# 1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look for mistkes. If you hve trouble, sk mth tech or someone else who understnds the topic. TOPIC : ELEMENTARY OPERATIONS with NUMERICAL nd ALGEBRAIC FRACTIONS A. Simplifying frctions (by reducing): exmple: (note tht you must be ble to find common fctor in this cse 9 in both the top nd bottom in order to reduce frction). exmple: b 4b 4b 4 b 4b (common fctor: ) Problems -8: Reduce: 4x+7y y b 5+c x 6xy 5by x 8. 4(x+)(x ) (x )(x ) B. Equivlent frctions (equivlent rtios): exmple: is equivlent to how mny eighths? 4 4 8, exmple: 6 5 5b, 6 b 6 6b 5 b 5 5b exmple: x+ x+ 4( x+), x+ 4 x+ x+8 x+ 4 x+ 4x+4 exmple: x x+ ( x+) ( x ), x ( x ) ( x ) x+ ( x ) ( x+) x x+ ( x+) ( x ) Problems 9-: Complete: b ( +b) ( b) 0. x 7 7y. x 6 6 x x+. x+ ( x ) ( x+) C. Finding the lowest common denomintor: (LCD) by finding the lest common multiple (LCM) of ll denomintors: exmple: 5 6 nd 8. First find LCM of 6 nd 5: LCM 5 0, so , nd exmple: 4 nd 6 : 4 6 LCM, so 9 4, nd 6 exmple: ( x+) nd x 6( x+) : ( x + ) ( x + ) 6( x +) x + LCM ( x +) ( x + ), so ( x+) ( x+) ( x+) x+ nd x 6 x+ 4( x+) 6( x+) ( x+), x( x+) 6( x+) ( x+) Problems 4-8: Find equivlent frctions with the lowest common denomintor: 4. nd 9 7. x nd 4 x 5. x nd 5 8. x 5( x ) x 4 6. nd x+ 7x y nd 0( x ) D. Adding nd subtrcting frctions: If denomintors re the sme, combine the numertors: x exmple: x x x x y y y y If denomintors re different, find equivlent frctions with common denomintors: exmple: exmple: x + ( x+) x+ ( x ) ( x+) + ( x ) ( x ) ( x+) ( x+6+x x ) ( x+) 4x+5 ( x ) ( x+) Problems 9-6: Find the sum or difference s indicted (reduce if possible): x x b. b b+ b+ x. + x x x 6. ( x ) ( x ) + x x 4. x x 5. x x x+ x x+ ( x ) ( x ) ( x )

2 E. Multiplying frctions: Multiply the top numbers, multiply the bottom numbers, reduce if possible: exmple: exmple: 7. b 7 8. ( x+) x 4 ( x+) ( x+) ( x ) x x ( x ) ( x+) ( x ) x+6 x ( +b) 9. ( x+4) 5y 5y x 6 x y ( x y) 5 p ( 5 p ) ( +b) F. Dividing frctions: Mke compound frction nd then multiply the top nd bottom (of the lrger frction) by the LCD of both: exmple: c b bd b d b d bc bd exmple: 7 c d c d x 5x 5x exmple: x y y x y y x y 5x 0. b. 4. x+7 x 9 x. b 4 5 4xy 4y Answers: x 5. 5b x+y y b 5 +c 7. 4( x+) 8. x xy. x (cn t reduce) ( x + ) ( ). + b. 4. 6, , 5x x x 6. x( x +) ( x +), ( x +) 7., 4 x x x( x ) 8. 0( x ) ( x ), x ( x ) ( y ) 0( x ) ( x ) b b x +x x 4 x b 4 8. y x 4 ( x+) ( x ) ( 5 p) 9. +b x y 0. 9 b. x +7 x b TOPIC : OPERATIONS with EXPONENTS nd RADICALS A. Definitions of powers nd roots: Problems -0: Find the vlue: ( 4) 4. x B. Lws of integer exponents: I. b c b+c II. b c III. IV. b c V. b ( b ) c bc ( b) c c b c c c b c VI. 0 (if 0) VII. b b Problems -0: Find x:. 4 x x. 4 x 5. 4 x. 4 x 6. 8 x

3 7. x 9. c x c 4 8. b0 b x 0. y x b 5 y Problems -4: Find the vlue:. 7x 0 8. x 4 x. 4 x 9. c+ x c. 4 8x 40. x x 4. 6x x + x 6. ( ) 4. x x 7. x c+ x c Problems 44-47: Write two given wys: Given No negtive powers No frction 44. d 4 d x y 46. bc b c 47. x y z x 5 y 6 z C. Lws of rtionl exponents, rdicls: Assume ll rdicls re rel numbers: I. If r is positive integer, p is n integer, nd 0, then p r r p r p, which is rel number. (Also true if r is positive odd integer nd < 0). r r r II. b b, or b r b r III. IV. r r b r b rs r s r, or ( b ) r r b r s r, or rs s Problems 48-5: Write s rdicl: x x x r ( r ) Problems 54-57: Write s frctionl power: s Problems 58-6: Find x: x x x 6. x 8 4 x Problems 6-64: Write with positive exponents: 6. 9x 6 y b D. Simplifiction of rdicls: exmple: 6 4 exmple: exmple: exmple: 8 Problems 65-78: Simplify (ssume ll rdicls re rel numbers): x x x + x + x x x E. Rtionliztion of denomintors: 5 exmple: exmple: exmple: Problems 79-87: Simplify: b Answers:

