Glossry Glossry solute mimum nd minimum [p. 455] For continuous function f defined on n intervl [, ]: the solute mimum is the vlue M of the function f such tht f ( M for ll [, ] the solute minimum is the vlue N of the function f such tht f ( N for ll [, ]. ccelertion [p. 39] the rte of chnge of prticle s velocity with respect to time ccelertion, verge [p. 39] The verge ccelertion of prticle for the time intervl [t, t 2 ] is given y v 2 v, where v 2 is the velocity t 2 t t time t 2 nd v is the velocity t time t. ccelertion, instntneous [pp. 39, 442] = dv dt = d2 dt 2 ddition rule for choices [p. 743] To determine the totl numer of choices from disjoint lterntives, simply dd up the numer of choices ville for ech lterntive. ddition rule for proility [p. 58] The proility of or or oth occurring is given y Pr( = Pr( + Pr( Pr( mplitude of trigonometric functions [p. 40] The distnce etween the men position nd the mimum position is clled the mplitude. The grph of y = sin hs n mplitude of. ngle of depression [p. 497] the ngle etween the horizontl nd direction elow the horizontl eye level cliff ngle of depression line of sight ngle of elevtion [p. 497] the ngle etween the horizontl nd direction ove the horizontl eye level line of sight ngle of elevtion nti-derivtive [p. 347] To find the generl nti-derivtive of f (: If F ( = f (, then f ( d = F( + c where c is n ritrry rel numer. rc [MM&2] Two points on circle divide the circle into rcs; the shorter is the minor rc, nd the longer is the mjor rc. SMPLE FINL PGES rc length, l [MM&2] The length of rc is given y l = rθ, where θ c = O. Note: The glossry contins some terms from Mthemticl Methods Units & 2 [MM&2]. D O θ r
Glossry 75 re of tringle [p. 494] given y hlf the product of the lengths of two sides nd the sine of the ngle included etween them. re = 2 h re = c sin 2 rithmetic sequence [MM&2] sequence in which ech successive term is found y dding fied mount to the previous term; e.g. 2, 5, 8,,.... n rithmetic sequence hs recurrence reltion of the form t n = t n + d, where d is the common difference. The nth term cn e found using t n = + (n d, where = t. rithmetic series [MM&2] the sum of the terms in n rithmetic sequence. The sum of the first n terms is given y the formul S n = n ( 2 + (n d 2 where = t nd d is the common difference. rrngements [p. 744] counted when order is importnt. The numer of wys of selecting nd rrnging r ojects from totl of n ojects is n! = n (n (n 2 (n r + (n r! symptote [MM&2] stright line is n symptote of the grph of function y = f ( if the grph of y = f ( gets ritrrily close to the stright line. n symptote cn e horizontl, verticl or olique. verge vlue [p. 402] The verge vlue of continuous function f for n intervl [, ] is defined s f ( d. ering [p. 498] the compss ering; the direction mesured from north clockwise ernoulli rndom vrile [p. 560] rndom vrile tht tkes only the vlues (indicting success nd 0 (indicting filure ernoulli sequence [p. 560] sequence of repeted trils with the following properties: Ech tril results in one of two outcomes, usully designted s success or filure. The proility of success on single tril, p, is constnt for ll trils. The trils re independent. (The outcome of tril is not ffected y outcomes of other trils. h inomil distriution [p. 562] The proility of oserving successes in n independent trils, ech with proility of success p, is given y ( n Pr(X = = p ( p n, = 0,,..., n ( n n! where =! (n! inomil epnsion [p. 748] n ( n ( + n = n k k k k=0 ( ( n n = n + n + n 2 2 + + n 2 ( n The (r + st term is n r r. r inomil eperiment [p. 562] ernoulli sequence of n independent trils, ech with proility of success p hin rule [p. 282] The chin rule cn e used to differentite complicted function y = f ( y trnsforming it into two simpler