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A7001
०० |
Test the absolute convergence of > (-1)" 5 ‏اد‎
‎tl (3k - 2)!
Find the Taylor series for ‏كك‎ at x=2.
1+
۸۸۱1١ 11

Answer any two questions, each carries 5 marks.
Find the local linear approximation L to f(x,y) =log(xy) at P(1,2) and compare
the error in approximating f by L at Q(1.01, 2.01) with the distance between P
and Q.
Let w=4x? +4y? +2°,x = psin 00050, y = psingsin 0,2 = pcos ‏,ل‎ Find

Locate all relative extrema and saddle points of f(x,y) = ‏بود4‎ ದಂಗೆ - ७.
Module 111

Answer any two questions, each carries 5 marks.
Find the equation of the tangent plane and parametric equation for the normal line
to the surface x? + y> + 22 = 25 21116 point (3,0, 4).
A particle is moving along the curve r(t) = (7 - ‏م2‎ + ) 2 -4)/ where / denotes
the time. Find the scalar tangential and normal components of acceleration at
t =1.Also find the vector tangential and normal components of acceleration at
t=l.

The graphs of ‏(/)ء‎ = {7 + | +3¢°kand (ಗಿ =(t-Di +22 + )5 —t)k are

intersect at the point P(1,1,3) .Find, to the nearest degree, the acute angle between
the tangent lines to the graphs of 7,(¢) & 7,(¢) at the point P(1.1,3).

Module 1V

Answer any two questions, each carries5 marks.
1 4

Change the order of integration and evaluate | | ०० dydx.

0 4x
Use triple integral to find the volume bounded by the cylinder x* ൦ =9 and
between the planes 2-1 and ++ 2 = 5.

Find the area of the region enclosed between the parabola y= ಗೆ and the line

॥ =2x.
Module V
Answer any three questions, each ೧0111055 marks.

Determine whether F(x,y)=(cosy+ycosx)it+(sinx-—xsiny)j is 8
conservative vector field. If so find the potential function for it.

(3,3) று x
Show that the integral | (€ logy -_ jax + (^ - €” logx)dy ,;where x and ந

x 2

(1.1)
are positive is independent of the path and find its value.
Find the work done by the force field F(x, y,z) =xyi+ ‏صر‎ j+xzk ona particle

that moves along the curve C:r(t)=ti+¢° j+k(O
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