4 x if x 0; x if x < y x c 8. x 9. x 6 4 x x 4. x 9 4. x d 8 d y 9 x 6 y 9x b 8b 47. y 9 z x y 9 z x x x x 5. x 5. x x y 64. b x x 7. x 7. 4 x x 74. if 0; if < b b TOPIC : LINEAR EQUATIONS nd INEQUALITIES A. Solving liner equtions: Add or subtrct the sme vlue on ech side of the eqution, or multiply or divide ech side by the sme vlue, with the gol of plcing the vrible lone on one side. If there re one or more frctions, it my be desirble to eliminte them by multiplying both sides by the common denomintor. If the eqution is proportion, you my wish to cross-multiply. Problems -5: Solve:. x 9 9. x 4 x +. 6x 5 0. x x+. x x x x 4. 4 x. x x 9. x+ 6. x x 5 x x 5 x x 6 x 5. x x+8 8. x x+ 6 7 B. Solving pir of liner equtions: The solution consists of n ordered pir, n infinite number of ordered pirs, or no solution. Problems 6-: Solve for the common solution(s) by substitution or liner combintions: 6. x + y 7 0. x y 5 x y 8 x + 5y 7. x + y 5. 4x y x y 4x + y 8. x y 9. x + y x 8 x + y 9. x y. x y y x 5 6x 9 y C. Anlytic geometry of one liner eqution: The grph of y mx + b is line with slope m nd y-intercept b. To drw the grph, find one point on it (such s (0, b)) nd then use the slope to find nother point. Drw the line joining the two points.

5 exmple: y x + 5 hs slope nd y-intercept 5. To grph the line, locte (0, 5). From tht point, go down (top of slope frction), nd over (right) (bottom of frction) to find second point. Drw the line joining the points. Problems 4-8: Find slope nd y-intercept, nd sketch grph: 4. y x x y 5. y x 8. x y + 6. y 4x 8 To find n eqution of non-verticl line, it is necessry to know its slope nd one of its points. Write the slope of the line through (x, y) nd the known point, then write n eqution which sys tht this slope equls the known slope. exmple: Find n eqution of the line through (-4, ) nd (-, 0). Slope 0 4+ Using (-, 0) nd (x, y), Slope y 0 ; cross multiply, x+ get y x +, or y x Problems 9-: Find n eqution of line: 9. Through (-, ) nd (-, -4) 0. Through (0, -) nd (-, -5). Through (, -) nd (5, -). Through (8, 0), with slope. Through (0, -5), with slope A verticl line hs no slope, nd its eqution cn be written so it looks like x k (where k is number). A horizontl line hs zero slope, nd its eqution looks like y k. exmple: Grph on the sme grph: x nd + y. The first eqution is x 4 The second is y 4 5 D. Anlytic geometry of two liner equtions: Two distinct lines in plne re either prllel or intersecting. They re prllel if nd only if they hve the sme slope, nd hence the equtions of the lines hve no common solutions. If the lines hve unequl slopes, they intersect in one point nd their equtions hve exctly one common solution. (They re perpendiculr if their slopes re negtive reciprocls, or one is horizontl nd the other is verticl.) If one eqution is multiple of the other, ech eqution hs the sme grph, nd every solution of one eqution is solution of the other. Problems 6-4: For ech pir of equtions in problems 6 to, tell whether the lines re prllel, perpendiculr, intersecting but not perpendiculr, or the sme line: 6. Problem Problem 0 7. Problem 7 4. Problem 8. Problem 8 4. Problem 9. Problem 9 4. Problem E. Liner inequlities: exmple: One vrible grph: solve nd grph on number line: x 7 (This is n bbrevition for {x: x 7 }) Subtrct, get x 6 Divide by, x Grph: Problems 44-50: Solve nd grph on number line: 44. x > x < x < x > x 46. x x > < x exmple: Two vrible grph: grph solution on number plne: x y > (This is n bbrevition for {( x, y): x y > }) Subtrct x, multiply by, get y < x. Grph y x, but drw dotted line, nd shde the side where y < x : - Problems 4-5: Grph nd write eqution for 4. The line through (-, 4) nd (-, ) 5. The horizontl line through (4, -) Problems 5-56: Grph on number plne: 5. y < 54. x < y + 5. y > x 55. x + y < 5. y x x y >

6 F. Absolute vlue equtions nd inequlities: exmple: x Since the bsolute vlue of both nd is, x cn be either or. Write these two equtions nd solve ech: x x x or x 5 x x 5 Grph: Problems 57-6: Solve nd grph on number line: 57. x 60. x x 6. x x exmple: x < The bsolute vlue of n number between nd (exclusive) is less thn. Write this inequlity nd solve: < x <. 6 Subtrct Multiply by, get 5 > x >. (Note tht this sys x > nd x < 5). Grph: exmple: x +. The bsolute vlue is greter thn or equl to for ny number or. So, x + x + x or x 4 x x Grph: Problems 6-66: Solve nd grph on number line: 6. x < 65. x + 6. < x x < x + < Answers: (9, -) 7. (, 4) 8. (8, 5) 9. (-4, -9) 0. ( 8 9, 9 ). ( 4, 0). no solution. (, ), where is ny number; infinite number of solutions 4. m, b m, b 6. m, b 4 7. m, b 8. m, b 9. y 5 x 0. y x. y. y x + 8. y x 5 4. x 5. y intersecting, not intersecting, not 9. intersecting, not 40. intersecting, not 4. intersecting, not 4. prllel 4. sme line 44. x > x < x x > x > x < x >