functions, which re chined together: h u g y Using Leiniz nottion, the chin rule is stted s dy d = dy du du d hnge of se [p. 2] log = log c log c ircle, generl eqution [p. 7] The generl eqution for circle is ( h 2 + (y k 2 = r 2, where the centre is (h, k nd the rdius is r. oefficient [p. 90] the numer tht multiplies power of in polynomil. E.g. for 2 5 7 2 + 4, the coefficient of 2 is 7. omintions [p. 744] see selections ommon difference, d [MM&2] the difference etween two consecutive terms of n rithmetic sequence, i.e. d = t n t n SMPLE FINL PGES ommon rtio, r [MM&2] the quotient of two consecutive terms of geometric sequence, i.e. r = t n t n ompss ering [p. 498] the direction mesured from north clockwise omplement, [p. 58] the set of outcomes tht re in the smple spce, ε, ut not in. The proility of the event is Pr( = Pr(. Glossry
752 Glossry Glossry omplementry reltionships [p. 38] ( π ( π sin 2 θ = cos θ sin 2 + θ = cos θ ( π ( π cos 2 θ = sin θ cos 2 + θ = sin θ omposition of functions [p. 23] For two functions f nd g, the function with rule h( = f (g( is the composition of f with g. We write h = f g. For emple, if f ( = 4 nd g( = +, then ( f g( = f (g( = ( + 4. ompound interest [MM&2] is clculted t regulr intervls on the totl of the mount originlly invested nd the interest ccumulted over the previous yers. If $P is invested t R% p.. compounded nnully, then the vlue of the investment fter n yers, $ n, is given y n = Pr n, where r = + R 00 oncvity [p. 447] If f ( > 0 for ll (,, then the grdient of the curve is incresing over the intervl; the curve is sid to e concve up. If f ( < 0 for ll (,, then the grdient of the curve is decresing over the intervl; the curve is sid to e concve down. onditionl proility [p. 526] the proility of n event occurring when it is known tht some event hs occurred, given y Pr( Pr( = Pr( onfidence intervl [p. 678] n intervl estimte for the popultion proportion p sed on the vlue of the smple proportion ˆp ongruence tests [p. 480] Two tringles re congruent if one of the following conditions holds: SSS the three sides of one tringle re equl to the three sides of the other tringle SS two sides nd the included ngle of one tringle re equl to two sides nd the included ngle of the other tringle S two ngles nd one side of one tringle re equl to two ngles nd the mtching side of the other tringle RHS the hypotenuse nd one side of right-ngled tringle re equl to the hypotenuse nd one side of nother right-ngled tringle. ongruent figures [p. 480] hve ectly the sme shpe nd size onstnt function [MM&2] function with rule of the form f ( = ; e.g. f ( = 7 ontinuous function [p. 274] function f is continuous t the point = if lim f ( = f (. ontinuous rndom vrile [p. 582] rndom vrile X tht cn tke ny vlue in n intervl of the rel numer line onvergent series [MM&2] n infinite series t + t 2 + t 3 + is convergent if the sum of the first n terms, S n, pproches limiting vlue s n. n infinite geometric series is convergent if < r <, where r is the common rtio. oordintes [MM&2] n ordered pir of numers tht identifies point in the rtesin plne; the first numer identifies the position with respect to the -is, nd the second numer identifies the position with respect to the y-is osine function [p. 3] cosine θ is defined s the -coordinte of the point P on the unit circle where OP forms n ngle of θ rdins with the positive direction of the -is. - O - y θ cos θ P(θ = (cos θ, sin θ sin θ osine rule [p. 490] For tringle : c 2 = 2 + 2 2 cos or equivlently cos = 2 + 2 c 2 2 SMPLE FINL PGES c The cosine rule is used to find unknown quntities in tringle given two sides nd the included ngle, or given three sides. uic function [p. 03] polynomil of degree 3. cuic function f hs rule of the form f ( = 3 + 2 + c + d, where 0. umultive distriution function [p. 60] gives the proility tht the rndom vrile X tkes vlue less thn or equl to ; tht is, F( = Pr(X = f (t dt
