dacs2212 engineering mathematics

dacs2212 engineering mathematics - UNIVERSITI TEKNIKAL...

Info icon This preview shows pages 1–12. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UNIVERSITI TEKNIKAL MALAYSIA MELAKA PEPERIKSAAN AKHIR SEMESTER 1 SESI 2009/2010 F AKULTI KEJURUTERAAN MEKANIKAL KOD MATAPELAJARAN : DACS 2212 MATA PELAJARAN : ENGINEERING MATHEMATICS PENYELARAS : CIK NURUL NADIA NORDIN KURSUS : DMC MASA : 3 JAM TARIKH : 6 NOVEMBER 2009 TEMPAT : KOMPLEKS SUKAN ARAHAN KEPADA CALON 1. SILA JAWAB SEMUA SOALAN _ 2. RUMUS / FORMULA DILANIPIRKAN DI MUKA SURAT 10, ll DAN 12 KERTAS SOALAN INI TERDIRI DARIPADA (12) MUKA SURAT SAHAJA (TERMASUK MUKA SURAT HADAPAN) SULIT (DACS 2212) SULIT Question 1 (a) Given a function z = f(x,y) = 12 — (y +1)2 — x2 (i) Sketch the level curves on the xy —plane for z = 3, 8, 12. (ii) Sketch the graph of the function. [5 marks] (b) The relationship between the focal length, f of a lens with an object distance u and its image distance v is given by l 1 l —=—+—. f u v If the maximum percentage error in measuring u and v is 2% each, use partial derivative to approximate the maximum percentage error in calculating f . [6 marks] (c) Given that f(x,y)= x3 +y3 +3x2 —3y2 — 6. (i) Find all the critical points of f (x, y). [4 marks] (ii) Classify each of the critical points whether is corresponds to local maximum, local minimum or saddle point. [5 marks] 2 SULIT (DACS 2212) SULIT Question 2 (a) If f(x,y) = cos (y2 + 2x), show that fxy = — 4ycos (y2 + 2x). Hence evaluate fxy (2,1). [4 marks] (b) If f is a function and z = x y + f (x2 + yz), show that 62 62 2 2 y— — x— = y — x . 6x 6y [5 marks] (0) Use the chain rule to find the value of Z—u at the point (x/Z J2, 1), given that I” u=zsin(xy), x=r+s, y=r—s and z=r2 +s2. [5 marks] (d) An open rectangular box is to have a volume of 32 m3 . Find the dimensions that will make the surface area minimum. [6 marks] Question 3 (a) Evaluate the following integral by converting to polar coordinates 2 O I I y dy dx. 0 —\/ 4—)r2 [6 marks] 3 SULIT (DACS 2212) SULIT (b) Find the mass ofa lamina which has the shape of the region bounded by y=x2 and y=2x~x2 with density function 6(x, y) = 100x kgm‘z. [6 marks] (c) Use appropriate coordinate system to find the volume of the solid outside x2 + y2 =1, bounded above by z = 1/4 — x2 —y2 and below by z = 0. [8 marks] Question 4 (a) Find the mass of the solid in the first octant bounded b the coordinate planes, the lane y P y + z = 2 and the cylinder x = 4 — yz. The density function is given by 6 (x, y, z) = 3 . [7 marks] (b) By using cylindrical or spherical coordinates, evaluate the following triple integral [7 marks] [7 marks] 4 _ SULIT (DACS 2212) SULIT Question 5 (a) The position vector ofa particle is given by r(t)= 3sin(t ~5)i +3cos(t —5)j + (t2 — 6t +3)k. At t = 5 , find the following : (i) The velocity v. [3 marks] (ii) The speed v. [2 marks] (iii)The unit tangent vector T. [2 marks] (b) Let F(x,y,z)=xey i+zsinyj+xylnzk . Find (i) div F ‘ [2 marks] (ii) curl F [2 marks] (iii) v(v - F) [3 marks] (c) Given that ¢ (x, y, z) = xzy + yzz +1 . (i) Find the directional derivative of ¢ at (2,1,3) in the direction of vector 2i—3j+5k. [3 marks] (ii) Determine the direction for which the directional derivative of ¢ is maximum at (2,1, 3), and find its maximum value. [3 marks] 5 SULIT (DACS 2212) SULIT Soalan 1 (a) Diberikan suatu fungsi z = f(x,y) = 12 — (y +1)2 — x2 (i) Lakarkan lengkung—lengkung aras pada satah - xy untuk z = 3, 8,12. (ii) Lakarkan graf fungsi tersebut. [5 markah] (b) Hubungan di antarajarak fokas, f bagi suatu kanta dengan jarak objek u dan jarak imej v diberikan oleh 