te nd iscuits? In n experiment, the temperture, in C, vried directly with the squre of the pressure, in tmospheres. The temperture ws 20 C when the pressure ws 5 tm. Wht will the temperture e t 2 tm? Wht will the pressure e t 80 C? The weight, in grms, of ll erings vries directly with the cue of the rdius mesured in millimetres. A ll ering of rdius 4 mm hs weight of g. Wht will ll ering of rdius 6 mm weigh? A ll ering hs weight of 48.6 g. Wht is its rdius? The energy, in J, of prticle vries directly with the squre of its speed in m/s. A prticle moving t 20 m/s hs 50 J of energy. How much energy hs prticle moving t 4 m/s? At wht speed is prticle moving if it hs 200 J of energy? The cost, in, of trip vries directly with the squre root of the numer of miles trvelled. The cost of 100-mile trip is 35. Wht is the cost of 500-mile trip (to the nerest 1)? Wht is the distnce of trip costing 70? 523
6 22.2 Inverse vrition This section will introduce you to: inverse vrition nd show you how to work out the constnt of proportionlity Key words constnt of proportionlity, k inverse proportion There is inverse vrition etween two vriles when one vrile is directly proportionl to the reciprocl of the other. Tht is, the product of the two vriles is constnt. So, s one vrile increses, the other decreses. For exmple, the fster you trvel over given distnce, the less time it tkes. So there is n inverse vrition etween speed nd time. We sy speed is inversely proportionl to time. 1 k S nd so S = T T which cn e written s ST = k. EXAMPLE 3 M is inversely proportionl to R. If M = 9 when R = 4, find the following. M when R = 2 R when M = 3 1 k Step 1: M M = where k is the constnt of proportionlity. R R k Step 2: When M = 9 nd R = 4, we get 9 = = k k = Step 3: So the formul is M = R 36 When R = 2, then M = = When M = 3, then 3 = 3R = 36 R = 12 R EXERCISE 22C In ech cse, first find the formul connecting the vriles. T is inversely proportionl to m. If T = 6 when m = 2, find the following. T when m = 4 m when T = 4.8 W is inversely proportionl to x. If W = 5 when x = 12, find the following. W when x = 3 x when W =
7 CHAPTER 22: VARIATION Q vries inversely with (5 t). If Q = 8 when t = 3, find the following. Q when t = 10 t when Q = 16 M vries inversely with t 2. If M = 9 when t = 2, find the following. M when t = 3 t when M = 1.44 W is inversely proportionl to ««T. If W = 6 when T = 16, find the following. W when T = 25 T when W = 2.4 The grnt ville to section of society ws inversely proportionl to the numer of people needing the grnt. When 30 people needed grnt, they received 60 ech. Wht would the grnt hve een if 120 people hd needed one? If the grnt hd een 50 ech, how mny people would hve received it? While doing underwter tests in one prt of n ocen, tem of scientists noticed tht the temperture in C ws inversely proportionl to the depth in kilometres. When the temperture ws 6 C, the scientists were t depth of 4 km. Wht would the temperture hve een t depth of 8 km? To wht depth would they hve hd to go to find the temperture t 2 C? A new engine ws eing tested, ut it hd serious prolems. The distnce it went, in km, without reking down ws inversely proportionl to the squre of its speed in m/s. When the speed ws 12 m/s, the engine lsted 3 km. Find the distnce covered efore rekdown, when the speed is 15 m/s. On one test, the engine roke down fter 6.75 km. Wht ws the speed? In lloon it ws noticed tht the pressure, in tmospheres, ws inversely proportionl to the squre root of the height, in metres. When the lloon ws t height of 25 m, the pressure ws 1.44 tm. Wht ws the pressure t height of 9 m? Wht would the height hve een if the pressure ws 0.72 tm? The mount of wste which firm produces, mesured in tonnes per hour, is inversely proportionl to the squre root of the size of the filter eds, mesured in m 2. At the moment, the firm produces 1.25 tonnes per hour of wste, with filter eds of size 0.16 m 2. The filter eds used to e only 0.01 m 2. How much wste did the firm produce then? How much wste could e produced if the filter eds were 0.75 m 2? 525
8 y is proportionl to x. Complete the tle. x y The energy, E, of n oject moving horizontlly is directly proportionl to the speed, v, of the oject. When the speed is 10 m/s the energy is Joules. Find n eqution connecting E nd v. Find the speed of the oject when the energy is Joules. y is inversely proportionl to the cue root of x. When y = 8, x = 1 8. Find n expression for y in terms of x, Clculte i the vlue of y when x = 1 125, ii the vlue of x when y = 2. The mss of cue is directly proportionl to the cue of its side. A cue with side of 4 cm hs mss of 320 grms. Clculte the side length of cue mde of the sme mteril with mss of grms y is directly proportionl to the cue of x. When y = 16, x = 3. Find the vlue of y when x = 6. d is directly proportionl to the squre of t. d = 80 when t = 4 Express d in terms of t. Work out the vlue of d when t = 7. c Work out the positive vlue of t when d = 45. Edexcel, Question 16, Pper 5 Higher, June 2005 The force, F, etween two mgnets is inversely proportionl