# Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

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1 Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector is quntity tht hs size (k mgnitude) nd direction Identify rel-life exmples of vectors such s, displcement, velocity, ccelertion, electric field strength nd force Know tht vector is represented geometriclly by directed line segment (i.e. line with n rrow t one end) Know tht vector is described lgebriclly by its components Know tht vector cn exist in n dimensions (n N) Know tht vector in D hs components nd one in 3D hs 3 components, nd such vectors re usully written s column vectors : v v def = v w def = w w w 3 where v is the first component (x coordinte) of v nd v the second component (y coordinte) of v (nd similrly for w) Know tht the mgnitude of vector is given by: v def = v v + v, v = v w def = w + w + w, w = 3 w w w 3 Clculte the mgnitude of vector, where pproprite leving the nswer s surd in simplest form Know tht unit vector is one tht hs mgnitude equl to Know tht vectors re equl if ll their respective components re equl Know tht zero vector, written 0, is one M Ptel (August 0) St. Mchr Acdemy

2 with ll components equl to 0 Know tht or more vectors cn be dded if they hve the sme number of components nd their respective components re dded ccording to the vector ddition lw : u u + v = u v + v u + u + v v def = r + s = r r r 3 + s s s 3 def = r + r + r + s s s 3 3 Know tht or more vectors cn be dded to produce resultnt vector Know tht vector my be multiplied by sclr, this being known s sclr multipliction Know tht to multiply vector v by sclr k R, ech component of v is multiplied by k : v v = v k v def kv = kv Know tht the negtive of vector v is obtined by sclr multiplying v by, in which cse the new vector is denoted by -v Know tht vectors u nd v re prllel if u = k v,for some sclr k Know tht if u = k v for some sclr k, then u nd v re prllel Know tht, for ny vector v, there exists prllel unit vector Given vector, find unit vector tht is prllel to it Know tht, reltive to n origin O, the point A is sid to hve position vector = OA Know tht if nd b re the position vectors of points A nd B respectively, then the vector joining A to B hs position vector AB = b - Know tht points re sid to be colliner if they lie on the sme stright line Know tht if AB = k BC (so tht AB nd BC re prllel line segments) nd B is point common to both AB nd BC, then A, B nd C re colliner Given 3 points, determine whether or not they re colliner Given 3 colliner points A, P nd B such tht P divides line segment AB in the rtio m : n, know tht: M Ptel (August 0) St. Mchr Acdemy

3 AP PB = m n nd tht this consequently leds to, m AP = n PB Use the lst eqution to find the position vector of P when P divides AB in the rtio m : n Given 3 colliner points, A, P nd B, the rtio m : n nd the coordintes of A nd P, find the coordintes of B Know the Section Formul for the position vector of point P which divides line segment AB in the rtio m : n : P = n + m + n m b m + n Use the Section Formul to find the position vector of point which divides line in given rtio Know tht 3D vector my be written in terms of 3 mutully perpendiculr unit vectors in the x, y nd z directions: v = v i + v j + v 3 k where def = v v v 3 i = 0, j = 0 0, k = nd i = j = k = Know tht the sclr (dot) product of vectors nd b is defined s: b def = b cos θ where θ is the ngle between nd b nd 0 θ 80 Know tht the sclr product is number, not vector Know tht the 3 unit vectors, i, j nd k stisfy: M Ptel (August 0) 3 St. Mchr Acdemy

4 i j = j k = k i = 0 nd i i = j j = k k = Know tht the component form of the sclr product of vectors nd b is: b = n b r r r = = b + b + + b n n where is the r th component of, nd similrly for b r r Know tht in D nd 3D, this becomes: b = b + b b = b + b + 3 b 3 Given the mgnitudes of vectors nd the ngle between them, clculte their sclr product Given vectors in component form nd the ngle between them, clculte their sclr product Given vectors in component form, clculte their sclr product Given vectors in component form, clculte the ngle between them Given 3 points, clculte ny ngle in the tringle formed from these points Know tht if vectors nd b re perpendiculr, then b = 0 Know tht if b = 0, then nd b re perpendiculr Given vectors, determine whether or not they re perpendiculr Know the sclr product properties: b = b (Commuttive Lw) (b + c) = b + c (Distributive Lw) Use the bove properties of the sclr product to solve problems M Ptel (August 0) 4 St. Mchr Acdemy