7 x ± (on number line) 58. no solution 59. x, x 6. x, 6. < x < x > or x < < x < x or x < x <0 0 0 TOPIC 4: POLYNOMIALS nd POLYNOMIAL EQUATIONS A. Solving qudrtic equtions by fctoring: If b 0, then 0 or b 0 exmple: If ( x) ( x + ) 0 then ( x) 0 or ( x + ) 0 nd thus x or x Note: there must be zero on one side of the eqution to solve by the fctoring method. exmple: 6x x Rewrite: 6x x 0 Fctor: x( x ) 0 So x 0 or x Thus, x 0 or x. 0. Problems -: Solve by fctoring:. x( x ) 0 7. x x 6 0. x x 0 8. x x. x x 9. 6x x + 4. x(x + 4) 0 0. x + x 6 5. ( x + ) x 6. ( x +) ( x ) 0. x x x 6x B. Monomil fctors: The distributive property sys b + c ( b + c) exmple: x x x(x ) exmple: 4x y + 6xy xy(x + ) Problems -7: Fctor:. + b 6. x y y x 4. b + b 7. 6x y 9x 4 y 5. 4xy +0x C. Fctoring: ( x ) ( x + b) + ( x ) ( x + c): The distributive property sys jm + jn j( m + n). Compre this eqution with the following: ( x +) ( x + ) + ( x +) ( x 4) x + ( x + ) + ( x 4) Note tht j x +, m (x + ), nd n (x 4), nd we get ( x +) ( x + 6x x 4) ( x +) x + 7x + 5 Problems 8-0: Find P which completes the eqution: 8. (x )(x ) (x )(x + ) (x )( P ) ( x ) + ( x 4) ( x ) 9. x + 4 ( x ) ( P ) 0. ( x ) ( x +) x x ( x + ) ( P ) D. The qudrtic formul: If qudrtic eqution looks like x + bx + c 0, then the roots (solutions) cn be found by using the qudrtic formul: x b± b 4c exmple: x + x 0,, b, nd c x ± 4 ( ) ± 4+ ± ± or 6 exmple: x x 0,, b, c ± +4 x ± nd 5 Problems -4: Solve: So there re two roots:. x x 6 0. x x 0. x + x 4. x x 4 0 E. Qudrtic inequlities: exmple: Solve x x < 6. First mke one side zero: x x 6 < 0. Fctor: ( x ) ( x + ) < 0. If ( x ) 0 or ( x + ) 0, then x or x.

8 These two numbers ( nd ) split the rel numbers into three sets (visulize the number line): x ( x ) ( x + ) ( x ) ( x + ) solution? x < < x < x > negtive negtive positive negtive positive positive positive negtive positive no yes no Therefore, if ( x ) ( x + )< 0, then < x < Note tht this solution mens tht x > nd x < Problems 5-9: Solve, nd grph on number line: 5. x x 6 > 0 8. x > x 6. x + x < 0 9. x + x > 0 7. x x < F. Completing the squre: x + bx will be the squre of binomil when c is dded, if c is found s follows: find hlf the coefficient of x, nd squre it this is c. Thus c b b 4, nd x + bx + c x + bx + b 4 x + b 8 exmple: x + 5x 5 4 Hlf of 5 is 5, nd 5, which must be dded to complete the squre: x + 5x x + 5 If the coefficient of x is not, fctor so it is. exmple: x x x x 6, nd ( 6) 6, so Hlf of is ( x x + 6) ( x 6), nd ( x x + 6) x x + 6. Thus, 6 (or must be dded to x x to complete the squre. Problems 0-: Complete the squre, nd tell wht must be dded: 0. x 0x. x x. x + x. x + 8x G. Grphing qudrtic functions: Problems 4-40: Sketch the grph: 4. y x 8. y (x +) 5. y x 9. y (x ) 6. y x y ( x + )(x ) 7. y x ) Answers:. 0,. 0,. 0, 4. 0, 4 5., 6., 7., 8., 9., 0.,..,. + b 4. b + b 5. x y + 5x 6. xy x y 7. x y(y x) 8. x x 9. x + x 6 0. x 4x + 4.,. ± , 4 5. x < or x > 6. < x < 0 7. no solution, no grph 8. 0 < x < 9. x < or x > 0. (x 5), dd 5, dd 4. x ( x 4), dd 9 6. (x + ), dd (,-) - -

