 |
ADC801S - ADVANCE CALCULUS - 1ST OPP - JUNE 2023 |
|
|
1 Page 1 |
▲back to top |
nAmlBIA UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
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:JUNE 2023
DURATION: 3 HOURS
PAPER:THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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)
|
|
2 Page 2 |
▲back to top |
ADC 801S
ADVANCED CALCULUS
June 2023
1. (a) If x = rcos0 and y = rsin0, find the (r,0) equations for¢ which obeys Laplace's
equation in two-dimensional caresian co-ordinates
(5)
(b) if Q = log(tanx +tarry+ tanz), show that
.
sm
2x-aau
X
+
.
sm
2y-aauy
+
.
sm
2z-aau
Z
=
2
(5)
(c) If u = x2tan ~, find
a2u I
axay (-1,2)
(5)
2. (a) Minimize J(x1, x2) = X1- X2+ 2xr + 2x1X2+ x~ by taking the starting from the
point X 1 = { ~} using Davidon-Fletcher-Powell (DFP) method with
[Bi] = [~ ~] , E = 0.01
(10)
(b) Minimize J(x1, X2) = X1- X2+ 2xr + 2X1X2+ x~ by taking the starting from the
point X 1 = { ~} , by using Newton's Method
(10)
3. (a) If
show that
r · V¢ = n<p
where n is constant
(8)
(b) Find the directional derivative of the function
cp(x,y, z) = x 2y - 3yz + 2xz
in the direction
n = 4i - 7j + 4k
at the point (3, -2, 1).
(8)
4. (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 wi~hin the boundaries 0 x 1. The density
function associated with this model is denoted as p = xy.
1
|
|
3 Page 3 |
▲back to top |
ADC 801S
ADVANCED CALCULUS
June 2023
i. Find the mass of the body.
(4)
11. Find the coordinates of the center of mass.
(5)
5. A curve is defined parametrically by
x(t) = aet cost, y(t) = aet sin t, and z(t) = v'2a(et - I).
Find the following for the curve:
(a) The tangent vector T
(5)
(b) The curvature "'
(5)
(c) The principal normal vector N
(5)
(d) The binormal vector 13
(5)
(e) The torsion T
(5)
End of Exam!
3