5 Further Clculus Skill Achieved? Know the derivtives of the sine nd cosine functions when x is expressed in rdins: d dx sin x = cos x d dx cos x = sin x Know the integrls of the sine nd cosine functions bsed on the bove derivtives: sin x dx = cos x + C cos x dx = sin x + C Know the Chin Rule, nmely, if g is differentible t x = nd f is differentible t g (), the composition f g is differentible t x = nd its derivtive is given by: (f g ) () = f (g ()) g () Know tht n lterntive wy of writing the Chin Rule, where the functionl dependence on x is emphsised, is: h (x) = f (g (x)) g (x) where h = f g, the composition of f with g Know tht the Chin Rule is written succinctly using Leibniz nottion: dy dx = dy du du dx where u = g (x) nd y = f (u) Know the specil form of the Chin Rule when g (x) = x + b (, b R ) nd f (x) = x n (n R ): d dx (x + b) n n = n (x + b) M Ptel (August 0) 5 St. Mchr Acdemy

6 Know the specil forms of the Chin Rule: d dx sin (x + b) = cos (x + b) d dx cos (x + b) = sin (x + b) Differentite functions using the Chin Rule, for exmple: f (x) = (5x 4) f (x) = (x + 3) 4 f (x) = (x ) 5 f (x) = (x 3 + 4) f (x) = 3 x + f (x) = (8 3 x ) f (x) = (4 3x ) f (d) = 4 3 (7 d d ) f (x) = ( + sin x) 4 f (x) = cos x 3 sin 4x f (x) = cos ( 5x) f (x) = sin x π 6 f (x) = sin 3 x f (x) = sin (cos x) f (t) 3 = cos t M Ptel (August 0) 6 St. Mchr Acdemy

7 Know the integrls: ( x + b) n dx = ( x + b) ( n + ) n + + C (n ) sin ( x + b) dx = cos (x + b) + C cos ( x + b) dx = sin (x + b) + C Evlute definite nd indefinite integrls using the bove rules, for exmple: dx (7 3 x ) (4 ) t dt 0 dx (3 x + ) 4 x + dx 0 sin (4 x + ) dx 0 4 sin (x + 3) dx 4 ( x + cos 5 x ) dx Clculte the re enclosed by the x xis nd the grphs of: f (x) = (3x 8) (x [0,]) f (x) = (3 x) 3 (x [0,]) M Ptel (August 0) 7 St. Mchr Acdemy

8 Clculte the re enclosed by the x xis, sine grph nd given limits (which my hve to be found) Clculte the re enclosed by the x xis, cosine grph nd given limits (which my hve to be found) Clculte the re enclosed between sine grphs (whose intersection points my hve to be found) Clculte the re enclosed between cosine grphs (whose intersection points my hve to be found) Clculte the re enclosed between sine nd cosine grph (whose intersection points my hve to be found) Clculte the re enclosed between other curves, for exmple: f (x) = 3 x nd g (x) = (x 9) (x [0,4 5]) f (x) = x + nd g (x) = 3 x x 3 (x [ 0 5,4]) Solve differentil equtions such s: f (x) = sin 3x when x = π 9 nd f (x) = dy dx = 3 sin x when x = 5π nd y = 3 dv dt = (4 t) when t = 0 nd v = 0 M Ptel (August 0) 8 St. Mchr Acdemy

9 Exponentil nd Logrithmic Functions Skill Achieved? Know tht for >, the grph of the exponentil function is clled growth curve Know tht for 0 < <, the grph of the exponentil function is clled decy curve Know tht the exponentil function to bse e, f (x) = e x, is the exponentil function which hs the property tht f (x) = e x Know tht the bse e for the bove exponentil function hs the irrtionl vlue: e = Know tht the inverse of the exponentil function to bse e is the logrithmic function to bse e nd hs the specil nottion: ln x def = log x e Know tht tking the exponentil to bse of number x nd then tking the logrithm to the bse results in x (nd vice vers): log x = x = x Know the specific cse of the bove when = e : log x ln e x = x e ln x = x Chnge between exponentil nd logrithmic nottion: p = x log x = p Know tht logrithm cn be viewed s power (index) Know the lws of exponentils nd the equivlent lws of logrithms : p q = p + q log xy = log x + log y p q x = p q log y = log x log y M Ptel (August 0) 9 St. Mchr Acdemy