Glossry 753 D Definite integrl [pp. 378, 384] f ( d denotes the signed re enclosed y the grph of y = f ( etween = nd =. Degree of polynomil [p. 90] given y the highest power of with non-zero coefficient; e.g. the polynomil 2 5 7 2 + 4 hs degree 5. Dependent vrile [MM&2] If one vrile, y, cn e epressed s function of nother vrile,, then the vlue of y depends on the vlue of. We sy tht y is the dependent vrile nd tht is the independent vrile. Derivtive function [p. 256] lso clled the grdient function. The derivtive f of function f is given y f f ( + h f ( ( = lim h 0 h Derivtives, sic [pp. 259, 27, 285 296] f ( f ( c 0 where c is constnt where R \ {0} e k ln(k sin(k cos(k ke k k cos(k k sin(k Difference of two cues [p. 99] 3 y 3 = ( y( 2 + y + y 2 Difference of two squres [MM&2] 2 y 2 = ( y( + y Differentile [p. 275] function f is sid to e differentile t the point = if f ( + h f ( lim eists. h 0 h Differentition rules [p. 259] Sum: f ( = g( + h(, f ( = g ( + h ( Multiple: f ( = k g(, f ( = k g ( see lso chin rule, product rule, quotient rule Diltion from the -is [p. 50] diltion of fctor from the -is is descried y the rule (, y (, y. The curve with eqution y = f ( is mpped to the curve with eqution y = f (. Diltion from the y-is [p. 50] diltion of fctor from the y-is is descried y the rule (, y (, y. The curve with eqution y = f ( is mpped to the curve with eqution y = f (. Discontinuity [p. 274] function is sid to e discontinuous t point if it is not continuous t tht point. Discrete rndom vrile [p. 534] rndom vrile X which cn tke only countle numer of vlues, usully whole numers Discriminnt,, of qudrtic [p. 82] the epression 2 4c, which is prt of the qudrtic formul. For the qudrtic eqution 2 + + c = 0: If 2 4c > 0, there re two solutions. If 2 4c = 0, there is one solution. If 2 4c < 0, there re no rel solutions. Disjoint [p. 2] Two sets nd re disjoint if they hve no elements in common, i.e. =. Displcement [p. 36] The displcement of prticle moving in stright line is defined s the chnge in position of the prticle. Distnce etween two points [p. 43] The distnce etween points (, y nd ( 2, y 2 is = ( 2 2 + (y 2 y 2 Domin [p. 6] the set of ll the first coordintes of the ordered pirs in reltion E Element [p. 2] memer of set. If is n element of set, we write. If is not n element of set, we write. Empty set, [p. 2] the set tht hs no elements Equting coefficients [p. 92] Two polynomils P nd Q re equl only if their corresponding coefficients re equl. For emple, two cuic polynomils P( = 3 3 + 2 2 + + 0 nd Q( = 3 3 + 2 2 + + 0 re equl if nd only if 3 = 3, 2 = 2, = nd 0 = 0. Euler s numer, e [p. 96] the nturl se for eponentil nd logrithmic functions: e = lim ( + n = 2.78 28... n n Even function [p. 8] function f is even if f ( = f ( for ll in the domin of f ; the grph is symmetric out the y-is. SMPLE FINL PGES Event [p. 56] suset of the smple spce (tht is, set of outcomes Epected vlue of rndom vrile, E(X [pp. 542, 595] lso clled the men, µ. For discrete rndom vrile X: E(X = Pr(X = = p( For continuous rndom vrile X: E(X = f ( d Eponentil function [p. 90] function f ( = k, where k is non-zero constnt nd the se is positive rel numer other thn Glossry D E