to the squre of the distnce, x, etween them. When x = 3, F = 4. Find n expression for F in terms of x. Clculte F when x = 2. c Clculte x when F = 64. Edexcel, Question 17, Pper 5 Higher, June Two vriles, x nd y, re known to e proportionl to ech other. When x = 10, y = 25. Find the constnt of proportionlity, k, if: y x y x 2 c y 1 x d ««y 1 x y is directly proportionl to the cue root of x. When x = 27, y = 6. Find the vlue of y when x = 125. Find the vlue of x when y = 3. The surfce re, A, of solid is directly proportionl to the squre of the depth, d. When d = 6, A = 12π. Find the vlue of A when d = 12. Give your nswer in terms of π. Find the vlue of d when A = 27π. r is inversely proportionl to t. r = 12 when t = 0.2 Clculte the vlue of r when t = 4. Edexcel, Question 4, Pper 13B Higher, Jnury 2003 The frequency, f, of sound is inversely proportionl to the wvelength, w. A sound with frequency of 36 hertz hs wvelength of metres. Clculte the frequency when the frequency nd the wvelength hve the sme numericl vlue. t is proportionl to m 3. When m = 6, t = 324. Find the vlue of t when m = 10. Also, m is inversely proportionl to the squre root of w. When t = 12, w = 25. Find the vlue of w when m = 4. P nd Q re positive quntities. P is inversely proportionl to Q 2. When P = 160, Q = 20. Find the vlue of P when P = Q. 526
9 CHAPTER 22: VARIATION WORKED EXAM QUESTION y is inversely proportionl to the squre of x. When y is 40, x = 5. Find n eqution connecting x nd y. Find the vlue of y when x = 10. Solution 1 y y = x 2 k x 2 k 40 = 25 k = = y = x 2 or yx 2 = When x = 10, y = = = First set up the proportionlity reltionship nd replce the proportionlity sign with = k. Sustitute the given vlues of y nd x into the proportionlity eqution to find the vlue of k. Sustitute the vlue of k to get the finl eqution connecting y nd x. Sustitute the vlue of x into the eqution to find y. The mss of solid, M, is directly proportionl to the cue of its height, h. When h = 10, M = The surfce re, A, of the solid is directly proportionl to the squre of the height, h. When h = 10, A = 50. Find A, when M = Solution M = kh = k 1000 k = 4 So, M = 4h 3 A = ph 2 50 = p 100 p = 1 So, A = h = 4h 3 h 3 = 8000 h = A = (20) = = First, find the reltionship etween M nd h using the given informtion. Next, find the reltionship etween A nd h using the given informtion. Find the vlue of h when M = Now find the vlue of A for tht vlue of h. 527
10 An electricity compny wnts to uild some offshore wind turines (s shown elow). The compny is concerned out how ig the turines will look to person stnding on the shore. It sks n engineer to clculte the ngle of elevtion from the shore to the highest point of turine, when it is rotting, if the turine ws plced t different distnces out to se. Help the engineer to complete the first tle elow. Distnce of Angle of elevtion turine out to se from shore 3km 2.29º 4km 5km 6km 7km 8km 50m 70m The power ville in the wind is mesured in wtts per metre squred of rotor re (W/m 2 ). Wind speed is mesured in metres per second (m/s). The power ville in the wind is proportionl to the cue of its speed. A wind speed of 7 m/s cn provide 210 W/m 2 of energy. Complete the tle elow to show the ville power t different wind speeds. Wind speed Aville power (m/s) (W/m 2 )
11 Vrition The engineer investigtes the different mounts of power produced y different length rotor ldes t different wind speeds. He clcultes the rotor re for ech lde length this is the re of the circle mde y the rotors nd then works out the power produced y these ldes t the different wind speeds shown. Help him to complete the tle. Rememer 1 W = 1 wtt 1000 W = 1 kw = 1 kilowtt 1000 kw = 1 MW = 1 megwtt Wind speed Aville power Rotor re for Power Rotor re for Power Rotor re for Power (m/s) (W/m 2 ) 50 m lde (m 2 ) (MW) 60 m lde (m 2 ) (MW) 70 m lde (m 2 ) (MW)
12 GRADE YOURSELF Ale to find formule descriing direct or inverse vrition nd use them to solve prolems Ale to solve direct nd inverse vrition prolems involving three vriles Wht you should know now How to recognise direct nd inverse vrition Wht constnt of proportionlity is, nd how to find it How to find formule descriing inverse or direct vrition How to solve prolems involving direct or inverse vrition 530
Linear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, 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 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 informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
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 informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
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 information8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
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 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 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 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 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 information2 Calculate the size of each angle marked by a letter in these triangles.