10 ( m ) n = mn log (x n ) = n log x 0 = log = 0 = log = where p = x nd q = y Simplify logrithmic expressions such s: 5 log + log 4 log log + log 50 log log e log 3e e e Simplify other logrithmic expressions such s writing: 3 log e log 3e e e in the form A + log B log C (A, B, C N) e e Solve logrithmic equtions such s: 4 log x 6 log x 4 = log (x ) + = 0 3 log (x + ) log 3 = 3 log (5 x) log (3 x) = 4 4 log (x ) = 0 3 log (q 4) = 3 log (x x + 0) log (5 x) = 3 Solve other logrithmic equtions such s: log x + log x = 3 9 M Ptel (August 0) 0 St. Mchr Acdemy

11 Identify the exponentil nd logrithmic buttons (or functions) on scientific clcultor Solve exponentil equtions such s: 6 = 3 4 x 54 = m 3 without using the exponentil or logrithmic buttons (or functions) Use the logrithmic button (or function) to solve exponentil equtions such s: 0 5 = e 0 00 t 0 = 5 e t t 0 88 = e Use the logrithmic button (or function) to solve problems involving inequtions such s, find the smllest nturl number n such tht: 3 n > n < 0 4 Use the logrithmic button (or function) to solve problems involving inequtions such s, find the lrgest nturl number n such tht: 8 n < 300 Use the logrithmic button (or function) to solve problems involving inequtions such s, find the lrgest nturl number n such tht: 0 6 n > 0 09 Verify the following equivlence (for ny bse r): y = k x n log r y = log r k + n log r x Know tht by letting, Y = log r y, X = log r x, c = log r k nd m = n, the RHS of the bove equivlence cn be written s: Y = mx + c Know tht if plot of log r y ginst log r x is stright line, then x nd y obey reltionship of the form, y = k x n Given stright line grph of log r y ginst log r x M Ptel (August 0) St. Mchr Acdemy

12 nd points on the line, determine k nd n Given stright line grph of log r y ginst log r x nd the grdient nd y - intercept, determine k nd n Given tble of x nd y vlues, nd reltionship of the form y = k x n, determine k nd n using simultneous equtions Verify the following equivlence (for ny bse r): y = b x log r y = log r + x log r b Know tht by letting, Y = log r y, X = x, c = log r nd m = log r b, the RHS of the bove equivlence cn be written s: Y = mx + c Know tht, if plot of log r y ginst x is stright line, then x nd y obey reltionship of the form, y = b x Given stright line grph of log r y ginst x nd points on the line, determine nd b Given stright line grph of log r y ginst x nd the grdient nd y - intercept, determine nd b Given tble of x nd y vlues, nd reltionship of the form y = b x, determine nd b using simultneous equtions Know tht reltionships of the form y = k x n nd y = b x re common mthemticl models Know tht the exponentil lw: N (t) = N e kt 0 where N, k R re constnts, is commonly 0 used to model decy nd growth Given the bove exponentil lw, clculte k given t nd the reltionship between N (t) nd N 0 Given the bove exponentil lw, clculte N given k, t nd N (t) 0 Solve contextul problems involving logrithms nd exponentils M Ptel (August 0) St. Mchr Acdemy

13 Wve Functions Skill Achieved? Know tht wve function is ny function of the form: f (x) = k sin (x ± α ) or f (x) = k cos (x ± α ) where k is the mplitude nd α is the phse ngle Express sin x + b cos x in the form: k cos (x α ) where k = + b nd tn α =, for exmple, b 3 cos x + 5 sin x (α [0, 90]) Express sin x + b cos x in the form: k cos (x + α ) where k = + b nd tn α =, for exmple, b 8 cos x 6 sin x (α (0, 360)) Express sin x + b cos x in the form: k sin (x α ) where k = + b nd tn α = b, for exmple, sin x cos x (x [0, π]) Express sin x + b cos x in the form: k sin (x + α ) where k = + b nd tn α = b, for exmple, M Ptel (August 0) 3 St. Mchr Acdemy

14 sin x + 5 cos x (α [0, 360 )) Use wve function to find the mximum nd/or minimum vlues or turning points of expressions of the form sin x + b cos x, for exmple: Mximum TP of cos x sin x (x [0, π]) Minimum TP of sin x + 5 cos x (x [0, 360]) Mx. nd Min. TP of cos x 5 sin x (x [0, 360)) Use wve function to solve trigonometric equtions such s: 3 cos x + 5 sin x = 4 (x [0, 90]) Given vlues for nd b, sketch the grph of y = sin x + b cos x, indicting mximum nd minimum vlues nd intersections with xes Given vlues for, b nd c, sketch the grph of y = sin x + b cos x + c, indicting mximum nd minimum vlues nd intersections with xes Solve contextul problems involving wve function M Ptel (August 0) 4 St. Mchr Acdemy

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