754 Glossry Glossry F I F Fctor [MM&2] numer or epression tht divides nother numer or epression without reminder Fctor theorem [p. 98] If β + α is fctor of polynomil P(, then P ( α β = 0. onversely, if P ( α β = 0, then β + α is fctor of P(. Fctorise [MM&2] epress s product of fctors Formul [MM&2] n eqution contining symols tht sttes reltionship etween two or more quntities; e.g. = lw (re = length width. The vlue of, the suject of the formul, cn e found y sustituting given vlues of l nd w. Function [p. 8] reltion such tht for ech -vlue there is only one corresponding y-vlue. This mens tht, if (, nd (, c re ordered pirs of function, then = c. Function, verticl-line test [p. 8] used to identify whether reltion is function or not. If verticl line cn e drwn nywhere on the grph nd it only ever intersects the grph mimum of once, then the reltion is function. Fundmentl theorem of clculus [p. 380] If f is continuous function on n intervl [, ], then f ( d = F( F( where F is ny nti-derivtive of f nd f ( d is the definite integrl from to. G Geometric sequence [MM&2] sequence in which ech successive term is found y multiplying the previous term y fied mount; e.g. 2, 6, 8, 54,.... geometric sequence hs recurrence reltion of the form t n = rt n, where r is the common rtio. The nth term cn e found using t n = r n, where = t. Geometric series [MM&2] the sum of the terms in geometric sequence. The sum of the first n terms is given y the formul S n = (rn = ( rn r r where = t nd r is the common rtio. Grdient function see derivtive function Grdient of line [p. 43] The grdient is m = rise run = y 2 y 2 where (, y nd ( 2, y 2 re the coordintes of two points on the line. The grdient of verticl line (prllel to the y-is is undefined. I Implied domin see miml domin Indefinite integrl see nti-derivtive Independence [p. 529] Two events nd re independent if nd only if Pr( = Pr( Pr( Independent vrile [MM&2] If one vrile, y, cn e epressed s function of nother vrile,, then the vlue of y depends on the vlue of. We sy tht y is the dependent vrile nd tht is the independent vrile. Inde lws [p. 200] To multiply two powers with the sme se, dd the indices: y = +y To divide two powers with the sme se, sutrct the indices: y = y To rise power to nother power, multiply the indices: ( y = y Rtionl indices: m n = ( n m For se R + \ {}, if = y, then = y. Inequlity [MM&2] mthemticl sttement tht contins n inequlity symol rther thn n equls sign; e.g. 2 + < 4 Infinite geometric series [MM&2] If < r <, then the sum to infinity is given y S = r where = t nd r is the common rtio. Integers [p. 3] Z = {..., 2,, 0,, 2,... } Integrls, sic [pp. 348 358] f ( f ( d r r+ r + + c where r Q \ { } ln( + + c + for + > 0 e k k ek + c sin(k cos(k k cos(k + c k sin(k + c SMPLE FINL PGES Integrtion, properties [p. 348] f ( + g( d = f ( d + g( d k f ( d = k f ( d Integrtion (definite, properties [p. 388] c f ( d = f ( d + f ( d f ( d = 0 f ( d = f ( d c
Glossry 755 Intersection of sets [pp. 2, 56] The intersection of two sets nd, written, is the set of ll elements common to nd. Intervl [p. 4] suset of the rel numers of the form [, ], [,, (,, etc. Irrtionl numer [p. 3] rel numer tht is not rtionl; e.g. π nd 2 Itertive rule [MM&2] see recurrence reltion K Krnugh mp [p. 52] proility tle L Lw of totl proility [p. 527] In the cse of two events, nd : Pr( = Pr( Pr( + Pr( Pr( Left-endpoint method [p. 373] gives n estimte for the re under the grph of y = f ( etween = nd = : L n = [ f (0 + f ( + + f ( n ] n Limit [p. 255] The nottion lim f ( = p sys tht the limit of f (, s pproches, is p. We cn lso sy: s pproches, f ( pproches p. Liner eqution [MM&2] polynomil eqution of degree ; e.g. 2 + = 0 Liner function [MM&2] function with rule of the form f ( = m + c; e.g. f ( = 3 + Liner function of rndom vrile [p. 607] E(X + = E(X + Vr(X + = 2 Vr(X Literl eqution [p. 6] n eqution for the vrile in which the coefficients of, including the constnts, re pronumerls; e.g. + = c Logrithm [p. 202] If R + \ {} nd R, then the sttements = y nd log y = re equivlent. Logrithm, nturl [p. 203] The nturl logrithm function is given y ln = log e where the se e is Euler s numer. Logrithm lws [p. 204] log (y = log + log y ( log = log y log y ( log = log log ( p = p log Logrithmic