Tuesday, 25 September 2018

9:07 PM

Machine generated alternative text: A function f is defined by y) = 8 .T6 y4 sin (7 x Enter a Maple expression for the function f using the arrow operator "->" in the box below. (Do not enter "f at the beginning or ";" at the end of your answer.) —5y)

 

Machine generated alternative text: f: (sy) *8x6y4 sin(7x 5y)

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Given that Find the partial derivative and enter your answer, in Maple syntax, in the box below 5) sin 2 y +6m 33

 

Machine generated alternative text: f: = (x , y) •sin ; f: 615 ) -10800 cos(2y6 — 1440 sin(2y6

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Find the partial fraction decomposition of where p (x) = 8 x8 +46 x7 — 211 x6 — 1015m + 4784 A — 11534 x3 + 8707 x 2 and x +6m —37 —77 x + 633 x 5 — 1777 x4 + 2085 x3 — 771 and enter it in the box below. To prevent typing errors, you can copy and paste p 01 1 869*X-91 56; q *xA2-1 323; into your Maple worksheet and copy and paste the Maple output into the answer box. - 1869 - 9156 - 1386 + 1323

Machine generated alternative text: x-9156; x+1323 ; convert (p/ q, parfrac , x) ; 4617 —21116 101515 478414 — 1153413 633 e 17771 2085 e 2 870712 1869 x 77112 13861 9156 1323 12—31 3

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Use Maple to find the solution of the initial value problem 2 d2y 1 dy y dx2 0 with initial conditions y (0) 1 and y' (0) — 7 Using Maple syntax, type in your answer in the box below, or copy (Ctrl-C) from your Maple worksheet and paste (Ctrl-V) in the answer box the solution Do NOT enter the y(x)= part of the Maple output.

 

Machine generated alternative text: dsolve ({y (x) *diff (y (x) ,x$2) —1/2* (diff (y (x) , x) ) y (O) —I, D (y) (O) =7}) ; 49 y(x) = x-2 71 1

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Suppose that a function f has derivatives of all orders ata The the series 00 is called the Taylor series for f about a, where f (n) is the n th order derivative of f Suppose that the Taylor series for e2x sin (4 x) about 0 is k! ao al.T4-a,2x + Enter the exact values of and a8 in the boxes below 32/15 _kav + .

 

Machine generated alternative text: taylor (exp (2*x) *sin (4 *x) , 4x 812 — 1614 — x: 15 + 176 x-6 45 2224 315 32 15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Suppose that 111 — 420 256 -540 565 -989 -437 150 -442 900 266 -480 -442 113 -39 309 -475 -681 -721 -481 815 -155 and -79 -280 331 376 114 -699 262 -44 -241 177 496 -396 242 To avoid typing errors; you can copy and past the following sequences to your Maple worksheet to form entries of the vectors 420, 256, -54€, 565, -989, -437, 150, 90€, 266, -48e, -721, -481, -699, 262, 815, -44, -442, -39, 3€9, -475, -155, -79, -28€, 331, -241, 177, 496, -396, -442 -681 376 242 (a) Find the dot product of and 112_ Enter your answer in the box below. 111 • 112 (b) Suppose that A is the matrix whose columns are these four vectors in order, Number Let v 39 be the vector -72 -64 Hence Av A = (ul 1 114) is a linear combination of the form Enter the values of Al, (c) Suppose that Av = (bl b2 A 4 in the boxes below 39 b8)T -72 Enter the value of bl in the box below bl = 128808

 

Machine generated alternative text: with (Linearmgebra) : ul : *420 , u2 : *900 , 256, -540, 266, -480, -481, 815, 262, -44, 565 , -989, -442, -39, -155, -79, -241, 177, -437, 150, 309, -475, -280, 331 , 496, -396, -442>: -681>: 376>: 242>: Transpose (8) [I] ; 128808 26446 - 70804 5391 -238 3526 -18063 -66025 128808

 

 

 

 

 

 

 

 

 

 

 

 