Cmridge Essentils Mthemtics Support 8 GM1.1 GM1.1 1 Clculte the size of ech ngle mrked y letter. c 2 Clculte the size of ech ngle mrked y letter in these tringles. c d 3 Clculte the size of ech ngle mrked
More informationLesson 8.1 Graphing Parametric Equations
Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find
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 informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
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 informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More information12.1 Introduction to Rational Expressions
. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To
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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.
8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims
More informationAlg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A
lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,
More informationAPPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationMA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1
MA 15910, Lessons nd Introduction to Functions Alger: Sections 3.5 nd 7.4 Clculus: Sections 1. nd.1 Representing n Intervl Set of Numers Inequlity Symol Numer Line Grph Intervl Nottion ) (, ) ( (, ) ]
More informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationHQPD - ALGEBRA I TEST Record your answers on the answer sheet.
HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationMATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs
MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching
More informationWhat s in Chapter 13?
Are nd volume 13 Wht s in Chpter 13? 13 01 re 13 0 Are of circle 13 03 res of trpeziums, kites nd rhomuses 13 04 surfce re of rectngulr prism 13 05 surfce re of tringulr prism 13 06 surfce re of cylinder
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationDate Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )
UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationWhat else can you do?
Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright
More information( ) Same as above but m = f x = f x - symmetric to y-axis. 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 information3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.
SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd
More informationMinnesota State University, Mankato 44 th Annual High School Mathematics Contest April 12, 2017
Minnesot Stte University, Mnkto 44 th Annul High School Mthemtics Contest April, 07. A 5 ft. ldder is plced ginst verticl wll of uilding. The foot of the ldder rests on the floor nd is 7 ft. from the wll.
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationThe area under the graph of f and above the x-axis 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 x-xis etween nd is denoted y f(x) dx nd clled the
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 information10.2 The Ellipse and the Hyperbola
CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point
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 informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationPhysics 1402: Lecture 7 Today s Agenda
1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationChapter E - Problems
Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current
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 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 informationAn Overview of Integration
An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is
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 informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
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 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 informationPolynomials and Division Theory
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
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationProblem Solving 7: Faraday s Law Solution
MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce
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 informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More information1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?
Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 17
CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
More informationMAC 1105 Final Exam Review
1. Find the distnce between the pir of points. Give n ect, simplest form nswer nd deciml pproimtion to three plces., nd, MAC 110 Finl Em Review, nd,0. The points (, -) nd (, ) re endpoints of the dimeter
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationdy ky, dt where proportionality constant k may be positive or negative
Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. -6 ì 4 = -6 ì 0 = 4 ì 0 = -6 ì 3 = -5 ì -3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = -5 ì -2 = -6 ì 2 = -6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity -B- Integer multipliction doc-69 Activity -B- More integer multipliction doc-698 Activity -B- Advnced integer multipliction doc-699 Multiplying integers FLUENCY
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 informationPhysics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:
Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You
More informationMath 7, Unit 9: Measurement: Two-Dimensional Figures Notes
Mth 7, Unit 9: Mesurement: Two-Dimensionl Figures Notes Precision nd Accurcy Syllus Ojective: (6.) The student will red the pproprite mesurement tool to the required degree of ccurcy. We often use numers
More information1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.
Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationSpecial Relativity solved examples using an Electrical Analog Circuit
1-1-15 Specil Reltivity solved exmples using n Electricl Anlog Circuit Mourici Shchter mourici@gmil.com mourici@wll.co.il ISRAE, HOON 54-54855 Introduction In this pper, I develop simple nlog electricl
More informationFINALTERM EXAMINATION 9 (Session - ) Clculus & Anlyticl Geometry-I Question No: ( Mrs: ) - Plese choose one f ( x) x According to Power-Rule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationIndividual Contest. English Version. Time limit: 90 minutes. Instructions:
Elementry Mthemtics Interntionl Contest Instructions: Individul Contest Time limit: 90 minutes Do not turn to the first pge until you re told to do so. Write down your nme, your contestnt numer nd your
More informationThe Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
More informationCHAPTER 20: Second Law of Thermodynamics
CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationSection 7.2 Velocity. Solution
Section 7.2 Velocity In the previous chpter, we showed tht velocity is vector becuse it hd both mgnitude (speed) nd direction. In this section, we will demonstrte how two velocities cn be combined to determine
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles
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 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 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 information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information