scle [p. 227] mesurement scle tht uses the logrithm of quntity; e.g. Richter scle for erthqukes M Mrgin of error, E [p. 682] the distnce etween the smple estimte nd the endpoints of the confidence intervl Miml domin [p. 6] When the rule for reltion is given nd no domin is specified, then the domin tken is the lrgest for which the rule hs mening. Mimum nd minimum vlue [p. 455] see solute mimum nd minimum Men of rndom vrile, µ [pp. 542, 595] see epected vlue of rndom vrile, E(X Medin of rndom vrile, m [p. 598] the middle vlue of the distriution. For continuous rndom vrile, the medin is the vlue m such m tht f ( d = 0.5. Midpoint of line segment [p. 43] If P(, y is the midpoint of the line segment joining (, y nd ( 2, y 2, then = + 2 2 nd y = y + y 2 2 Model [p. 236] mthemticl representtion of rel-world sitution. For emple, n eqution tht descries the reltionship etween two physicl quntities is mthemticl model. Multipliction rule for choices [p. 743] When sequentil choices re involved, the totl numer of possiilities is found y multiplying the numer of options t ech successive stge. Multipliction rule for proility [p. 526] the proility of events nd oth occurring is Pr( = Pr( Pr( Multi-stge eperiment [p. 527] n eperiment tht could e considered to tke plce in more thn one stge; e.g. tossing two coins SMPLE FINL PGES Mutully eclusive [p. 58] Two events re sid to e mutully eclusive if they hve no outcomes in common. N n! [p. 744] red s n fctoril, the product of ll the nturl numers from n down to : n! = n (n (n 2 (n 3 2 Nturl logrithm [p. 203] see logrithm Nturl numers [p. 3] N = {, 2, 3, 4,... } Glossry K N
756 Glossry Glossry O P n r [p. 744] the numer of comintions of n ojects in groups of size r: n n! r = r! (n r! ( n n lterntive nottion for n r is. r Norml distriution [p. 625] symmetric, ell-shped distriution tht often occurs for mesure in popultion (e.g. height, weight, IQ; its centre is determined y the men, µ, nd its width y the stndrd devition, σ. Norml line, eqution [p. 305] Let (, y e point on the curve y = f (. If f is differentile t =, the eqution of the norml t (, y is y y = ( f ( O Odd function [p. 8] function f is odd if f ( = f ( for ll in the domin of f ; the grph hs rottionl symmetry out the origin. Optimistion prolem [p. 458] prolem where quntity is to e mimised or minimised under given constrints; e.g. to mimise the re of lnd enclosed y fied length of fencing Ordered pir [p. 6] pir of elements, denoted (, y, where is the first coordinte nd y is the second coordinte P Pscl s tringle [p. 748] tringulr pttern of numers formed y the inomil coefficients n r 2 3 3 4 6 4 Percentile [p. 597] For continuous rndom vrile X, the vlue p such tht Pr(X p = q% is clled the qth percentile of X, nd is found y p solving f ( d = q 00. Period of function [p. 40] function f with domin R is periodic if there is positive constnt such tht f ( + = f ( for ll. The smllest such is clled the period of f. Sine nd cosine hve period 2π. Tngent hs period π. function of the form y = cos(n + ε + or y = sin(n + ε + hs period 2π n. Permuttions [p. 744] see rrngements Piecewise-defined function [p. 7] function which hs different rules for different susets of its domin Point estimte [p. 678] If the vlue of the smple proportion ˆp is used s n estimte of the popultion proportion p, then it is clled point estimte of p. Point of inflection [p. 447] point where curve chnges from concve up to concve down or from concve down to concve up. Tht is, point of inflection occurs where the sign of the second derivtive chnges. Polynomil function [p. 90] polynomil hs rule of the type y = n n + n n + + + 0, n N {0} where 0,,..., n re numers clled coefficients. Popultion [p. 655] the set of ll eligile memers of group which we intend to study Popultion