9 TOPIC 5: FUNCTIONS 9 A. Wht functions re nd how to write them: The re of squre depends on the side length s, nd given s, we cn find the re A for tht vlue of s. The side nd re cn be thought of s n ordered pir: (s, A). For exmple, (5, 5) is n ordered pir. Think of function s set of ordered pirs with one restriction: no two different ordered pirs my hve the sme first element. Thus {(s, A) : A is the re of the squre with side length s} is function consisting of n infinite set of ordered pirs. A function cn lso be thought of s rule: for exmple, A s is the rule for finding the re of squre, given side. The re depends on the given side nd we sy the re is function of the side. ' A f (s)' is red ' A is function of s', or ' A f of s'. There re mny functions of s, the one here is s. We write this f (s) s nd cn trnslte: the function of s is s. Sometimes we write A(s) s. This sys the re is function of s, nd specificlly, it is s. In some reltions, s x + y 5, y is not function of x, since both (, 4) nd (, -4) mke the reltion true. Problems -7: Tell whether or not ech set of ordered pirs is function: {(, ) ( 0, ) }. {( 0, 5) } {(, ) (,0) }.,., 4. {(x, y) : y x nd x is ny rel number} 5. {(x, y) : x y nd y is ny rel number} 6. {(x, y) : y nd x is ny rel number} 7. {(x, y) : x 4 nd y is ny rel number} Problems 8-: Is y function of x? 8. y 4x + x 0. ( x + y) 6 9. y x. y x B. Function vlues nd substitution: If A(s) s, A(), red A of, mens replce every s in A(s) s with, nd find the re when s is. When we do this, we find A() 9. exmple: g( x) is given: y g(x) πx exmple: g() π 9π exmple: g(7) π 7 49π exmple: g() π exmple: g(x + h) π(x + h) πx + πxh + πh. Given y f ( x) x ; complete these ordered pirs: (, ), (0, ), (, ), (, 0), (, -), ( x, ) Problems -7: Given f (x) x x+ Find:. f 6. f ( ) 4. f ( ) 7. f ( x ) 5. f ( 0) C. Composition of functions: exmple: If f (x) x, nd g( x) x, f g( x) every x in f (x) x with g( x) giving f ( g( x) ) ( g( x ), which equls (x ) x 6x + 9. exmple: g f ( ( )) g( ) ) g( 4) 4 4 is red f of g of x, nd mens replce Problems 8-6: Use f nd g s bove: 8. g( f( x) ). f( x) g( x) 9. f( g ) 4. f( x) g x) 0. g( g( x) ) 5. g( x ). f( x)+ g( x) 6. ( g( x ). f( x) g( x) exmple: If k( x) x 4x, for wht x is k( x) 0? If k( x) 0, then x 4x 0 nd since x 4x x( x 4) 0, x cn be either 0 or 4. (These vlues of x: 0 nd 4, re clled zeros of the function, becuse ech mkes the function zero.) Problems 7-0: Find x so: 7. k( x) 4 9. x is zero of x( x +) 8. k( x) 5 0. x is zero of (x x ) D. Grphing functions: An esy wy to tell whether reltion between two vribles is function or not is by grphing it: if verticl line cn be drwn which hs two or more

10 0 points in common with the grph, the reltion is not function. If no verticl line touches the grph more thn once, then it is function. exmple: x y hs this grph: Not function (the verticl line hits it more thn once). Problems -9: Tell whether or not ech of the following is function: Since y f( x), the vlues of y re the vlues of the function, which correspond to specific vlues of x. The heights of the grph bove (or below) the x-xis re the vlues of y nd so lso of the function. Thus for this grph, f is the height (vlue) of the function t x nd vlue is : At x, the vlue (height) of f ( x)is zero; in other words, f ( ) 0. Note tht f > f ( ), since > 0, nd tht f ( 0) < f( ), since f ( ) nd f ( 0) <. Problems 40-44: For this grph, tell whether the sttement is true or flse: 40. g( ) g( 0) 4. g( )> g 4. g( 0) g 44. g< g( 0) < g( 4) 4. g> g( ) To grph y f( x), determine the degree of f( x) if it is polynomil. If it is liner (first degree) the grph is line, nd you merely plot two points (select ny x nd find the corresponding y) nd drw their line. If f( x)is qudrtic (second degree), its grph is prbol, opening upwrd if the coefficient of x is positive, downwrd if negtive. To plot ny grph, it cn be helpful to find the following: ) The y-intercept (find f ( 0) to locte y-xis crossing) b) The x-intercept (find x for which f ( x) 0 x-xis crossing) c) Wht hppens to y when x is very lrge (positive) or very smll negtive? d) Wht hppens to y when x is very close to number which mkes the bottom of frction zero? e) Find x in terms of y, nd find wht hppens to x s y pproches number which mkes the bottom of frction zero. (d, e, nd sometimes c bove will help find verticl nd horizontl symptotes.) exmple: y x nlysis: qudrtic function (due to x ) y-intercept: f ( 0) x-intercepts: y x 0 so x ± exmple: y x+ ) y-intercepts: f ( 0) b) x-intercept: none, since there is no solution to y 0 x+ c) Lrge x: negtive y pproches zero; very negtive x mkes y positive nd going to zero. (So y 0, the x-xis, is n symptote line.) d) The bottom of the frction, x +, is zero if x. As x moves to from the left, y gets very lrge positive, nd if x pproches from bove, y becomes very negtive. (The line x is n symptote.)