Let A be an m x n matrix. The kernel of A is the setof vectors ker(A) = {x which is a vector space. The dimension of ker(A) is called the nullity of A, denoted by nullity(A) (a) Find the nullity of the matrix -76 88 -96 176 and enter in the box below -210 -40 -58 -46 196 120 -90 14 -180 44 -186 -8 -618 -24 -230 178 nullity(A) — To avoid typing errors, you may copy and paste the following sequence of entries -76, 88, -96, 176, -210, -40, -58, -46, 196, 120; -90; 14, -180, 44 -186, -8, -618, -24, -230, 178

 

(b) For the matrix A in (a); select all the statements below which are true. 0 is in ker(Ab -86 -42 is in ker(A)_ -5 -67 You may copy and paste the entries -86, -42, -5, -67 -2 is in ker(A) is in ker(Ab is in ker (Ab -124 is in ker(A). -53 21 You may copy and paste the entries -124 2, -53, 21 D ker(A) is subspace of R4 4.1 ker(A) is subspace of R5

 

with (LinearAIgebra) : A: , 88, -96, s ion (A) Nu IISpace (A) ; LinearSoIve (A , 120 , -90 , -180, 44, -186, 2 -2 -618 , -24, -230, —Rank (A) ;

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Let A be an m x n matrix. The image of A is the setof vectors im(A) which is a vector space. The dimension of im(A) is called the rank of A, denoted by rank(A) (a) Find the rank of the matrix and enter in the box below Ax for some x e IR n} 170 -158 -94 -150 133 -275 -171 -225 -172 -54 174 32 3 220 -142 -298 -182 821 -59 -867 -353 rank(A) Note: The Maple command for find the rank of A is Rank(A), upper case R. To avoid typing errors, you may copy and paste the following sequence of entries 170; -158; -94 -150; 133, -275, -171, -225, -172, -54, 174, 32, 220; -142; -298; -182, 821, -59, -867, -353

 

Machine generated alternative text: (b) For the matrix A in (a); select all the statements below which are true. D Im(A) is subspace ofR5 24 -98 is in im(A)_ -62 -75 You may copy and paste the entries 24, -98, -62, -75 0 is in im(A) is in im(Ab is in im(A) 4.1 im(A) is subspace ofR4 is in im(Ab 87 -68 is in im(A)_ 82 -92 You may copy and paste the entries 87, -68, 82, -92

 

Machine generated alternative text: -158, -142 , -150> 1<133 , <220 , > Rank (A) ; -298 , -275, -59, -171, -867 , -54, 174, > Linearsolve (A, , -98 , -62 , -75>) : > LinearSoIve (A, , 4, —2 , 4, —2>) : Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side Matrix. 4. does not match number of rows of right hand. 5 > LinearSoIve (A, , O , O , O , : Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side Matrix. 4. does not match number of rows of right hand. 5 > LinearSoIve (A, , —2 , —4, 4, —2>) : Error. (in LinearAIgeEra : —Linearsolve) number of rows of left hand side Matrix. 4. does not match number of rows of right hand. 5 > LinearSoIve (A, , O , O , : > Linearsolve (A, , -68 , 82 , -92>) : Error. ( in LinearAIgeEra : —Linearsolve) inconsistent system

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Suppose that S P2(x) — P4(x) — = {PI' P2' P.3' P4}, where 3 147 + 103 + 109 + 19 x 39+ +43m —281 x — -94+209x-224x -262x +21 x 5 5 - 232 5 + 269 x -15—93X+192x +203X +231 X + 166T , and — -4287 + 13475 x - 533 + 1696 + 16633x +32087 x 5 To avoid typing errors, you can copy and past the following sequences to your Maple worksheet 147, 103, 109, 272, -19, 3 39, 272, 43, 43, -281, -232 -94, 209, -224, -262, 21, 269 -15, -93, 192, -4287, 13475, 203, 231, 166 -533, 1696, 16633, 32087 (a) The number of elements of S is less than the dimensions of IP5 Hence S cannot be a spanning set of IP5 (b) The polynomial p is a linear combination of S written in the form , the set of polynomials of degree 5 or less apl + ßP2 + + Find a possible set of values for a, B, 7, ö_ Enter the values ofa, B, 7, as a sequence in the box below [a, ß, 7, ö] [27, 4, 78, 721

 

Machine generated alternative text: PI : *147 , p 4: 15 103, 272, 209, -93 , 109, 272, -19, 43, 43, -281, -232>: -224, -262, 21, 192, 203, 231, 13475, -533, 1696, LinearSoIve (<pl Ip2 Ip3 Ip4> , p) ; convert (8 , list) ; 269>: 166>: 16633 , 27 4 78 72 32087>: [2724: 78: 72]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Suppose that S — {111 , 112, 113, 114, 115, 116} C R 111 — -41 -28 56 53 -26 where u2 -91 -1 -66 -60 93 113 578 89 162 141 -387 -9 50 114 -43 -33 -11 -16 91 27 -86 and 116 -5 182 -323 -183 73 To avoid typing errors, you can copy and past the following sequences to your Maple worksheet -41, -91, 578, -9, -11, -5, -28, —I, 89, -16, 182, 56, -66, 162, -47, 91, -323, 53, -60, 141, -43, 27, -26 93 -387 -33 -86 -183, 73 Since the number of vectors in S is greater that the dimension of R the set S Find a possible setof values for Al, A 4, not all zero, such that must be linearly dependent Enter the valuesof Al, .\2, as a sequence in the box below [Al' •\2' A4, '\5, [3, 5, 1, -3/2, 1, 1/2] Hint: There are infinitely many solutions for Al, . The solution given by Maple will be in terms of parameters. To get one possible setof values, not all zero, choose some nice values for the parameters.

 

Machine generated alternative text: with (Linearmgebra) : -28, 56, 91 u3 : *578 , 11 -66 , 162 , 89, 50, 53, -26>: -387>: , -33>: -16, 91, -60 , 141, 27, -86>: 182, -323, -183, 73>: _ 41 -28 56 53 _ 26 _ 91 -66 -60 93 578 89 162 141 -387 2 2 -9 _ 47 -43 -33 -11 -16 91 27 -86 182 -323 -183 73 LinearSoIve (S

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Use Maple to find the eigenvalues of -8040 2160 -180 2520 720 -25471 7971 27472 25617 -9238 -9410 9930 1460 13110 -2180 15807 -11187 -16224 -24489 4506 21676 -13056 -22702 -23772 1138 Enter the eigenvalues in the box below as a Maple set (Eg, ifthe eigenvalues were 1, 2, 3; 4, you should enter {1 To avoid typing errors; you may copy and paste the following sequence of entries of A -8040; 2160, -180, 2520, 720, -25471; 7971 , 27472; 25617, -9238, -9410; 9930, 1460; 13110, -2180; 15807, -11187, -16224, -24489; 4506, 21676; -13056, -22702, -23772; 1138 and edit it appropriately to create a matrix. {-10980, -9150, -7320, -5490, 10980}

 

Machine generated alternative text: 2160, -180, 1<-25471, 7971, 27472, 2520, 25617 , 720> -9238> 1<-9410 , 1<15807 , 1<21676, 9930, 1460, 13110, -2180> -11187, -13056, -16224, -22702 , -24489, -23772, 4506> E igenvalues (A) ; convert (8 , set) ; -5490 -10980 - 7320 -9150 10980 -10980: -9150: -7320: -5490: 10980)

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Given the three points A(-2, 4; 5); B(0i 5, 6), C(7, 7, 7), let: • Sl be the sphere with centre A and radius 4, • S2 be the sphere which has the line segment BC as a diameter, • T be the circle of intersection of Sl and S2, • E be the centre of T, • Ll be the line through B and E, • L 2 be the line through A parallel to Using the geom3d package, or othenvise: 4 1 3 (i) Find the coordinates of E and enter them in the box below You should enclose the coordinates with square brackets, eg [1 2,3], and your answer should be exact ie not a decimal approximation. To prevent typing errors you can copy and paste the answer from your Maple worksheet [137/146, 370/73, 847/146] (ii) Find a decimal approximation to the angle (in radians) between Ll and L 2 0.8527069987 Your answer should be correct to 10 significant figures Enter your answer in the box below. (iii) Find the distance between Ll and L 2. Your answer should be exact, not a decimal approximation. can copy and paste the answer from your Maple worksheet Enter your answer in the box below using Maple syntax. To prevent typing errors you