prmeter [p. 659] sttisticl mesure tht is sed on the whole popultion; the vlue is constnt for given popultion Popultion proportion, p [p. 658] the proportion of individuls in the entire popultion possessing prticulr ttriute Position [p. 36] For prticle moving in stright line, the position of the prticle reltive to point O on the line is determined y its distnce from O nd whether it is to the right or left of O. The direction to the right of O is positive. Power function [p. 26] function of the form f ( = r, where r is non-zero rel numer Proility [p. 56] numericl vlue ssigned to the likelihood of n event occurring. If the event is impossile, then Pr( = 0; if the event is certin, then Pr( = ; otherwise 0 < Pr( <. Proility density function [p. 585] usully denoted f (; descries the proility distriution of continuous rndom vrile X such tht Pr( < X < = f ( d SMPLE FINL PGES Proility function (discrete [p. 535] denoted y p( or Pr(X =, function tht ssigns proility to ech vlue of discrete rndom vrile X. It cn e represented y rule, tle or grph, nd must give proility p( for every vlue tht X cn tke. Proility tle [p. 52] tle used for illustrting proility prolem digrmmticlly
Glossry 757 Product of functions [p. 2] ( f g( = f ( g( nd dom( f g = dom f dom g Product rule [p. 298] If h( = f (g( then h ( = f (g ( + f (g( In Leiniz nottion: If y = uv, then dy d = u dv d + v du d Pythgoren identity [p. 38] cos 2 θ + sin 2 θ = Q Q [p. 3] the set of ll rtionl numers Qudrtic, turning point form [p. 77] The turning point form of qudrtic function is y = ( h 2 + k, where (h, k is the turning point. Qudrtic formul [p. 8] The solutions of the qudrtic eqution 2 + + c = 0, where 0, re given y = ± 2 4c 2 Qudrtic function [p. 76] qudrtic hs rule of the form y = 2 + + c, where, nd c re constnts nd 0. Qurtic function [p. 07] polynomil of degree 4. qurtic function f hs rule of the form f ( = 4 + 3 + c 2 + d + e, where 0. Quotient rule [p. 302] If h( = f ( g( then h ( = f (g( f (g ( ( 2 g( In Leiniz nottion: du If y = u dy v, then v d = d u dv d v 2 R R + R [p. 3] { : > 0 }, positive rel numers [p. 3] { : < 0 }, negtive rel numers R \ {0} [p. 3] the set of rel numers ecluding 0 [p. 46] { (, y :, y R }; i.e. R 2 is the set of ll ordered pirs of rel numers R 2 Rdin [p. 29] One rdin (written c is the ngle sutended t the centre of the unit circle y n rc of length unit: c = 80 π nd = πc 80 Rndom eperiment [p. 56] n eperiment, such s the rolling of die, in which the outcome of single tril is uncertin ut oservle Rndom smple [p. 655] smple of size n is clled simple rndom smple if it is selected from the popultion in such wy tht every suset of size n hs n equl chnce of eing chosen s the smple. In prticulr, every memer of the popultion must hve n equl chnce of eing included in the smple. Rndom vrile [p. 534] vrile tht tkes its vlue from the outcome of rndom eperiment; e.g. the numer of heds oserved when coin is tossed three times Rnge [p. 6] the set of ll the second coordintes of the ordered pirs in reltion Rtionl numer [p. 3] numer tht cn e written s p, for some integers p nd q with q 0 q Rectngulr hyperol [p. 27] The sic rectngulr hyperol hs eqution y =. Recurrence reltion [MM&2] rule which enles ech susequent term of sequence to e found from previous terms; e.g. t =, t n = t n + 2 Reflection in the -is [p. 52] reflection in the -is is descried y the rule (, y (, y. The curve with eqution y = f ( is mpped to the curve with eqution y = f (. Reflection in the y-is [p. 52] reflection in the y-is is descried y the rule (, y (, y. The curve with eqution y = f ( is mpped to the curve with eqution y = f (. Regression [p. 236] the process of fitting mthemticl model to dt Reltion [p. 6] set of ordered pirs; e.g. { (, y : y = 2 } Reltive growth rte [pp. 222, 33] The reltive growth rte of function f is f ( f (. For n eponentil function f ( = e k, the constnt k is the reltive growth rte. SMPLE FINL PGES Reminder theorem [p. 97] When polynomil P( is divided y β + α, the reminder is P ( α β. Right-endpoint method [p. 373] gives n estimte for the re under the grph of y = f ( etween = nd = : R n = [ f ( + f ( 2 + + f ( n ] n Glossry Q R