11 Grph: e) To solve for x, multiply by x + nd divide by y to get x + y or x y y y Note tht y close to zero results in very positive or negtive x nd mens y 0 is n symptote, which we lredy found bove in prt c. Problems 45-54: Sketch the grph: 45. f ( x) x 50. y x 46. y 4 x 5. f ( x) 47. y x 5. y x 48. y x 5. y x 49. y x + x y x x Answers:. yes. no. yes 4. yes 5. no 6. yes 7. no 8. yes 9. no ( intersecting lines) 0. no ( prllel lines). yes. 7, -,, 4,, x none (undefined) 7. x x 8. x x 6. x + x. x x. x x + 4. x x 5. x 6. x 6x , 5 9., 0 0.,. yes. no. yes 4. no 5. yes 6. yes 7. yes 8. no 9. yes 40. F 4. T 4. T 4. T 44. T TOPIC 6: TRIGONOMETRY A. Trig functions in right tringles: The sine rtio for n cute ngle of right tringle is defined to be the length of the opposite leg to the length of the hypotenuse. Thus the sine rtio for ngle B, bbrevited sin B, is b c. c B A b C The reciprocl of the sine rtio is the cosecnt (csc), so csc B c b. The other four trig rtios (ll functions) re cosine cos djcent leg hypotenuse secnt sec cos hypotenuse djcent leg tngent tn cotngent cot ctn opposite leg djcent leg djcent leg opposite leg Problems -8: For this right tringle, give the following rtios: x. tn x. sin x 6 0 cos x θ. cos θ 8

12 4. sinθ cosθ 5. cos x, which mens (cos x) 6. sin x 8. sinθ cosθ 7. cosx sinθ B. Circulr trig definitions: Given circle with rdius r, centered on (0, 0). Drw the rdius connecting line from the vertex to ny point on the circle, mking n ngle θ with the positive x-xis (θ my be ny rel number, positive mesure is counterclockwise). The coordintes (x, y) of point P together with rdius r re used to define the functions: sinθ y r, cosθ x r, tnθ y x, nd the reciprocl functions s before. (Note tht for 0<θ < π, these definitions gree P( x, y) with the right tringle definitions. Also note tht sinθ, cosθ, nd tnθ cn be ny rel number.) Problems 9-: For the point (-, 4) on the bove circle, give: 9. x y r 0. tnθ. cosθ. cotθ sinθ Note tht for ny given vlue of trig function, (in its rnge), there re infinitely mny vlues of θ. Problems -4: Find two positive nd two negtive vlues θ for which:. sinθ 4. tnθ tn45 Problems 5-6: Given sin θ 5 nd π < θ < π, then: 5. tnθ 6. cosθ C. Pythgoren reltions (identities): + b c (or x + y r ) bove, cn be divided by c (or r ) to give c + b c c c, or sin A + cos A, (or sin θ + cos θ ), clled n identity becuse it is true for ll vlues of A for which it is defined. y r x θ Problems 7-8: Get similr identity by dividing + b c by: 7. b 8. D. Similr tringles: If ABC ~ DEF, nd if tn A 4, then tn D 4 lso, since EF : DF BC : AC Find DC, given DB 5 nd sin E.4 E. Rdins nd degrees: For ngle θ, there is point P on the circle, nd n rc from A counter-clockwise to P. The length of the rc is θ 60 C θ 60 πr, nd the rtio of the length of rc to rdius is π θ, where θ is 80 the number of degrees (nd the rtio hs no units). This is the rdin mesure ssocited with point P. So P cn be locted two wys: by giving the centrl ngle θ in degrees, or in number of rdii to be wrpped round the circle from point A (the rdin mesure). Converting: rdins π 80 degrees or degrees 80 π rdins. π exmple: (rdins) π π exmple: π 7 π (rdins) 80 (which mens tht it would tke little over 7 rdii to wrp round the circle from A to 40.) Problems 0-: Find the rdin mesure for centrl ngle of: Problems 4-6: Find the degree mesure, which corresponds to rdin mesure of: 4. π π 6 Problems 7-9: Find the following vlues by sketching the circle, centrl ngle, nd verticl segment from point P to the x-xis. (Rdin

