Multivariable Calculus BS 5 Semester/Term UOS — University of Sargodha 2024
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Paper text
30226
University of Sargodha
BS 5 Term Examination 2024
Subject: CS
Paper: Multivariable Calculus (MATH-301)
Time Allowed: 02:30 Hours
Maximum Marks: 60
Note: Objective part is compulsory. Attempt any three questions from subjective part.
Objective Part
(Compuisory)
Q. 1.
Write short answers of the following in 2-3 lines cach on your answer sheet.
i.
i.
i.
iv.
.٧
vi.
vii.
viii.
ix.
X.
X1.
xii.
Find 92
when z = xexy.
Show that x z
= 0 for ½=*
Define partial derivative.
Find the gradient of f(x, y, z) = exY cos(z) at the point (0,2,0).
Define the homogeneous function.
Find the Laplace transform of e- sin (2t).
Find the inverse Laplace transform of (s-a)(a+b)
Define the Heaviside function.
Define the directional derivative.
Find xx, if f(x, y) = xy? + xy - y3.
If w = x2 + x, x = sin(t), y = cost), then find aw
Compute the double integral fo f(2x + 2y)dydx.
(2*12)
Subjective Part
(3*12)
Q.2.
Q.3.
Q.4
a) Find the Laplace transform of f(t) = sinh(at) cos(at)
b) Find all second order partial derivatives of f(x, y) = In(xy) - exy + xy 3
a) Find the point on the plane x + 2y + 3z = 6 that is closest to origin.
b) Find dz
dt
when z = xe×y, x = t', y = t-1.
a) Find the tangent plane to the sphere x2 + y2 + z2 = 14 at (1,2,3).
b) Compute the differentials of u = tp%
Q.5.
Q.6.
b) Find the linearization of z = yvx at the point (9,-2).
a) Show that the function f(x, y) = In(x2 + y½) satisfies Laplace's equation.
b) Find and classify ail critical points of f(x, y) = 4 + x3 + y - 3xy.
LK-6793/04-06-24----