 

Machine generated alternative text: with (geom3d) : point (A, [—2 , 4, point (E , [0, point (C , [7 , 5, 7, sphere ( SI , [A, sphere ( S2 , [E , intersection (T , 5]): c]): sl, S2) : point (E , coordinates ( center (T) ) ) : line (Ll, [E, E]) : line (L2, [A, [4, 1, coordinates (E) ; 137 146 evalf (FindAngIe (LI , L2) ) ; 370 73 847 146 0.8527069987 distance (LI , L2) ; 104 659 4613

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: Given the three points A (—2, —3, 0), B (—1, 0, 4), C (—3, 1, —2), let: • Ll be the line through A parallel to 5 5 -2 -4 • P be the plane through B with normal 5 • E be the point of intersection of Ll and P, • S be the sphere through A, B; C, E; • F be the centre of S, • L2 be the line through C and F. Using the geom3d package, or othepwise: (i) Find a decimal approximation to the angle between Ll and P, correct to 10 significant figures. Enter your answer in the box below. To prevent typing errors you can copy and paste the answer from your Maple worksheet 0.1064637912 (ii) Find the coordinates of F and enter them in the box below You should enclose the coordinates with square brackets; eg [1 and your answer should be exact, ie not a decimal approximation. To prevent typing errors you can copy and paste the answer from your Maple worksheet [3699/670, 519/670, -979/670] (iii) Find the distance beftveen A and L2_ Your answer should be exact, not a decimal approximation. can copy and paste the answer from your Maple worksheet Enter your answer in the box below using Maple syntax To prevent typing errors you

 

Machine generated alternative text: restart ; with (geom3d) : point (A, [—2, —3, O]): point (E point (C, I, : line (Ll, [A, plane (P , [E , intersection (E , LI , P) : sphere (S, : point (F , coordinates (center (S) ) ) : line (1.2 , [C,F]) : evalf (FindAngIe (LI , P) , 10) coordinates (F) ; dis tance (A , L2) ; 0.1064637912 3699 670 519 670 638170638 979 670 32745803 32745803

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: The skeleton "for loop" below is used to evaluate the sum 18 for k from 4 to 70 Select the correct item from each drop down box so that the output from the loop contains exactly 67 lines and displays the sum for each integer value of k from 4 to 70 for k , from 4 , to 70 , do n=18..23) end do; The terms of a sequence an are generated by the recurrence relation an+l = an — 2 an _1 + an _2 forn = 3, 4, 5, ... Using your Maple worksheet, write a for loop to find the value of am given that al = 4, = 0 and q — Copy (Ctrl-C) the correct value of am from your Maple worksheet and paste (Crtl-V) it in th answer box. -290694793 -1

 

Machine generated alternative text: for n from 4 to 70 do end do : a[70] ; 3] -290694793

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Machine generated alternative text: A simple iteration procedure with ao 0 and 2 an 2 is being used to find an approximate solution to the equation x = sin Select the correct expressions from the drop down menus to define a procedure which takes a positive integer m and uses a for loop to calculate am. The procedure should return am if lam — am Digits t proc(m) local a, i; for i from —Il < 10—17 and m —1 otherwise. All calculations are done using 30 significant figures. do end do; if abs (a [m] a[m] -a[m-l)) < Ion (—17) then

 

Machine generated alternative text: end if end proc; Use this procedure to calculate f (5) and f (13) and enter your answers in the box below as decimal approximations correct to 30 significant figures. To prevent typing errors, you can copy and paste your answers from your Maple worksheet. The value of f (5) is -1 The value of f (13) is 0.99996540607060892587.

 

Machine generated alternative text: Digits : ; f : —proc (m) local a , i ; for i from I to m do evalf (sin ( / 4) ) end do ; if abs (a[m] < IOA (—17) then else end if end proc ; f (13) ; proc(m) local i; a[O] = O; for i to m do a[i] — Digits 3D 1])A2)) end do, if abs(a[m] — a[m 0.999965406070608925874302546481 1]) < 1/100000000000000000 then a[m] else I end if end proc

 

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