13 mesure if no units re given.) Use no tbles or clcultor. 7. cos 5π 9. sin( 5 ) 6 8. tn( 5 ) Problems 0-: Sketch to evlute without tble or clcultor: 0. sec80. sinπ. cot( π ). cos π F. Trigonometric equtions: exmple: Solve, given tht 0 θ < π : tn θ tnθ 0. Fctoring, we get tnθ(tnθ ) 0, which mens tht tnθ 0 or tnθ. Thus θ 0 (degrees or rdins) plus ny multiple of 80 (or π ), which is n 80 (or n π ), or θ 45 ( π 4 rdins), or 45 + n 80 (or π 4 + n π ). Thus, θ cn be 0, π 4, π, or 5π 4, which ll check in the originl eqution. Problems 4-9: Solve, for 0 θ π : 4. sinθ cosθ 7. sinθ cotθ 5. sin θ + cosθ 8. cosθ tnθ 6. sin θ cos θ 9. sinθ tnθ secθ G. Grphs of trig functions: By finding vlues of sin x when x is multiple of π, we cn get quick sketch of y sin x. The sine is periodic (it repets every π, its period). sin x never exceeds one, so the mplitude of sin x is, nd we get this grph: To grph y sinx, we note tht for given vlue of x, sy x, the vlue of y is found on the grph of y sin x three times s fr from the y-xis s. Thus ll points of the grph. y sinx re found by moving ech point of y sin x grph to its previous distnce from the y- xis, showing the new grph repets times in the period of y sin x, so the period of y sin x, so the period of y sinx is π. Problems 40-46: Sketch ech grph nd find its period nd mplitude: 40. y cos x 44. y 4sin x 4. y cosx 45. y sin x 4. y tn x 46. y + cos x 4. y tn x H. Identities: exmple: Find formul for cosa, given cos A + B Substitute A for B: cos A cos A + A cos A sin A cos A cosb sin Asin B. cos A cos A sin Asin A Problems 47-49: Use sin x + cos x nd the bove to show: 47. cosa cos A 48. cosa sin A 49. cos x + cos x 50. Given cos A, sina tn A, nd seca cosa sin A + cos A, show tht tn x + sec x. 5. Given sin A sin Acos A, show 8sin x cos x 4sin x. Answers: , 4, π, π, π, 7π 4. 5, 5, 45, tn A + sec A 8. + cot A csc A π 5. π 4. π. 7π ( 540 π)

14 π 4, 5π 4 5. π, π, π 6. 0 θ < π 7. π, 5π 8. π 9. π, π 40. π 4. - π - π P π A P π A π 4 π π π - π π - π π P π no A P π no A P π A 4 P π A P π A 47. cosa cos A sin A cos A ( cos A) cos A 48. cosa cos A sin A ( sin A) sin A sin A 49. From problem 47, cosx cos x. Add, divide by. 50. tn A + sin A cos A + sin A +cos A cos A cos A sec A 5. Multiply given by 4, then let A x TOPIC 7: LOGARITHMIC nd EXPONENTIAL FUNCTIONS A. Logrithms nd exponents: Exponentil form: 8 Logrithmic form: log 8 Both of the equtions bove sy the sme thing. log 8 is red log bse two of eight equls three nd trnsltes the power of which gives 8 is. Problems -4: Write the following informtion in both exponentil nd logrithmic forms:. The power of, which gives 9 is.. The power of x, which gives x is.. 0 to the power is 00 is the power of 69 which gives. 4. exponent rules: (ll quntities rel) b c b+c b c b c ( b ) c bc ( b) c c b c c c b b c 0 (if 0) b b p p r r p r (think of p r s power root ) log rules: (bse ny positive rel number except ) log b log + logb log log logb b log b blog log b b (log b) b log b log c b log c (bse chnge rule) Problems 5-5: Use the exponent nd log rules to find the vlue of x: x x 6. x 9. log 7 x 7. x log 6 x 8. log 0 x log log 0. log x 5 5. log 4 log x. log 6. log x log x log 6 x 7. log (7 4 ) x 8. log( x 6) log( 6 x) 9. log 4 64 x. 7 x x x. log 8 log log 7 8 x 7 5. x 4. log 4 0 x log 4 0 Problems 6-: Find the vlue: log log log 6 6. log Find log, given: log log 0.0 Problems -4: Given log 04 0, find:. log log 04 Problems 5-46: Solve for x in terms of y nd z: 5. x y z 6. 9 y z x

15 7. x y 9. x y 8. x y 40. log x log y 4. log x log y logz 4. log x log y 4. log x log y + log z 44. log x + log y logz 45. log 7 y; log 7 z x log 46. y log 9; x log B. Inverse functions nd grphing: If y f( x)nd y g( x) re inverse functions, then n ordered pir (, b) stisfies y f( x) if nd only if the ordered pir (b, ) stisfies y g( x). In other words, f nd g re inverses of ech other mens f ( ) b if nd only if g( b) To find the inverse of function y f( x) ) Interchnge x nd y ) Solve this eqution for y in terms of x, so y g( x) ) Then if g is function, f nd g re inverses of ech other. The effect on the grph of y f( x) when x nd y re switched, is to reflect the grph over the 45 line (bisecting qudrnts I nd III). This reflected grph represents y g( x). exmple: Find the inverse of f ( x) x or y x ) Switch x nd y: x y (note y nd x 0) l f 5 ) Solve for y : x y, so y x + ( x 0 is still true) ) Thus g(x) x + (with x 0) is the inverse function, nd hs this grph: Note tht the f nd g grphs re reflections of ech other in the 45 line, nd tht the ordered pir (,5) stisfies g nd (5,) stisfies f. exmple: Find the inverse of y f (x) x nd grph both functions on one grph: ) Switch: x y ) Solve: log x y g(x), the inverse to get the grph of g(x) log x, reflect the f grph over the 45 line: Problems 47-48: Find the inverse function nd sketch the grphs of both: 47. f ( x) x 48. f (x) log ( x) (note tht x must be positive, which mens x must be negtive) Problems 49-56: Sketch the grph: 49. y x 4 5. y 4 x 50. y 4 x 54. y log 4 x 5. y 4 x 55. y 4 x 5. y log 4 x 56. y log 4 (x ) l f f l l g l g Answers:. 9, log 9. x x,log x x. 0 00,log , log ny rel number > 0 nd log 4 log 5. (ny bse; if log bse, x log ) y + z 6. z y 7. y 8. log y 9. y log 40. y y y