1 1 1 — = —+—. fuv Jika peratus ralat maksimum dalam mengira u dan v ialah 2% dalam setiap pengukuran, gunakan terbitan separa untuk menganggarkan peratus ralat maksimum dalam mengira f . [6 markah] (c) Diberikan bahawa f(x,y) = x3 + y3 + 3x2 — 3y2 — 6 . (i) Dapatkan semua titik genting bagi f (x, y). [4 markah] (ii) Kelaskan setiap titik genting tersebut sama ada ianya titik maksimum, titik minimum atau titik pelana. [5 markah] 6 SULIT (DACS 2212) ' SULIT Soalan 2 (a) Jika f(x,y) = kos (y2 + 2x), tunjukkan bahawa fxy = — 4y kos (y2 + 2x). Seterusnya nilaikan fxy (2,1). [4 markah] (b) J ika f adalah suatu fungsi dan z = xy + f (x2 + yz), tunjukkan bahawa y%—xg=y2 —x2. 6x 6y [5 markah] (c) Gunakan petua rantai untuk mendapatkan nilai Z—u pada titik (5, J7; 1),jika diberikan r u=zsin(xy), X=r+s, y=r—s dan z=r2+s2. [5 markah] (d) Sebuah kotak segiempat tepat, terbuka pada bahagian atasnya, mempunyai isipadu 32 m3 . Dapatkan dimensi bagi kotak ini supaya jumlah luas permukaannya adaIah minimum. [6 markah] Soalan 3 (a) Nilaikan kamiran berikut dengan menukarkannya ke dalam koordinat kutub 2 0 I I y dy dx. 0 43:7 [6 markah] 7 SULIT (DAC S 2212) SULIT (b) Tentukan jisim bongkah dalam oktan pertama yang dibatasi oleh satah-satah koordinat, satah y+z=2 dan silinder x=4—y2. Fungsi ketumpatannya diberikan oleh 6 (x, y, z) = 3 . [7 markah] (c) Dengan menggunakan sistem koordinat yang sesuai, dapatkan isipadu bongkah yang berada di luar x2 + y2 =1, dibatasi atas oleh z = 4 — x2 - y2 dan di bawah oleh z = 0. [7 markah] Soalan 4 (a) Dapatkan jisim bagi suatu lamina yang mempunyai bentuk suatu rantau yang dibatasi oleh y=x2 and y=2x—x2 dengan fungsi ketumpatan 6(x, y) = 100x kgm'2 [6 markah] (b) Dengan menggunakan koordinat silinder atau sfera, nilaikan kamiran ganda ti ga berikut 2 3 J53? (HUI! J x 2+y2 dydxdz. O 0 0 [7 markah] 1/3x2 +3>y2 +322 dz dx dy. [7 markah] 8 - SULIT (DACS 2212) SULIT Soalan 5 (a) Vektor kedudukan bagi suatu zarah diberikan oleh r(t)= 3sin(t — 5)i+ 3k0s(t — 5)j + (:2 — 6t + 3)k Pada t = 5 , dapatkan yang berikut : (i) Halaju v. [3 markah] (ii) Laju v. [2 markah] (iii)Vector tangent unit T . [2 markah] (b) Katalah F(x,y,z) = xey i+ zsin y j + xyln z k . Cari (i) div F [2 markah] (ii) curl F [2 markah] (iii) V(V - F) [3 markah] (c) Diberikan bahawa ¢ (x, y, z) = x2y + yzz +1. (i) Cari terbitan berarah bagi ¢ pada (2,1, 3) dalam arah vektor 2 i ~ 3 j + 5 k . [3 markah] (ii) Tentukan arah dengan terbitan berarah bagi ¢ adalah maksimum pada (2,1, 3),dan dapatkan nilai maksimumnya. [3 markah] 9 SULIT Trigonometry 1 cscx = . sm x sin x tan x = cosx sin2 x+cos2 x =1 sin(— x) = —sin x csc(— x) = — csc(x) Differentiation Derivative Formula: Product Rule: Quotient Rule: secx = cos x cos x cot x = . sm x tan2x+1 =5602 x (DACS 2212) FORMULAE cos(— x) = cos x SGC(— x) = sec(x) cotx = tan x l+cot2 x = csc2 x tan(— x): —tanx cot(— x) = — cot x , fl . f(x+h)—f(x) farm—~71— d[uv] dx d? V dx l = u'v + uv' u'v—uv V 10 SULIT Formulae PamiT (DACS 2212) SULIT Derivatives of Some Common Functions: d _ f ’(X) d_xlnf(x) ‘ f(x) 19m) 2 , f0) dx f (x)e g—sin f(X)= f’(x)cosf(X) icosffi): _f'(x)sin f(x) d Etan f(x) = f'(x)sec2 f(x) ~:—xcot f(x) = —f'(x)cos eczf(x) d —secx = secxtanx Integration Integration of Some Common Functions: n+1 x jxwx: +C ,n¢—1 n+1 dx —=Inx+C x emx Iemxdx: +C m sinmx Icosmxdxz +C m . cosmx Ismmxdx=— +C m Isecz xdx=tanx+C Formulae Page 2 of3 1 1 SULIT (DACS 2212) SULIT Multiple Integral I!Jf(xayvz)dV = Iflf(rcos€,rsin 6)dz]dA I”f(x,y,z)dV = I”f(psin¢cos€,psin¢sin 6,,0cos¢),02 sin¢ dp d¢ d6 G G 12 ' SULIT Formulae Page 3 of 3 ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern