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ADC801S - ADVANCE CALCULUS - 2ND OPP - JULY 2023 |
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n Am I BI A u n IVER s ITY
OF SCIEn CE Ano TECHn OLOGY
FACULTYOF HEALTH, NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENTOF MATHEMATICS, STATISTICSAND ACTUARIALSCIENCE
QUALIFICATION:BACHELOR OF SCIENCEHONOURS IN APPLIED MATHEMATICS
QUALIFICATIONCODE: 08BSHM
LEVEL: 8
COURSECODE: ADC801S
COURSENAME: ADVANCED CALCULUS
SESSION:JULY 2023
DURATION: 3 HOURS
PAPER:THEORY
MARKS: 87
SUPPLEMENTARY /SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
Prof A.S Eegunjobi
MODERATOR
Prof O.D Makinde
INSTRUCTIONS
1. Answer ALL the questions.
2. Write clearly and neatly.
3. Number the answers clearly.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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ADC 801S
ADVANCED CALCULUS
July 2023
1. (a) Determine the minimum distance between the origin and the hyperbola defined by
x2 + 8xy + 7y2 = 226
(6)
(b) Show that V · ("vgm)= m(m + l)gm- 2, if g =xi+ yj + zk.
(9)
(c) A material body's geometric representation is a planar area R, delimited by the
curves y = x2 and y = - x2 within the boundaries O :S x '.S1. The density
function associated with this model is denoted as p = xy.
i. Find the mass of the body.
(4)
ii. Find the coordinates of the center of mass.
(5)
(d) Determine the flux of F = i - j + xyzk through the circular region S obtained by
cutting the sphere x2 + y2 + z2 = 4 with a plane y = x.
(6)
(e) Find the volume of the solid region bounded above the paraboliod z = l - x2 - y2
and below the plane z = l - y.
(6)
2. (a) if Q = log(tanx + tany + tanz), show that
sin 2x 8u + sin 2y 8u + sin 2z Bu = l
2 ox 2 By 2 oz
(5)
(b) If x = rcos0 and y = rsin0, find the (r,0) equations for¢ which obeys Laplace's
equation in two-dimensional caresian co-ordinates
(5)
(c) If A, B and Care vectors, show that
-dA·b
X C=
dA
- ·BX
C+A·-+Ad·BB
dt
dt
dt
dC
X-
dt
(5)
3. (a) Niinimize f(x 1, x2 ) = x 1 - x2 + 2xi + 2x 1x2 + x~ by taking the starting from the
point X 1 = { ~} using Davidon-Fletcher-Powell (DFP) method with
[B1]= [~ ~] , E = 0.01
(b) Minimize f(x 1, x2 ) = x 1 - x2 + 2xi + 2x 1x2 + x~ by taking the starting from the
point X 1 ={~},by using Newton's Method
(10)
4. (a) Evaluate the integral
1 1 % cos2 xdx
0 ( 2 cos x + sin x )2 given
% cosxdx
0 a cos x + sin x
mr
Ina
2(a 2 + 1) --- a 2 + 1
2
(8)
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ADC 801S
ADVANCED CALCULUS
July 2023
(b) Find the maximum possible volume of a rectangular box that is completely enclosed
by the surface of the ellipsoid defined by the equation 2x2 + 3y2 + z2 = 18, where
each of its edges is parallel to one of the coordinate axes.
(8)
End of Exam!
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