16 4. y z 4. y 4. yz z y z y 46. y 47. g(x) x g(x) x g f f g (,) TOPIC 8: MATHEMATICAL MODELING WORD PROBLEMS Word Problems:. of 6 of 4 of number is. Wht is the number?. On the number line, points P nd Q nd units prt. Q hs coordinte x. Wht re the possible coordintes of P?. Wht is the number, which when multiplied by, gives 46? 4. If you squre certin number, you get 9. Wht is the number? 5. Wht is the power of 6 tht gives 6? 6. Point X is on ech of two given intersecting lines. How mny such points X re there? 7. Point Y is on ech of two given circles. How mny such points Y? 8. Point Z is on ech of given circle nd given ellipse. How mny such Z? 9. Point R is on the coordinte plne so its distnce from given point A is less thn 4. Show in sketch where R could be. Problems 0-: A 0. If the length of chord AB is x nd length of CB is 6, wht is AC?. If AC y nd CB z, how long is AB (in terms of y nd z)?. This squre is cut into two smller squres nd two non-squre rectngles s shown. C O B Before being cut, the lrge squre hd re ( + b). The two smller squres hve res nd b. Find the totl re of the two nonsqure rectngles. Do the res of the 4 prts dd up to the re of the originl squre?. Find x nd y: 4. When constructing n equilterl tringle with n re tht is 00 times the re of given equilterl tringle, wht length should be used for side? Problems 5-6: x nd y re numbers, nd two x s equl three y s: 5. Which of x or y must be lrger? 6. Wht is the rtio of x to y? Problems 7-: A plne hs certin speed in still ir. In still ir, it goes 50 miles in hours: 7. Wht is its (still ir) speed? 8. How long does it tke to fly 000 miles? 9. How fr does the plne go in x hours? 0. If the plne flies ginst 50 mph hedwind, wht is its ground speed?. If it hs fuel for 7.5 hours of flying time, how fr cn it go ginst this hedwind? Problems -5: Georgie nd Porgie bke pies. Georgie cn complete 0 pies n hour.. How mny cn he mke in one minute?. How mny cn he mke in 0 minutes? 4. How mny cn he mke in x minutes? 5. How long does he tke to mke 00 pies?

17 7 Problems 6-8: Porgie cn finish 45 pies n hour: 6. How mny cn she mke in one minute? 7. How mny cn she mke in 0 minutes? 8. How mny cn she mke in x minutes? Problems 9-: If they work together, how mny pies cn they produce in: 9. minute. 80 minutes 0. x minutes. hours Problems -4: A nurse needs to mix some lcohol solutions, given s percent by weight of lcohol in wter. Thus in % solution, % of the weight would be lcohol. She mixes x grms of % solution, y grms of 0% solution, nd 0 grms of pure wter to get totl of 40 grms of solution which is 8% lcohol:. In terms of x, how mny grms of lcohol re in the % solution? 4. The y grms of 0% solution would include how mny grms of lcohol? 5. How mny grms of solution re in the finl mix (the 8% solution)? 6. Write n expression in terms of x nd y for the totl number of grms in the 8% solution contributed by the three ingredients (the %, 0%, nd wter). 7. Use your lst two nswers to write totl grms eqution. 8. How mny grms of lcohol re in the 8%. 9. Write n expression in terms of x nd y for the totl number of grms of lcohol in the finl solution. 40. Use the lst two nswers to write totl grms of lcohol eqution. 4. How mny grms of ech solution re needed? 4. Hlf the squre of number is 8. Wht is the number? 4. If the squre of twice number is 8, wht is the number? 44. Given positive number x. The squre of positive number y is t lest 4 times x. How smll cn y be? 45. Twice the squre of hlf of number is x. Wht is the number? Problems 46-48: Hlf of x is the sme s onethird of y. 46. Which of x nd y is the lrger? 47. Write the rtio x: y s the rtio of two integers. 48. How mny x s equl 0 y s? Problems 49-50: A gthering hs twice s mny women s men. If W is the number of women nd M is the number of men 49. Which is correct: MW or MW W 50. Write the rtio s the rtio of two M +W integers. Problems 5-5: If A is incresed by 5%, it equls B. 5. Which is lrger, B or the originl A? 5. B is wht percent of A? 5. A is wht percent of B? Problems 54-56: If C is decresed by 40%, it equls D. 54. Which is lrger, D or the originl C? 55. C is wht percent of D? 56. D is wht percent of C? Problems 57-58: The length of rectngle is incresed by 5% nd its width is decresed by 40%. 57. Its new re is wht percent of its old re? 58. By wht percent hs the old re incresed or decresed? Problems 59-6: Your wge is incresed by 0%, then the new mount is cut by 0% (of the new mount). 59. Will this result in wge, which is higher thn, lower thn, or the sme s the originl wge? 60. Wht percent of the originl wge is this finl wge? 6. If the bove steps were reversed (0% cut followed by 0% increse), the finl wge would be wht percent of the originl wge? Problems 6-75: Write n eqution for ech of the following sttement bout rel numbers nd tell whether it is true or flse: 6. The product of the squres of two numbers is the squre of the product of the two numbers. 6. The squre of the sum of two numbers is the sum of the squres of the two numbers 64. The squre of the squre root of number is the squre root of the squre of the number. 65. The squre root of the sum of the squres of two numbers is the sum of the two numbers. 66. The sum of the bsolute vles of two numbers is the bsolute vlue of the sum of the two numbers.

18 67. The product of the bsolute vlues of two numbers is the bsolute vlue of the product of the two numbers. 68. The negtive of the bsolute vlue of number is the bsolute vlue of the negtive of the number. 69. The cube root of the squre of number is the cube of the squre root of the number. 70. The reciprocl of the negtive of number is the negtive of the reciprocl of the number. 7. The reciprocl of the sum of two numbers is the sum of the reciprocls of the two numbers The reciprocl of the product of two numbers is the product of the reciprocls of the two numbers. 7. The reciprocl of the quotient of two numbers is the quotient of the reciprocls of the two numbers. 74. The multiple of x which gives xy is the multiple of y which gives yx. 75. The power of, which gives xy is the power of, which gives yx. Answers:. 44. x +, x ,, or 8. 0,,,, or 4 9. Inside circle of rdius 4 centered on A 0. x 6. y + z. b, yes: ( + b) + b + b. x 40 7 ; y times the originl side 5. x mph hours x miles mph. 000 miles x min x x x 4..y x + y x + y x +.y 40..0x +.y. 4. x 80 7 ; y , , x 45. x 46. y 47. : M W B 5. 5% 5. 80% 54. C % % % 58. 5% decrese 59. lower % 6. sme (96%) 6. b (b) T 6. ( + b) + b F T (if ll expressions re rel) b + b F b + b F 67. b b T 68. F 69. F 70. T 7. +b + b F 7. b b T 7. b b T 74. y x F 75. xy yx T

### Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

### PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

### NCERT INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS. Trigonometric Ratios of the angle A in a triangle ABC right angled at B are defined as:

INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: side opposite to A BC sine of A = sin A = hypotenuse

### Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

### Math 135 Circles and Completing the Square Examples

Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

### Section 5-4 Trigonometric Functions

5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

### Factoring Polynomials

Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

### Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

### Mathematics Higher Level

Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

### Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

### Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

### Section 7-4 Translation of Axes

62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

### SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

### MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

### Curve Sketching. 96 Chapter 5 Curve Sketching

96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

### Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

### Warm-up for Differential Calculus

Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

### LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

### FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

### Section A-4 Rational Expressions: Basic Operations

A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

### PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

### THE RATIONAL NUMBERS CHAPTER

CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted

### www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

### AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

### Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

### Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

### Volumes of solids of revolution

Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the x-xis. There is strightforwrd technique which enbles this to be done, using

### 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

### Unit 6: Exponents and Radicals

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

### Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

### Basic Analysis of Autarky and Free Trade Models

Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

### Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

### Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

### P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

### 6.2 Volumes of Revolution: The Disk Method

mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

### Double Integrals over General Regions

Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

### 1 Numerical Solution to Quadratic Equations

cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

### Vectors 2. 1. Recap of vectors

Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

### Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006

dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.

### 4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

### addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

### QUANTITATIVE REASONING

Guide For Exminees Inter-University Psychometric Entrnce Test QUNTITTIVE RESONING The Quntittive Resoning domin tests your bility to use numbers nd mthemticl concepts to solve mthemticl problems, s well

### Lesson 12.1 Trigonometric Ratios

Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

### Applications to Physics and Engineering

Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

### EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

### 5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

### Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

### Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I

Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the

### 9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

### Answer, Key Homework 4 David McIntyre Mar 25,

Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his print-out should hve 18 questions. Multiple-choice questions my continue on the next column or pe find ll choices before mkin your selection.

### Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

### Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

### Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

### . At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

### Experiment 6: Friction

Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

### Lecture 5. Inner Product

Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

### PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

### ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS

CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common

### Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

### Helicopter Theme and Variations

Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

### Plotting and Graphing

Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

### Right Triangles and Trigonometry

9 Right Tringles nd Trigonometry 9.1 The Pythgoren Theorem 9. Specil Right Tringles 9.3 Similr Right Tringles 9.4 The Tngent Rtio 9.5 The Sine nd osine Rtios 9.6 Solving Right Tringles 9.7 Lw of Sines

### The remaining two sides of the right triangle are called the legs of the right triangle.

10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

### MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

### Or more simply put, when adding or subtracting quantities, their uncertainties add.

Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

### Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

### Square & Square Roots

Squre & Squre Roots Squre : If nuber is ultiplied by itself then the product is the squre of the nuber. Thus the squre of is x = eg. x x Squre root: The squre root of nuber is one of two equl fctors which

### The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

### Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

### A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

### SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS)

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470 - COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions

### The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

### Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

### CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

### Linear Equations in Two Variables

Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then

### Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

### Pure C4. Revision Notes

Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

### and thus, they are similar. If k = 3 then the Jordan form of both matrices is

Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

### Review guide for the final exam in Math 233

Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

### 9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

### 2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

### Lecture 3 Gaussian Probability Distribution

Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

### Integration by Substitution

Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### SAT Subject Math Level 2 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

### 2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

### Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

### LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

### Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.

### 5.6 POSITIVE INTEGRAL EXPONENTS

54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

### Exponential and Logarithmic Functions

Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

### 15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time