758 Glossry Glossry S S Smple [p. 655] suset of the popultion which we select in order to mke inferences out the whole popultion Smple proportion, ˆp [p. 659] the proportion of individuls in prticulr smple possessing prticulr ttriute. The smple proportions ˆp re the vlues of rndom vrile ˆP. Smple spce, ε [p. 56] the set of ll possile outcomes for rndom eperiment Smple sttistic [p. 659] sttisticl mesure tht is sed on smple from the popultion; the vlue vries from smple to smple Smpling distriution [p. 665] the distriution of sttistic which is clculted from smple Scientific nottion [MM&2] numer is in stndrd form when written s product of numer etween nd 0 nd n integer power of 0; e.g. 6.626 0 34. Secnt [p. 254] stright line tht psses through two points (, f ( nd (, f ( on the grph of function y = f ( Second derivtive [p. 44] The second derivtive of function f with rule f ( is denoted y f nd hs rule f (. The second derivtive of y with respect to is denoted y d2 y d. 2 Second derivtive test [p. 449] If f ( = 0 nd f ( > 0, then the point (, f ( is locl minimum. If f ( = 0 nd f ( < 0, then the point (, f ( is locl mimum. If f ( = 0, then further investigtion is necessry. Sector [MM&2] Two rdii nd n rc define region clled sector. In this digrm, the shded region is minor sector nd the unshded region is mjor sector. re of sector = 2 r2 θ where θ c = O Selections [p. 744] counted when order is not importnt. The numer of wys of selecting r ojects from totl of n ojects is n n! r = r! (n r! ( n n lterntive nottion for n r is. r D O θ Sequence [MM&2] list of numers, with the order eing importnt; e.g.,, 2, 3, 5, 8, 3,... The numers of sequence re clled its terms, nd the nth term is often denoted y t n. see lso rithmetic sequence, geometric sequence Series [MM&2] the sum of the terms in sequence see lso rithmetic series, geometric series Set difference [p. 3] The set \ contins ll the elements of tht re not in. For emple, R \ {0} is the set of ll rel numers ecluding 0. Set nottion [p. 2] mens is n element of mens is not n element of mens is suset of mens intersection mens union is the empty set, contining no elements Sets of numers [p. 3] N is the set of nturl numers Z is the set of integers Q is the set of rtionl numers R is the set of rel numers Signed re [p. 383] Regions ove the -is re defined to hve positive signed re. Regions elow the -is re defined to hve negtive signed re. For emple, the signed re of the shded region in the following grph is 2 + 3 4. 0 y 3 2 4 SMPLE FINL PGES Signs of trigonometric functions [p. 34] st qudrnt ll re positive 2nd qudrnt sin is positive 3rd qudrnt tn is positive 4th qudrnt cos is positive S T y
Glossry 759 Simple interest [MM&2] is lwys clculted on the mount originlly invested (the principl. If $P is invested t R% p.., then the vlue of the investment fter n yers, $ n, is given y n = P + np R 00 Simultion [p. 660] using technology (clcultors or computers to repet rndom process mny times; e.g. rndom smpling Simultneous equtions [p. 7] equtions of two or more lines or curves in the rtesin plne, the solutions of which re the points of intersection of the lines or curves Sine function [p. 3] sine θ is defined s the y-coordinte of the point P on the unit circle where OP forms n ngle of θ rdins with the positive direction of the -is. - O - y θ cos θ Sine rule [p. 486] For tringle : sin = sin = c sin c P(θ = (cos θ, sin θ sin θ The sine rule is used to find unknown quntities in tringle given one side nd two ngles, or given two sides nd non-included ngle. Speed [p. 37] the mgnitude of velocity Speed, verge [p. 37] totl distnce trvelled verge speed = totl time tken Stndrd devition of rndom vrile, σ [pp. 545, 602] mesure of the spred or vriility, given y sd(x = Vr(X Stndrd form [MM&2] see scientific nottion Stndrd norml distriution [p. 623] specil cse of the norml distriution where µ = 0 nd σ = Sttionry point [p. 322] point (, f ( on curve y = f ( is sttionry point if f ( = 0. Stright line, eqution given two points [p. 43] y y = m(, where m = y 2 y 2 Stright line, grdient intercept form [p. 43] y = m + c, where m is the grdient nd c is the y-is intercept Stright lines, prllel [MM&2] Two non-verticl stright lines re prllel to ech other if nd only if they hve the sme grdient. Stright lines, perpendiculr [p. 43] Two stright lines re perpendiculr to ech other if nd only if the product of their grdients is (or if one is horizontl nd the other verticl. Strictly decresing [pp. 26, 265] function f is strictly decresing on n intervl if 2 > implies f ( 2 < f (. Strictly incresing [pp. 26, 265] function f is strictly incresing on n intervl if 2 > implies f ( 2 > f (. Suset [p. 2] set is clled suset of set if every element of is lso n element of. We write. Sum of functions [p. 2] ( f + g( = f ( + g( nd dom( f + g = dom f dom g Sum of two cues [p. 99] 3 + y 3 = ( + y( 2 y + y 2 Sum to infinity [MM&2] The sum to infinity of n infinite geometric series eists provided < r < nd is given y S = r where = t nd r is the common rtio. T Tngent function [p. 3] The tngent function is given y SMPLE FINL PGES tn θ = sin θ cos θ Tngent line, eqution [p. 305] Let (, y e point on the curve y = f (. Then, if f is differentile t =, the eqution of the tngent t (, y is given y y y = f ( (. Totl chnge [p. 405] Given the rule for f (, the totl chnge in the vlue of f ( etween = nd = cn e found using f ( f ( = f ( d Glossry T
760 Glossry Glossry U Z Trnsltion [p. 46] trnsltion of h units in the positive direction of the -is nd k units in the positive direction of the y-is is descried y the rule (, y ( + h, y + k, where h, k > 0. The curve with eqution y = f ( is mpped to the curve with eqution y k = f ( h. Trpezoidl rule [p. 375] gives n estimte for the re under the grph of y = f ( etween = nd = : T n = [ f (0 + 2 f ( + + 2 f ( n + f ( n ] 2n Tree digrm [p. 527] digrm representing the outcomes of multi-stge eperiment Trigonometric functions [p. 3] the sine, cosine nd tngent functions Trigonometric functions, ect vlues [p. 33] θ c θ sin θ cos θ tn θ 0 0 0 0 π 3 6 30 2 2 3 π 4 45 2 2 π 3 3 60 3 2 2 π 2 90 0 undefined Trigonometric rtios [p. 48] sin θ = opposite hypotenuse cos θ = djcent hypotenuse tn θ = opposite djcent hypotenuse θ djcent side opposite side U Uniform distriution [pp. 558, 62] discrete rndom vrile X with n vlues, 2, 3,..., n hs uniform distriution if ech vlue of X is eqully likely, nd therefore Pr(X = = n, for =, 2, 3,..., n continuous rndom vrile X hs uniform distriution if its proility density function is given y if f ( = 0 otherwise where nd re rel constnts with <. Union of sets [pp. 2, 56] The union of two sets nd, written, is the set of ll elements which re in or or oth. V Vrince of rndom vrile, σ 2 [pp. 545, 602] mesure of the spred or vriility, defined y Vr(X = E[(X µ 2 ]. n lterntive (computtionl formul is Vr(X = E(X 2 [ E(X ] 2 Velocity [p. 37] the rte of chnge of prticle s position with respect to time Velocity, verge [p. 37] chnge in position verge velocity = chnge in time Velocity, instntneous [p. 37] v = d dt Verticl-line test [p. 8] see function Z Z [p. 3] the set of ll integers Zero polynomil [p. 90] The numer 0 is clled the zero polynomil. SMPLE FINL PGES